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== 6.1#h(0.6em)Add and Subtract Polynomials

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Simplify: #math.equation(block: false, alt: "8 x plus 3 x .")[$8 x + 3 x .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "11 x")[$11 x$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract: #math.equation(block: false, alt: "open parenthesis 5 n plus 8 close parenthesis minus open parenthesis 2 n minus 1 close parenthesis .")[$( 5 n + 8 ) − ( 2 n − 1 ) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "3 n plus 9")[$3 n + 9$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write in expanded form: #math.equation(block: false, alt: "a to the power 5 .")[$a^(5) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "a times a times a times a times a")[$a ⋅ a ⋅ a ⋅ a ⋅ a$]
]
]


=== Identify Polynomials, Monomials, Binomials and Trinomials

You have learned that a #emph[term] is a constant or the product of a constant and one or more variables. The constant is called a coefficient. When it is of the form #math.equation(block: false, alt: "a x to the power m")[$a x^(m)$], where #math.equation(block: false, alt: "a")[$a$] is a constant and #math.equation(block: false, alt: "m")[$m$] is a whole number, it is called a monomial. Some examples of monomial are #math.equation(block: false, alt: "8 , −2 x squared , 4 y cubed , and 11 z to the power 7")[$8 , −2 x^(2) , 4 y^(3) , #h(0.2em) "and" #h(0.2em) 11 z^(7)$].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Monomials]

A #strong[monomial] is a term of the form #math.equation(block: false, alt: "a x to the power m")[$a x^(m)$], where #math.equation(block: false, alt: "a")[$a$] is a constant and #math.equation(block: false, alt: "m")[$m$] is a positive whole number.
]

A monomial, or two or more monomials combined by addition or subtraction, is a polynomial. Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term. A binomial has exactly two terms, and a trinomial has exactly three terms. There are no special names for polynomials with more than three terms.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Polynomials]

#strong[polynomial]—A monomial, or two or more monomials combined by addition or subtraction, is a polynomial.

- #strong[monomial]—A polynomial with exactly one term is called a monomial.
- #strong[binomial]—A polynomial with exactly two terms is called a binomial.
- #strong[trinomial]—A polynomial with exactly three terms is called a trinomial.
]

Here are some examples of polynomials.

#math.equation(block: true, alt: "Polynomial, b plus 1, 4 y squared minus 7 y plus 2, 4 x to the power 4 plus x cubed plus 8 x squared minus 9 x plus 1; Monomial, 14, 8 y squared, −9 x cubed y to the power 5, −13; Binomial, a plus 7, 4 b minus 5, y squared minus 16, 3 x cubed minus 9 x squared; Trinomial, x squared minus 7 x plus 12, 9 y squared plus 2 y minus 8, 6 m to the power 4 minus m cubed plus 8 m, z to the power 4 plus 3 z squared minus 1")[$"Polynomial" &  &  & #h(2em) b + 1 &  &  & #h(2em) 4 y^(2) − 7 y + 2 &  &  & #h(2em) 4 x^(4) + x^(3) + 8 x^(2) − 9 x + 1 &  &  &  \ "Monomial" &  &  & #h(2em) 14 &  &  & #h(2em) 8 y^(2) &  &  & #h(2em) −9 x^(3) y^(5) &  &  & #h(2em) −13 \ "Binomial" &  &  & #h(2em) a + 7 &  &  & #h(2em) 4 b − 5 &  &  & #h(2em) y^(2) − 16 &  &  & #h(2em) 3 x^(3) − 9 x^(2) \ "Trinomial" &  &  & #h(2em) x^(2) − 7 x + 12 &  &  & #h(2em) 9 y^(2) + 2 y − 8 &  &  & #h(2em) 6 m^(4) − m^(3) + 8 m &  &  & #h(2em) z^(4) + 3 z^(2) − 1$]

Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words #emph[monomial], #emph[binomial], and #emph[trinomial] when referring to these special polynomials and just call all the rest #emph[polynomials].

#examplebox("Example 1")[][
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial.

+ ⓐ #math.equation(block: false, alt: "4 y squared minus 8 y minus 6")[$4 y^(2) − 8 y − 6$]
+ ⓑ #math.equation(block: false, alt: "−5 a to the power 4 b squared")[$−5 a^(4) b^(2)$]
+ ⓒ #math.equation(block: false, alt: "2 x to the power 5 minus 5 x cubed minus 9 x squared plus 3 x plus 4")[$2 x^(5) − 5 x^(3) − 9 x^(2) + 3 x + 4$]
+ ⓓ #math.equation(block: false, alt: "13 minus 5 m cubed")[$13 − 5 m^(3)$]
+ ⓔ #math.equation(block: false, alt: "q")[$q$]

#solutionbox[
#figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([], [#emph[Polynomial]], [#emph[Number of terms]], [#emph[Type]]),
  [ⓐ], [#math.equation(block: false, alt: "4 y squared minus 8 y minus 6")[$4 y^(2) − 8 y − 6$]], [#math.equation(block: false, alt: "3")[$3$]], [Trinomial],
  [ⓑ], [#math.equation(block: false, alt: "−5 a to the power 4 b squared")[$−5 a^(4) b^(2)$]], [#math.equation(block: false, alt: "1")[$1$]], [Monomial],
  [ⓒ], [#math.equation(block: false, alt: "2 x to the power 5 minus 5 x cubed minus 9 x squared plus 3 x plus 4")[$2 x^(5) − 5 x^(3) − 9 x^(2) + 3 x + 4$]], [#math.equation(block: false, alt: "5")[$5$]], [Polynomial],
  [ⓓ], [#math.equation(block: false, alt: "13 minus 5 m cubed")[$13 − 5 m^(3)$]], [#math.equation(block: false, alt: "2")[$2$]], [Binomial],
  [ⓔ], [#math.equation(block: false, alt: "q")[$q$]], [#math.equation(block: false, alt: "1")[$1$]], [Monomial],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

ⓐ #math.equation(block: false, alt: "5 b")[$5 b$] ⓑ #math.equation(block: false, alt: "8 y cubed minus 7 y squared minus y minus 3")[$8 y^(3) − 7 y^(2) − y − 3$] ⓒ #math.equation(block: false, alt: "−3 x squared minus 5 x plus 9")[$−3 x^(2) − 5 x + 9$] ⓓ #math.equation(block: false, alt: "81 minus 4 a squared")[$81 − 4 a^(2)$] ⓔ #math.equation(block: false, alt: "−5 x to the power 6")[$−5 x^(6)$]

#solutionbox[
ⓐ monomial ⓑ polynomial ⓒ trinomial ⓓ binomial ⓔ monomial
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

ⓐ #math.equation(block: false, alt: "27 z cubed minus 8")[$27 z^(3) − 8$] ⓑ #math.equation(block: false, alt: "12 m cubed minus 5 m squared minus 2 m")[$12 m^(3) − 5 m^(2) − 2 m$] ⓒ #math.equation(block: false, alt: "the fraction 5 over 6")[$frac(5, 6)$] ⓓ #math.equation(block: false, alt: "8 x to the power 4 minus 7 x squared minus 6 x minus 5")[$8 x^(4) − 7 x^(2) − 6 x − 5$] ⓔ #math.equation(block: false, alt: "− n to the power 4")[$"−" n^(4)$]

#solutionbox[
ⓐ binomial ⓑ trinomial ⓒ monomial ⓓ polynomial ⓔ monomial
]
]


=== Determine the Degree of Polynomials

The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.

A #strong[monomial] that has no variable, just a constant, is a special case. The degree of a constant is 0—it has no variable.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Degree of a Polynomial]

The #strong[degree of a term] is the sum of the exponents of its variables.

The #strong[degree of a constant] is 0.

The #strong[degree of a polynomial] is the highest degree of all its terms.
]

Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.

#figure(figph[This table has 11 rows and 5 columns. The first column is a header column, and it names each row. The first row is named “Monomial,” and each cell in this row contains a different monomial. The second row is named “Degree,” and each cell in this row contains the degree of the monomial above it. The degree of 14 is 0, the degree of 8y squared is 2, the degree of negative 9x cubed y to the fifth power is 8, and the degree of negative 13a is 1. The third row is named “Binomial,” and each cell in this row contains a different binomial. The fourth row is named “Degree of each term,” and each cell contains the degrees of the two terms in the binomial above it. The fifth row is named “Degree of polynomial,” and each cell contains the degree of the binomial as a whole.” The degrees of the terms in a plus 7 are 0 and 1, and the degree of the whole binomial is 1. The degrees of the terms in 4b squared minus 5b are 2 and 1, and the degree of the whole binomial is 2. The degrees of the terms in x squared y squared minus 16 are 4 and 0, and the degree of the whole binomial is 4. The degrees of the terms in 3n cubed minus 9n squared are 3 and 2, and the degree of the whole binomial is 3. The sixth row is named “Trinomial,” and each cell in this row contains a different trinomial. The seventh row is named “Degree of each term,” and each cell contains the degrees of the three terms in the trinomial above it. The eighth row is named “Degree of polynomial,” and each cell contains the degree of the trinomial as a whole. The degrees of the terms in x squared minus 7x plus 12 are 2, 1, and 0, and the degree of the whole trinomial is 2. The degrees of the terms in 9a squared plus 6ab plus b squared are 2, 2, and 2, and the degree of the trinomial as a whole is 2. The degrees of the terms in 6m to the fourth power minus m cubed n squared plus 8mn to the fifth power are 4, 5, and 6, and the degree of the whole trinomial is 6. The degrees of the terms in z to the fourth power plus 3z squared minus 1 are 4, 2, and 0, and the degree of the whole trinomial is 4. The ninth row is named “Polynomial,” and each cell contains a different polynomial. The tenth row is named “Degree of each term,” and the eleventh row is named “Degree of polynomial.” The degrees of the terms in b plus 1 are 1 and 0, and the degree of the whole polynomial is 1. The degrees of the terms in 4y squared minus 7y plus 2 are 2, 1, and 0, and the degree of the whole polynomial is 2. The degrees of the terms in 4x to the fourth power plus x cubed plus 8x squared minus 9x plus 1 are 4, 3, 2, 1, and 0, and the degree of the whole polynomial is 4.], alt: "This table has 11 rows and 5 columns. The first column is a header column, and it names each row. The first row is named “Monomial,” and each cell in this row contains a different monomial. The second row is named “Degree,” and each cell in this row contains the degree of the monomial above it. The degree of 14 is 0, the degree of 8y squared is 2, the degree of negative 9x cubed y to the fifth power is 8, and the degree of negative 13a is 1. The third row is named “Binomial,” and each cell in this row contains a different binomial. The fourth row is named “Degree of each term,” and each cell contains the degrees of the two terms in the binomial above it. The fifth row is named “Degree of polynomial,” and each cell contains the degree of the binomial as a whole.” The degrees of the terms in a plus 7 are 0 and 1, and the degree of the whole binomial is 1. The degrees of the terms in 4b squared minus 5b are 2 and 1, and the degree of the whole binomial is 2. The degrees of the terms in x squared y squared minus 16 are 4 and 0, and the degree of the whole binomial is 4. The degrees of the terms in 3n cubed minus 9n squared are 3 and 2, and the degree of the whole binomial is 3. The sixth row is named “Trinomial,” and each cell in this row contains a different trinomial. The seventh row is named “Degree of each term,” and each cell contains the degrees of the three terms in the trinomial above it. The eighth row is named “Degree of polynomial,” and each cell contains the degree of the trinomial as a whole. The degrees of the terms in x squared minus 7x plus 12 are 2, 1, and 0, and the degree of the whole trinomial is 2. The degrees of the terms in 9a squared plus 6ab plus b squared are 2, 2, and 2, and the degree of the trinomial as a whole is 2. The degrees of the terms in 6m to the fourth power minus m cubed n squared plus 8mn to the fifth power are 4, 5, and 6, and the degree of the whole trinomial is 6. The degrees of the terms in z to the fourth power plus 3z squared minus 1 are 4, 2, and 0, and the degree of the whole trinomial is 4. The ninth row is named “Polynomial,” and each cell contains a different polynomial. The tenth row is named “Degree of each term,” and the eleventh row is named “Degree of polynomial.” The degrees of the terms in b plus 1 are 1 and 0, and the degree of the whole polynomial is 1. The degrees of the terms in 4y squared minus 7y plus 2 are 2, 1, and 0, and the degree of the whole polynomial is 2. The degrees of the terms in 4x to the fourth power plus x cubed plus 8x squared minus 9x plus 1 are 4, 3, 2, 1, and 0, and the degree of the whole polynomial is 4.", caption: none)

A polynomial is in #strong[standard form] when the terms of a polynomial are written in descending order of degrees. Get in the habit of writing the term with the highest degree first.

#examplebox("Example 2")[][
Find the degree of the following polynomials.

+ ⓐ #math.equation(block: false, alt: "10 y")[$10 y$]
+ ⓑ #math.equation(block: false, alt: "4 x cubed minus 7 x plus 5")[$4 x^(3) − 7 x + 5$]
+ ⓒ #math.equation(block: false, alt: "−15")[$−15$]
+ ⓓ #math.equation(block: false, alt: "−8 b squared plus 9 b minus 2")[$−8 b^(2) + 9 b − 2$]
+ ⓔ #math.equation(block: false, alt: "8 x y squared plus 2 y")[$8 x y^(2) + 2 y$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ 
 #linebreak() The exponent of #math.equation(block: false, alt: "y")[$y$] is one. #math.equation(block: false, alt: "y equals y to the power 1")[$y = y^(1)$]], [#math.equation(block: false, alt: "10 y")[$10 y$]
 #linebreak() The degree is 1.]),
  [ⓑ 
 #linebreak() The highest degree of all the terms is 3.], [#math.equation(block: false, alt: "4 x cubed minus 7 x plus 5")[$4 x^(3) − 7 x + 5$]
 #linebreak() The degree is 3.],
  [ⓒ 
 #linebreak() The degree of a constant is 0.], [#math.equation(block: false, alt: "−15")[$−15$]
 #linebreak() The degree is 0.],
  [ⓓ 
 #linebreak() The highest degree of all the terms is 2.], [#math.equation(block: false, alt: "−8 b squared plus 9 b minus 2")[$−8 b^(2) + 9 b − 2$]
 #linebreak() The degree is 2.],
  [ⓔ 
 #linebreak() The highest degree of all the terms is 3.], [#math.equation(block: false, alt: "8 x y squared plus 2 y")[$8 x y^(2) + 2 y$]
 #linebreak() The degree is 3.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the degree of the following polynomials:

ⓐ #math.equation(block: false, alt: "−15 b")[$−15 b$] ⓑ #math.equation(block: false, alt: "10 z to the power 4 plus 4 z squared minus 5")[$10 z^(4) + 4 z^(2) − 5$] ⓒ #math.equation(block: false, alt: "12 c to the power 5 d to the power 4 plus 9 c cubed d to the power 9 minus 7")[$12 c^(5) d^(4) + 9 c^(3) d^(9) − 7$] ⓓ #math.equation(block: false, alt: "3 x squared y minus 4 x")[$3 x^(2) y − 4 x$] ⓔ #math.equation(block: false, alt: "−9")[$−9$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "1")[$1$] ⓑ #math.equation(block: false, alt: "4")[$4$] ⓒ #math.equation(block: false, alt: "12")[$12$] ⓓ 3 ⓔ 0
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the degree of the following polynomials:

ⓐ #math.equation(block: false, alt: "52")[$52$] ⓑ #math.equation(block: false, alt: "a to the power 4 b minus 17 a to the power 4")[$a^(4) b − 17 a^(4)$] ⓒ #math.equation(block: false, alt: "5 x plus 6 y plus 2 z")[$5 x + 6 y + 2 z$] ⓓ #math.equation(block: false, alt: "3 x squared minus 5 x plus 7")[$3 x^(2) − 5 x + 7$] ⓔ #math.equation(block: false, alt: "− a cubed")[$"−" a^(3)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "0")[$0$] ⓑ #math.equation(block: false, alt: "5")[$5$] ⓒ #math.equation(block: false, alt: "1")[$1$] ⓓ 2 ⓔ 3
]
]


=== Add and Subtract Monomials

You have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficient.

#examplebox("Example 3")[][
Add: #math.equation(block: false, alt: "25 y squared plus 15 y squared")[$25 y^(2) + 15 y^(2)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "25 y squared plus 15 y squared")[$25 y^(2) + 15 y^(2)$]]),
  [Combine like terms.], [#math.equation(block: false, alt: "40 y squared")[$40 y^(2)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "12 q squared plus 9 q squared .")[$12 q^(2) + 9 q^(2) .$]

#solutionbox[
#math.equation(block: true, alt: "21 q squared")[$21 q^(2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "−15 c squared plus 8 c squared .")[$−15 c^(2) + 8 c^(2) .$]

#solutionbox[
#math.equation(block: true, alt: "−7 c squared")[$−7 c^(2)$]
]
]

#examplebox("Example 4")[][
Subtract: #math.equation(block: false, alt: "16 p minus open parenthesis −7 p close parenthesis")[$16 p − ( −7 p )$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "16 p minus open parenthesis −7 p close parenthesis")[$16 p − ( −7 p )$]]),
  [Combine like terms.], [#math.equation(block: false, alt: "23 p")[$23 p$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract: #math.equation(block: false, alt: "8 m minus open parenthesis −5 m close parenthesis .")[$8 m − ( −5 m ) .$]

#solutionbox[
#math.equation(block: true, alt: "13 m")[$13 m$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract: #math.equation(block: false, alt: "−15 z cubed minus open parenthesis −5 z cubed close parenthesis .")[$−15 z^(3) − ( −5 z^(3) ) .$]

#solutionbox[
#math.equation(block: true, alt: "−10 z cubed")[$−10 z^(3)$]
]
]

Remember that like terms must have the same variables with the same exponents.

#examplebox("Example 5")[][
Simplify: #math.equation(block: false, alt: "c squared plus 7 d squared minus 6 c squared")[$c^(2) + 7 d^(2) − 6 c^(2)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "c squared plus 7 d squared minus 6 c squared")[$c^(2) + 7 d^(2) − 6 c^(2)$]]),
  [Combine like terms.], [#math.equation(block: false, alt: "−5 c squared plus 7 d squared")[$−5 c^(2) + 7 d^(2)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "8 y squared plus 3 z squared minus 3 y squared .")[$8 y^(2) + 3 z^(2) − 3 y^(2) .$]

#solutionbox[
#math.equation(block: true, alt: "5 y squared plus 3 z squared")[$5 y^(2) + 3 z^(2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "3 m squared plus n squared minus 7 m squared .")[$3 m^(2) + n^(2) − 7 m^(2) .$]

#solutionbox[
#math.equation(block: true, alt: "−4 m squared plus n squared")[$−4 m^(2) + n^(2)$]
]
]

#examplebox("Example 6")[][
Simplify: #math.equation(block: false, alt: "u squared v plus 5 u squared minus 3 v squared")[$u^(2) v + 5 u^(2) − 3 v^(2)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "u squared v plus 5 u squared minus 3 v squared")[$u^(2) v + 5 u^(2) − 3 v^(2)$]]),
  [There are no like terms to combine.], [#math.equation(block: false, alt: "u squared v plus 5 u squared minus 3 v squared")[$u^(2) v + 5 u^(2) − 3 v^(2)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "m squared n squared minus 8 m squared plus 4 n squared .")[$m^(2) n^(2) − 8 m^(2) + 4 n^(2) .$]

#solutionbox[
There are no like terms to combine.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "p q squared minus 6 p minus 5 q squared .")[$p q^(2) − 6 p − 5 q^(2) .$]

#solutionbox[
There are no like terms to combine.
]
]


=== Add and Subtract Polynomials

We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.

#examplebox("Example 7")[][
Find the sum: #math.equation(block: false, alt: "open parenthesis 5 y squared minus 3 y plus 15 close parenthesis plus open parenthesis 3 y squared minus 4 y minus 11 close parenthesis .")[$( 5 y^(2) − 3 y + 15 ) + ( 3 y^(2) − 4 y − 11 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Identify like terms.], [#figure(figph[5 y squared minus 3 y plus 15, plus 3 y squared minus 4 y minus 11.], alt: "5 y squared minus 3 y plus 15, plus 3 y squared minus 4 y minus 11.", caption: none)]),
  [Rearrange to get the like terms together.], [#figure(figph[5y squared plus 3y squared, identified as like terms, minus 3y minus 4y, identified as like terms, plus 15 minus 11, identified as like terms.], alt: "5y squared plus 3y squared, identified as like terms, minus 3y minus 4y, identified as like terms, plus 15 minus 11, identified as like terms.", caption: none)],
  [Combine like terms.], [#figure(figph[8 y squared minus 7y plus 4.], alt: "8 y squared minus 7y plus 4.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the sum: #math.equation(block: false, alt: "open parenthesis 7 x squared minus 4 x plus 5 close parenthesis plus open parenthesis x squared minus 7 x plus 3 close parenthesis .")[$( 7 x^(2) − 4 x + 5 ) + ( x^(2) − 7 x + 3 ) .$]

#solutionbox[
#math.equation(block: true, alt: "8 x squared minus 11 x plus 8")[$8 x^(2) − 11 x + 8$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the sum: #math.equation(block: false, alt: "open parenthesis 14 y squared plus 6 y minus 4 close parenthesis plus open parenthesis 3 y squared plus 8 y plus 5 close parenthesis .")[$( 14 y^(2) + 6 y − 4 ) + ( 3 y^(2) + 8 y + 5 ) .$]

#solutionbox[
#math.equation(block: true, alt: "17 y squared plus 14 y plus 1")[$17 y^(2) + 14 y + 1$]
]
]

#examplebox("Example 8")[][
Find the difference: #math.equation(block: false, alt: "open parenthesis 9 w squared minus 7 w plus 5 close parenthesis minus open parenthesis 2 w squared minus 4 close parenthesis .")[$( 9 w^(2) − 7 w + 5 ) − ( 2 w^(2) − 4 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[9 w squared minus 7 w plus 5, minus 2 w squared minus 4.], alt: "9 w squared minus 7 w plus 5, minus 2 w squared minus 4.", caption: none)]),
  [Distribute and identify like terms.], [#figure(figph[9 w squared and 2 w squared are like terms. 5 and 4 are also like terms.], alt: "9 w squared and 2 w squared are like terms. 5 and 4 are also like terms.", caption: none)],
  [Rearrange the terms.], [#figure(figph[9 w squared minus 2 w squared minus 7 w plus 5 plus 4.], alt: "9 w squared minus 2 w squared minus 7 w plus 5 plus 4.", caption: none)],
  [Combine like terms.], [#figure(figph[7 w squared minus 7 w plus 9.], alt: "7 w squared minus 7 w plus 9.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the difference: #math.equation(block: false, alt: "open parenthesis 8 x squared plus 3 x minus 19 close parenthesis minus open parenthesis 7 x squared minus 14 close parenthesis .")[$( 8 x^(2) + 3 x − 19 ) − ( 7 x^(2) − 14 ) .$]

#solutionbox[
#math.equation(block: true, alt: "x squared plus 3 x minus 5")[$x^(2) + 3 x − 5$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the difference: #math.equation(block: false, alt: "open parenthesis 9 b squared minus 5 b minus 4 close parenthesis minus open parenthesis 3 b squared minus 5 b minus 7 close parenthesis .")[$( 9 b^(2) − 5 b − 4 ) − ( 3 b^(2) − 5 b − 7 ) .$]

#solutionbox[
#math.equation(block: true, alt: "6 b squared plus 3")[$6 b^(2) + 3$]
]
]

#examplebox("Example 9")[][
Subtract: #math.equation(block: false, alt: "open parenthesis c squared minus 4 c plus 7 close parenthesis")[$( c^(2) − 4 c + 7 )$] from #math.equation(block: false, alt: "open parenthesis 7 c squared minus 5 c plus 3 close parenthesis")[$( 7 c^(2) − 5 c + 3 )$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A math problem asks to subtract the polynomial (c² - 4c + 7) from the polynomial (7c² - 5c + 3).], alt: "A math problem asks to subtract the polynomial (c² - 4c + 7) from the polynomial (7c² - 5c + 3).", caption: none)]),
  [], [#figure(figph[7 c squared minus 5 c plus 3, minus c squared minus 4c plus 7.], alt: "7 c squared minus 5 c plus 3, minus c squared minus 4c plus 7.", caption: none)],
  [Distribute and identify like terms.], [#figure(figph[7 c squared and c squared are like terms. Minus 5c and 4c are like terms. 3 and minus 7 are like terms.], alt: "7 c squared and c squared are like terms. Minus 5c and 4c are like terms. 3 and minus 7 are like terms.", caption: none)],
  [Rearrange the terms.], [#figure(figph[7 c squared minus c squared minus 5 c plus 4 c plus 3 minus 7.], alt: "7 c squared minus c squared minus 5 c plus 4 c plus 3 minus 7.", caption: none)],
  [Combine like terms.], [#figure(figph[6 c squared minus c minus 4.], alt: "6 c squared minus c minus 4.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract: #math.equation(block: false, alt: "open parenthesis 5 z squared minus 6 z minus 2 close parenthesis")[$( 5 z^(2) − 6 z − 2 )$] from #math.equation(block: false, alt: "open parenthesis 7 z squared plus 6 z minus 4 close parenthesis")[$( 7 z^(2) + 6 z − 4 )$].

#solutionbox[
#math.equation(block: true, alt: "2 z squared plus 12 z minus 2")[$2 z^(2) + 12 z − 2$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract: #math.equation(block: false, alt: "open parenthesis x squared minus 5 x minus 8 close parenthesis")[$( x^(2) − 5 x − 8 )$] from #math.equation(block: false, alt: "open parenthesis 6 x squared plus 9 x minus 1 close parenthesis")[$( 6 x^(2) + 9 x − 1 )$].

#solutionbox[
#math.equation(block: true, alt: "5 x squared plus 14 x plus 7")[$5 x^(2) + 14 x + 7$]
]
]

#examplebox("Example 10")[][
Find the sum: #math.equation(block: false, alt: "open parenthesis u squared minus 6 u v plus 5 v squared close parenthesis plus open parenthesis 3 u squared plus 2 u v close parenthesis")[$( u^(2) − 6 u v + 5 v^(2) ) + ( 3 u^(2) + 2 u v )$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis u squared minus 6 u v plus 5 v squared close parenthesis plus open parenthesis 3 u squared plus 2 u v close parenthesis")[$( u^(2) − 6 u v + 5 v^(2) ) + ( 3 u^(2) + 2 u v )$]]),
  [Distribute.], [#math.equation(block: false, alt: "u squared minus 6 u v plus 5 v squared plus 3 u squared plus 2 u v")[$u^(2) − 6 u v + 5 v^(2) + 3 u^(2) + 2 u v$]],
  [Rearrange the terms, to put like terms together.], [#math.equation(block: false, alt: "u squared plus 3 u squared minus 6 u v plus 2 u v plus 5 v squared")[$u^(2) + 3 u^(2) − 6 u v + 2 u v + 5 v^(2)$]],
  [Combine like terms.], [#math.equation(block: false, alt: "4 u squared minus 4 u v plus 5 v squared")[$4 u^(2) − 4 u v + 5 v^(2)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the sum: #math.equation(block: false, alt: "open parenthesis 3 x squared minus 4 x y plus 5 y squared close parenthesis plus open parenthesis 2 x squared minus x y close parenthesis")[$( 3 x^(2) − 4 x y + 5 y^(2) ) + ( 2 x^(2) − x y )$].

#solutionbox[
#math.equation(block: true, alt: "5 x squared minus 5 x y plus 5 y squared")[$5 x^(2) − 5 x y + 5 y^(2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the sum: #math.equation(block: false, alt: "open parenthesis 2 x squared minus 3 x y minus 2 y squared close parenthesis plus open parenthesis 5 x squared minus 3 x y close parenthesis")[$( 2 x^(2) − 3 x y − 2 y^(2) ) + ( 5 x^(2) − 3 x y )$].

#solutionbox[
#math.equation(block: true, alt: "7 x squared minus 6 x y minus 2 y squared")[$7 x^(2) − 6 x y − 2 y^(2)$]
]
]

#examplebox("Example 11")[][
Find the difference: #math.equation(block: false, alt: "open parenthesis p squared plus q squared close parenthesis minus open parenthesis p squared plus 10 p q minus 2 q squared close parenthesis")[$( p^(2) + q^(2) ) − ( p^(2) + 10 p q − 2 q^(2) )$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis p squared plus q squared close parenthesis minus open parenthesis p squared plus 10 p q minus 2 q squared close parenthesis")[$( p^(2) + q^(2) ) − ( p^(2) + 10 p q − 2 q^(2) )$]]),
  [Distribute.], [#math.equation(block: false, alt: "p squared plus q squared minus p squared minus 10 p q plus 2 q squared")[$p^(2) + q^(2) − p^(2) − 10 p q + 2 q^(2)$]],
  [Rearrange the terms, to put like terms together.], [#math.equation(block: false, alt: "p squared minus p squared minus 10 p q plus q squared plus 2 q squared")[$p^(2) − p^(2) − 10 p q + q^(2) + 2 q^(2)$]],
  [Combine like terms.], [#math.equation(block: false, alt: "−10 p q plus 3 q squared")[$−10 p q + 3 q^(2)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the difference: #math.equation(block: false, alt: "open parenthesis a squared plus b squared close parenthesis minus open parenthesis a squared plus 5 a b minus 6 b squared close parenthesis")[$( a^(2) + b^(2) ) − ( a^(2) + 5 a b − 6 b^(2) )$].

#solutionbox[
#math.equation(block: true, alt: "minus 5 a b plus 7 b squared")[$− 5 a b + 7 b^(2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the difference: #math.equation(block: false, alt: "open parenthesis m squared plus n squared close parenthesis minus open parenthesis m squared minus 7 m n minus 3 n squared close parenthesis")[$( m^(2) + n^(2) ) − ( m^(2) − 7 m n − 3 n^(2) )$].

#solutionbox[
#math.equation(block: true, alt: "4 n squared plus 7 m n")[$4 n^(2) + 7 m n$]
]
]

#examplebox("Example 12")[][
Simplify: #math.equation(block: false, alt: "open parenthesis a cubed minus a squared b close parenthesis minus open parenthesis a b squared plus b cubed close parenthesis plus open parenthesis a squared b plus a b squared close parenthesis")[$( a^(3) − a^(2) b ) − ( a b^(2) + b^(3) ) + ( a^(2) b + a b^(2) )$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis a cubed minus a squared b close parenthesis minus open parenthesis a b squared plus b cubed close parenthesis plus open parenthesis a squared b plus a b squared close parenthesis")[$( a^(3) − a^(2) b ) − ( a b^(2) + b^(3) ) + ( a^(2) b + a b^(2) )$]]),
  [Distribute.], [#math.equation(block: false, alt: "a cubed minus a squared b minus a b squared minus b cubed plus a squared b plus a b squared")[$a^(3) − a^(2) b − a b^(2) − b^(3) + a^(2) b + a b^(2)$]],
  [Rearrange the terms, to put like terms together.], [#math.equation(block: false, alt: "a cubed minus a squared b plus a squared b minus a b squared plus a b squared minus b cubed")[$a^(3) − a^(2) b + a^(2) b − a b^(2) + a b^(2) − b^(3)$]],
  [Combine like terms.], [#math.equation(block: false, alt: "a cubed minus b cubed")[$a^(3) − b^(3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis x cubed minus x squared y close parenthesis minus open parenthesis x y squared plus y cubed close parenthesis plus open parenthesis x squared y plus x y squared close parenthesis")[$( x^(3) − x^(2) y ) − ( x y^(2) + y^(3) ) + ( x^(2) y + x y^(2) )$].

#solutionbox[
#math.equation(block: true, alt: "x cubed minus y cubed")[$x^(3) − y^(3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis p cubed minus p squared q close parenthesis plus open parenthesis p q squared plus q cubed close parenthesis minus open parenthesis p squared q plus p q squared close parenthesis")[$( p^(3) − p^(2) q ) + ( p q^(2) + q^(3) ) − ( p^(2) q + p q^(2) )$].

#solutionbox[
#math.equation(block: true, alt: "p cubed minus 2 p squared q plus q cubed")[$p^(3) − 2 p^(2) q + q^(3)$]
]
]


=== Evaluate a Polynomial for a Given Value

We have already learned how to evaluate expressions. Since polynomials are expressions, we’ll follow the same procedures to evaluate a #strong[polynomial]. We will substitute the given value for the variable and then simplify using the order of operations.

#examplebox("Example 13")[][
Evaluate #math.equation(block: false, alt: "5 x squared minus 8 x plus 4")[$5 x^(2) − 8 x + 4$] when

+ ⓐ #math.equation(block: false, alt: "x equals 4")[$x = 4$]
+ ⓑ #math.equation(block: false, alt: "x equals −2")[$x = −2$]
+ ⓒ #math.equation(block: false, alt: "x equals 0")[$x = 0$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ #math.equation(block: false, alt: "x equals 4")[$x = 4$]], []),
  [], [#figure(figph[5 x squared minus 8 x plus 4.], alt: "5 x squared minus 8 x plus 4.", caption: none)],
  [#figure(figph[The text reads 'Substitute 4 for x.' on a white background.], alt: "The text reads 'Substitute 4 for x.' on a white background.", caption: none)], [#figure(figph[5 times 4 squared minus 8 times 4 plus 4.], alt: "5 times 4 squared minus 8 times 4 plus 4.", caption: none)],
  [Simplify the exponents.], [#figure(figph[5 times 16 minus 8 times 4 plus 4.], alt: "5 times 16 minus 8 times 4 plus 4.", caption: none)],
  [Multiply.], [#figure(figph[A mathematical expression '80 - 32 + 4' is displayed on a white background.], alt: "A mathematical expression '80 - 32 + 4' is displayed on a white background.", caption: none)],
  [Simplify.], [#figure(figph[The number 52.], alt: "The number 52.", caption: none)],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ #math.equation(block: false, alt: "x equals −2")[$x = −2$]], []),
  [], [#figure(figph[5 x squared minus 8 x plus 4.], alt: "5 x squared minus 8 x plus 4.", caption: none)],
  [#figure(figph[Substitute negative 2 for x.], alt: "Substitute negative 2 for x.", caption: none)], [#figure(figph[5 times negative 2 squared minus 8 times negative 2 plus 4.], alt: "5 times negative 2 squared minus 8 times negative 2 plus 4.", caption: none)],
  [Simplify the exponents.], [#figure(figph[5 times 4 minus 8 times negative 2 plus 4.], alt: "5 times 4 minus 8 times negative 2 plus 4.", caption: none)],
  [Multiply.], [#figure(figph[The image displays the addition problem '20 + 16 + 4' in black text against a white background.], alt: "The image displays the addition problem '20 + 16 + 4' in black text against a white background.", caption: none)],
  [Simplify.], [#figure(figph[The number 40.], alt: "The number 40.", caption: none)],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓒ #math.equation(block: false, alt: "x equals 0")[$x = 0$]], []),
  [], [#figure(figph[5 x squared minus 8 x plus 4.], alt: "5 x squared minus 8 x plus 4.", caption: none)],
  [#figure(figph[The image shows the text 'Substitute 0 for x.' on a white background.], alt: "The image shows the text 'Substitute 0 for x.' on a white background.", caption: none)], [#figure(figph[5 times 0 squared minus 8 times 0 plus 4.], alt: "5 times 0 squared minus 8 times 0 plus 4.", caption: none)],
  [Simplify the exponents.], [#figure(figph[5 times 0 minus 8 times 0 plus 4.], alt: "5 times 0 minus 8 times 0 plus 4.", caption: none)],
  [Multiply.], [#figure(figph[A simple arithmetic expression displays '0 + 0 + 4' on a white background, clearly showing the sum of zero plus zero plus four.], alt: "A simple arithmetic expression displays '0 + 0 + 4' on a white background, clearly showing the sum of zero plus zero plus four.", caption: none)],
  [Simplify.], [#figure(figph[The number 4.], alt: "The number 4.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate: #math.equation(block: false, alt: "3 x squared plus 2 x minus 15")[$3 x^(2) + 2 x − 15$] when

+ ⓐ #math.equation(block: false, alt: "x equals 3")[$x = 3$]
+ ⓑ #math.equation(block: false, alt: "x equals −5")[$x = −5$]
+ ⓒ #math.equation(block: false, alt: "x equals 0")[$x = 0$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "18")[$18$] ⓑ #math.equation(block: false, alt: "50")[$50$] ⓒ #math.equation(block: false, alt: "−15")[$−15$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate: #math.equation(block: false, alt: "5 z squared minus z minus 4")[$5 z^(2) − z − 4$] when

+ ⓐ #math.equation(block: false, alt: "z equals −2")[$z = −2$]
+ ⓑ #math.equation(block: false, alt: "z equals 0")[$z = 0$]
+ ⓒ #math.equation(block: false, alt: "z equals 2")[$z = 2$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "18")[$18$] ⓑ #math.equation(block: false, alt: "−4")[$−4$] ⓒ #math.equation(block: false, alt: "14")[$14$]
]
]

#examplebox("Example 14")[][
The polynomial #math.equation(block: false, alt: "−16 t squared plus 250")[$−16 t^(2) + 250$] gives the height of a ball #math.equation(block: false, alt: "t")[$t$] seconds after it is dropped from a 250 foot tall building. Find the height after #math.equation(block: false, alt: "t equals 2")[$t = 2$] seconds.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "−16 t squared plus 250")[$−16 t^(2) + 250$]]),
  [Substitute #math.equation(block: false, alt: "t equals 2")[$t = 2$].], [#math.equation(block: false, alt: "−16 open parenthesis 2 close parenthesis squared plus 250")[$−16 attach(( 2 ), t: 2) + 250$]],
  [Simplify.], [#math.equation(block: false, alt: "−16 times 4 plus 250")[$−16 · 4 + 250$]],
  [Simplify.], [#math.equation(block: false, alt: "−64 plus 250")[$−64 + 250$]],
  [Simplify.], [#math.equation(block: false, alt: "186")[$186$]],
  [], [After 2 seconds the height of the ball is 186 feet.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The polynomial #math.equation(block: false, alt: "−16 t squared plus 250")[$−16 t^(2) + 250$] gives the height of a ball #math.equation(block: false, alt: "t")[$t$] seconds after it is dropped from a 250-foot tall building. Find the height after #math.equation(block: false, alt: "t equals 0")[$t = 0$] seconds.

#solutionbox[
#math.equation(block: true, alt: "250")[$250$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The polynomial #math.equation(block: false, alt: "−16 t squared plus 250")[$−16 t^(2) + 250$] gives the height of a ball #math.equation(block: false, alt: "t")[$t$] seconds after it is dropped from a 250-foot tall building. Find the height after #math.equation(block: false, alt: "t equals 3")[$t = 3$] seconds.

#solutionbox[
#math.equation(block: true, alt: "106")[$106$]
]
]

#examplebox("Example 15")[][
The polynomial #math.equation(block: false, alt: "6 x squared plus 15 x y")[$6 x^(2) + 15 x y$] gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side #emph[x] feet and sides of height #emph[y] feet. Find the cost of producing a box with #math.equation(block: false, alt: "x equals 4")[$x = 4$] feet and #math.equation(block: false, alt: "y equals 6")[$y = 6$] feet.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[6 x squared plus 15 x y.], alt: "6 x squared plus 15 x y.", caption: none)]),
  [#figure(figph[Substitute x equals 4 and y equals 6.], alt: "Substitute x equals 4 and y equals 6.", caption: none)], [#figure(figph[6 times 4 squared plus 15 times 4 times 6.], alt: "6 times 4 squared plus 15 times 4 times 6.", caption: none)],
  [Simplify.], [#figure(figph[6 times 16 plus 15 times 4 times 6.], alt: "6 times 16 plus 15 times 4 times 6.", caption: none)],
  [Simplify.], [#figure(figph[96 plus 360.], alt: "96 plus 360.", caption: none)],
  [Simplify.], [#figure(figph[The number 456.], alt: "The number 456.", caption: none)],
  [], [The cost of producing the box is \$456.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The polynomial #math.equation(block: false, alt: "6 x squared plus 15 x y")[$6 x^(2) + 15 x y$] gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side #emph[x] feet and sides of height #emph[y] feet. Find the cost of producing a box with #math.equation(block: false, alt: "x equals 6")[$x = 6$] feet and #math.equation(block: false, alt: "y equals 4")[$y = 4$] feet.

#solutionbox[
\$576
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The polynomial #math.equation(block: false, alt: "6 x squared plus 15 x y")[$6 x^(2) + 15 x y$] gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side #emph[x] feet and sides of height #emph[y] feet. Find the cost of producing a box with #math.equation(block: false, alt: "x equals 5")[$x = 5$] feet and #math.equation(block: false, alt: "y equals 8")[$y = 8$] feet.

#solutionbox[
\$750
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Access these online resources for additional instruction and practice with adding and subtracting polynomials.

- #link("https://openstax.org/l/25Addsubtrpoly1")[Add and Subtract Polynomials 1]
- #link("https://openstax.org/l/25Addsubtrpoly2")[Add and Subtract Polynomials 2]
- #link("https://openstax.org/l/25Addsubtrpoly3")[Add and Subtract Polynomial 3]
- #link("https://openstax.org/l/25Addsubtrpoly4")[Add and Subtract Polynomial 4]
]


=== Key Concepts

- #strong[Monomials]
- A monomial is a term of the form  #math.equation(block: false, alt: "a x to the power m")[$a x^(m)$], where #math.equation(block: false, alt: "a")[$a$] is a constant and #math.equation(block: false, alt: "m")[$m$] is a whole number
- #strong[Polynomials]
- #strong[polynomial]—A monomial, or two or more monomials combined by addition or subtraction is a polynomial.
- #strong[monomial]—A polynomial with exactly one term is called a monomial.
- #strong[binomial]—A polynomial with exactly two terms is called a binomial.
- #strong[trinomial]—A polynomial with exactly three terms is called a trinomial.
- #strong[Degree of a Polynomial]
- The #strong[degree of a term] is the sum of the exponents of its variables.
- The #strong[degree of a constant] is 0.
- The #strong[degree of a polynomial] is the highest degree of all its terms.


==== Practice Makes Perfect

#strong[Identify Polynomials, Monomials, Binomials, and Trinomials]

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

ⓐ #math.equation(block: false, alt: "81 b to the power 5 minus 24 b cubed plus 1")[$81 b^(5) − 24 b^(3) + 1$]
ⓑ #math.equation(block: false, alt: "5 c cubed plus 11 c squared minus c minus 8")[$5 c^(3) + 11 c^(2) − c − 8$]
ⓒ #math.equation(block: false, alt: "the fraction 14 over 15 y plus the fraction 1 over 7")[$frac(14, 15) y + frac(1, 7)$] #linebreak() 
ⓓ 5 #linebreak() 
ⓔ #math.equation(block: false, alt: "4 y plus 17")[$4 y + 17$]

#solutionbox[
ⓐ trinomial ⓑ polynomial ⓒ binomial ⓓ monomial ⓔ binomial
]

ⓐ #math.equation(block: false, alt: "x squared minus y squared")[$x^(2) − y^(2)$]
ⓑ #math.equation(block: false, alt: "−13 c to the power 4")[$−13 c^(4)$]
ⓒ #math.equation(block: false, alt: "x squared plus 5 x minus 7")[$x^(2) + 5 x − 7$]
ⓓ #math.equation(block: false, alt: "x squared y squared minus 2 x y plus 8")[$x^(2) y^(2) − 2 x y + 8$]
ⓔ 19

ⓐ #math.equation(block: false, alt: "8 minus 3 x")[$8 − 3 x$]
ⓑ #math.equation(block: false, alt: "z squared minus 5 z minus 6")[$z^(2) − 5 z − 6$]
ⓒ #math.equation(block: false, alt: "y cubed minus 8 y squared plus 2 y minus 16")[$y^(3) − 8 y^(2) + 2 y − 16$]
ⓓ #math.equation(block: false, alt: "81 b to the power 5 minus 24 b cubed plus 1")[$81 b^(5) − 24 b^(3) + 1$]
ⓔ #math.equation(block: false, alt: "−18")[$−18$]

#solutionbox[
ⓐ binomial ⓑ trinomial ⓒ polynomial ⓓ trinomial ⓔ monomial
]

ⓐ #math.equation(block: false, alt: "11 y squared")[$11 y^(2)$]
ⓑ #math.equation(block: false, alt: "−73")[$−73$]
ⓒ #math.equation(block: false, alt: "6 x squared minus 3 x y plus 4 x minus 2 y plus y squared")[$6 x^(2) − 3 x y + 4 x − 2 y + y^(2)$]
ⓓ #math.equation(block: false, alt: "4 y plus 17")[$4 y + 17$]
ⓔ #math.equation(block: false, alt: "5 c cubed plus 11 c squared minus c minus 8")[$5 c^(3) + 11 c^(2) − c − 8$]

#strong[Determine the Degree of Polynomials]

In the following exercises, determine the degree of each polynomial.

ⓐ #math.equation(block: false, alt: "6 a squared plus 12 a plus 14")[$6 a^(2) + 12 a + 14$]
ⓑ #math.equation(block: false, alt: "18 x y squared z")[$18 x y^(2) z$]
ⓒ #math.equation(block: false, alt: "5 x plus 2")[$5 x + 2$]
ⓓ #math.equation(block: false, alt: "y cubed minus 8 y squared plus 2 y minus 16")[$y^(3) − 8 y^(2) + 2 y − 16$]
ⓔ #math.equation(block: false, alt: "−24")[$−24$]

#solutionbox[
ⓐ 2 ⓑ 4 ⓒ 1 ⓓ 3 ⓔ 0
]

ⓐ #math.equation(block: false, alt: "9 y cubed minus 10 y squared plus 2 y minus 6")[$9 y^(3) − 10 y^(2) + 2 y − 6$]
ⓑ #math.equation(block: false, alt: "−12 p to the power 4")[$−12 p^(4)$]
ⓒ #math.equation(block: false, alt: "a squared plus 9 a plus 18")[$a^(2) + 9 a + 18$]
ⓓ #math.equation(block: false, alt: "20 x squared y squared minus 10 a squared b squared plus 30")[$20 x^(2) y^(2) − 10 a^(2) b^(2) + 30$]
ⓔ 17

ⓐ #math.equation(block: false, alt: "14 minus 29 x")[$14 − 29 x$]
ⓑ #math.equation(block: false, alt: "z squared minus 5 z minus 6")[$z^(2) − 5 z − 6$]
ⓒ #math.equation(block: false, alt: "y cubed minus 8 y squared plus 2 y minus 16")[$y^(3) − 8 y^(2) + 2 y − 16$]
ⓓ #math.equation(block: false, alt: "23 a b squared minus 14")[$23 a b^(2) − 14$]
ⓔ #math.equation(block: false, alt: "−3")[$−3$]

#solutionbox[
ⓐ 1 ⓑ 2 ⓒ 3 ⓓ 3 ⓔ 0
]

ⓐ #math.equation(block: false, alt: "62 y squared")[$62 y^(2)$]
ⓑ 15 ⓒ #math.equation(block: false, alt: "6 x squared minus 3 x y plus 4 x minus 2 y plus y squared")[$6 x^(2) − 3 x y + 4 x − 2 y + y^(2)$]
ⓓ #math.equation(block: false, alt: "10 minus 9 x")[$10 − 9 x$]
ⓔ #math.equation(block: false, alt: "m to the power 4 plus 4 m cubed plus 6 m squared plus 4 m plus 1")[$m^(4) + 4 m^(3) + 6 m^(2) + 4 m + 1$]

#strong[Add and Subtract Monomials]

In the following exercises, add or subtract the monomials.

#math.equation(block: true, alt: "7x squared plus 5 x squared")[$attach(#h(0.2em) "7x", t: 2) + 5 x^(2)$]

#solutionbox[
#math.equation(block: true, alt: "12 x squared")[$12 x^(2)$]
]

#math.equation(block: true, alt: "4y cubed plus 6 y cubed")[$attach(#h(0.2em) "4y", t: 3) + 6 y^(3)$]

#math.equation(block: true, alt: "−12 w plus 18 w")[$−12 w + 18 w$]

#solutionbox[
#math.equation(block: true, alt: "6 w")[$6 w$]
]

#math.equation(block: true, alt: "−3 m plus 9 m")[$−3 m + 9 m$]

#math.equation(block: true, alt: "4a minus 9 a")[$"4a" − 9 a$]

#solutionbox[
#math.equation(block: true, alt: "−5 a")[$−5 a$]
]

#math.equation(block: true, alt: "− y minus 5 y")[$"−" y − 5 y$]

#math.equation(block: true, alt: "28 x minus open parenthesis −12 x close parenthesis")[$28 x − ( −12 x )$]

#solutionbox[
#math.equation(block: true, alt: "40 x")[$40 x$]
]

#math.equation(block: true, alt: "13 z minus open parenthesis −4 z close parenthesis")[$13 z − ( −4 z )$]

#math.equation(block: true, alt: "−5 b minus 17 b")[$−5 b − 17 b$]

#solutionbox[
#math.equation(block: true, alt: "−22 b")[$−22 b$]
]

#math.equation(block: true, alt: "−10 x minus 35 x")[$−10 x − 35 x$]

#math.equation(block: true, alt: "12 a plus 5 b minus 22 a")[$12 a + 5 b − 22 a$]

#solutionbox[
#math.equation(block: true, alt: "−10a plus 5 b")[$"−10a" + 5 b$]
]

#math.equation(block: true, alt: "14x minus 3 y minus 13 x")[$"14x" − 3 y − 13 x$]

#math.equation(block: true, alt: "2 a squared plus b squared minus 6 a squared")[$2 a^(2) + b^(2) − 6 a^(2)$]

#solutionbox[
#math.equation(block: true, alt: "−4 a squared plus b squared")[$−4 a^(2) + b^(2)$]
]

#math.equation(block: true, alt: "5 u squared plus 4 v squared minus 6 u squared")[$5 u^(2) + 4 v^(2) − 6 u^(2)$]

#math.equation(block: true, alt: "x y squared minus 5 x minus 5 y squared")[$x y^(2) − 5 x − 5 y^(2)$]

#solutionbox[
#math.equation(block: true, alt: "x y squared minus 5 x minus 5 y squared")[$x y^(2) − 5 x − 5 y^(2)$]
]

#math.equation(block: true, alt: "p q squared minus 4 p minus 3 q squared")[$p q^(2) − 4 p − 3 q^(2)$]

#math.equation(block: true, alt: "a squared b minus 4 a minus 5 a b squared")[$a^(2) b − 4 a − 5 a b^(2)$]

#solutionbox[
#math.equation(block: true, alt: "a squared b minus 4 a minus 5 a b squared")[$a^(2) b − 4 a − 5 a b^(2)$]
]

#math.equation(block: true, alt: "x squared y minus 3 x plus 7 x y squared")[$x^(2) y − 3 x + 7 x y^(2)$]

#math.equation(block: true, alt: "12a plus 8 b")[$"12a" + 8 b$]

#solutionbox[
#math.equation(block: true, alt: "12a plus 8 b")[$"12a" + 8 b$]
]

#math.equation(block: true, alt: "19y plus 5 z")[$"19y" + 5 z$]

Add: #math.equation(block: false, alt: "4 a , −3 b , −8 a")[$4 a , −3 b , −8 a$]

#solutionbox[
#math.equation(block: true, alt: "−4 a minus 3 b")[$−4 a − 3 b$]
]

Add: #math.equation(block: false, alt: "4x , 3 y , −3 x")[$#h(0.2em) "4x" , 3 y , −3 x$]

Subtract #math.equation(block: false, alt: "5 x to the power 6 from minus 12 x to the power 6")[$5 x^(6) "from" − 12 x^(6)$].

#solutionbox[
#math.equation(block: true, alt: "−17 x to the power 6")[$−17 x^(6)$]
]

Subtract #math.equation(block: false, alt: "2 p to the power 4 from minus 7 p to the power 4")[$2 p^(4) "from" − 7 p^(4)$].

#strong[Add and Subtract Polynomials]

In the following exercises, add or subtract the polynomials.

#math.equation(block: true, alt: "open parenthesis 5 y squared plus 12 y plus 4 close parenthesis plus open parenthesis 6 y squared minus 8 y plus 7 close parenthesis")[$( 5 y^(2) + 12 y + 4 ) + ( 6 y^(2) − 8 y + 7 )$]

#solutionbox[
#math.equation(block: true, alt: "11 y squared plus 4 y plus 11")[$11 y^(2) + 4 y + 11$]
]

#math.equation(block: true, alt: "open parenthesis 4 y squared plus 10 y plus 3 close parenthesis plus open parenthesis 8 y squared minus 6 y plus 5 close parenthesis")[$( 4 y^(2) + 10 y + 3 ) + ( 8 y^(2) − 6 y + 5 )$]

#math.equation(block: true, alt: "open parenthesis x squared plus 6 x plus 8 close parenthesis plus open parenthesis −4 x squared plus 11 x minus 9 close parenthesis")[$( x^(2) + 6 x + 8 ) + ( −4 x^(2) + 11 x − 9 )$]

#solutionbox[
#math.equation(block: true, alt: "−3 x squared plus 17 x minus 1")[$−3 x^(2) + 17 x − 1$]
]

#math.equation(block: true, alt: "open parenthesis y squared plus 9 y plus 4 close parenthesis plus open parenthesis −2 y squared minus 5 y minus 1 close parenthesis")[$( y^(2) + 9 y + 4 ) + ( −2 y^(2) − 5 y − 1 )$]

#math.equation(block: true, alt: "open parenthesis 8 x squared minus 5 x plus 2 close parenthesis plus open parenthesis 3 x squared plus 3 close parenthesis")[$( 8 x^(2) − 5 x + 2 ) + ( 3 x^(2) + 3 )$]

#solutionbox[
#math.equation(block: true, alt: "11 x squared minus 5 x plus 5")[$11 x^(2) − 5 x + 5$]
]

#math.equation(block: true, alt: "open parenthesis 7 x squared minus 9 x plus 2 close parenthesis plus open parenthesis 6 x squared minus 4 close parenthesis")[$( 7 x^(2) − 9 x + 2 ) + ( 6 x^(2) − 4 )$]

#math.equation(block: true, alt: "open parenthesis 5 a squared plus 8 close parenthesis plus open parenthesis a squared minus 4 a minus 9 close parenthesis")[$( 5 a^(2) + 8 ) + ( a^(2) − 4 a − 9 )$]

#solutionbox[
#math.equation(block: true, alt: "6 a squared minus 4 a minus 1")[$6 a^(2) − 4 a − 1$]
]

#math.equation(block: true, alt: "open parenthesis p squared minus 6 p minus 18 close parenthesis plus open parenthesis 2 p squared plus 11 close parenthesis")[$( p^(2) − 6 p − 18 ) + ( 2 p^(2) + 11 )$]

#math.equation(block: true, alt: "open parenthesis 4 m squared minus 6 m minus 3 close parenthesis minus open parenthesis 2 m squared plus m minus 7 close parenthesis")[$( 4 m^(2) − 6 m − 3 ) − ( 2 m^(2) + m − 7 )$]

#solutionbox[
#math.equation(block: true, alt: "2 m squared minus 7 m plus 4")[$2 m^(2) − 7 m + 4$]
]

#math.equation(block: true, alt: "open parenthesis 3 b squared minus 4 b plus 1 close parenthesis minus open parenthesis 5 b squared minus b minus 2 close parenthesis")[$( 3 b^(2) − 4 b + 1 ) − ( 5 b^(2) − b − 2 )$]

#math.equation(block: true, alt: "open parenthesis a squared plus 8 a plus 5 close parenthesis minus open parenthesis a squared minus 3 a plus 2 close parenthesis")[$( a^(2) + 8 a + 5 ) − ( a^(2) − 3 a + 2 )$]

#solutionbox[
#math.equation(block: true, alt: "11 a plus 3")[$11 a + 3$]
]

#math.equation(block: true, alt: "open parenthesis b squared minus 7 b plus 5 close parenthesis minus open parenthesis b squared minus 2 b plus 9 close parenthesis")[$( b^(2) − 7 b + 5 ) − ( b^(2) − 2 b + 9 )$]

#math.equation(block: true, alt: "open parenthesis 12 s squared minus 15 s close parenthesis minus open parenthesis s minus 9 close parenthesis")[$( 12 s^(2) − 15 s ) − ( s − 9 )$]

#solutionbox[
#math.equation(block: true, alt: "12 s squared minus 16 s plus 9")[$12 s^(2) − 16 s + 9$]
]

#math.equation(block: true, alt: "open parenthesis 10 r squared minus 20 r close parenthesis minus open parenthesis r minus 8 close parenthesis")[$( 10 r^(2) − 20 r ) − ( r − 8 )$]

Subtract #math.equation(block: false, alt: "open parenthesis 9 x squared plus 2 close parenthesis")[$( 9 x^(2) + 2 )$] from #math.equation(block: false, alt: "open parenthesis 12 x squared minus x plus 6 close parenthesis")[$( 12 x^(2) − x + 6 )$].

#solutionbox[
#math.equation(block: true, alt: "3 x squared minus x plus 4")[$3 x^(2) − x + 4$]
]

Subtract #math.equation(block: false, alt: "open parenthesis 5 y squared minus y plus 12 close parenthesis")[$( 5 y^(2) − y + 12 )$] from #math.equation(block: false, alt: "open parenthesis 10 y squared minus 8 y minus 20 close parenthesis")[$( 10 y^(2) − 8 y − 20 )$].

Subtract #math.equation(block: false, alt: "open parenthesis 7 w squared minus 4 w plus 2 close parenthesis")[$( 7 w^(2) − 4 w + 2 )$] from #math.equation(block: false, alt: "open parenthesis 8 w squared minus w plus 6 close parenthesis")[$( 8 w^(2) − w + 6 )$].

#solutionbox[
#math.equation(block: true, alt: "w squared plus 3 w plus 4")[$w^(2) + 3 w + 4$]
]

Subtract #math.equation(block: false, alt: "open parenthesis 5 x squared minus x plus 12 close parenthesis")[$( 5 x^(2) − x + 12 )$] from #math.equation(block: false, alt: "open parenthesis 9 x squared minus 6 x minus 20 close parenthesis")[$( 9 x^(2) − 6 x − 20 )$].

Find the sum of #math.equation(block: false, alt: "open parenthesis 2 p cubed minus 8 close parenthesis")[$( 2 p^(3) − 8 )$] and #math.equation(block: false, alt: "open parenthesis p squared plus 9 p plus 18 close parenthesis")[$( p^(2) + 9 p + 18 )$].

#solutionbox[
#math.equation(block: true, alt: "2 p cubed plus p squared plus 9 p plus 10")[$2 p^(3) + p^(2) + 9 p + 10$]
]

Find the sum of #linebreak() #math.equation(block: false, alt: "open parenthesis q squared plus 4 q plus 13 close parenthesis")[$( q^(2) + 4 q + 13 )$] and #math.equation(block: false, alt: "open parenthesis 7 q cubed minus 3 close parenthesis")[$( 7 q^(3) − 3 )$].

Find the sum of #math.equation(block: false, alt: "open parenthesis 8 a cubed minus 8 a close parenthesis")[$( 8 a^(3) − 8 a )$] and #math.equation(block: false, alt: "open parenthesis a squared plus 6 a plus 12 close parenthesis")[$( a^(2) + 6 a + 12 )$].

#solutionbox[
#math.equation(block: true, alt: "8 a cubed plus a squared minus 2 a plus 12")[$8 a^(3) + a^(2) − 2 a + 12$]
]

Find the sum of #linebreak() #math.equation(block: false, alt: "open parenthesis b squared plus 5 b plus 13 close parenthesis")[$( b^(2) + 5 b + 13 )$] and #math.equation(block: false, alt: "open parenthesis 4 b cubed minus 6 close parenthesis")[$( 4 b^(3) − 6 )$].

Find the difference of #linebreak() #math.equation(block: false, alt: "open parenthesis w squared plus w minus 42 close parenthesis")[$( w^(2) + w − 42 )$] and #linebreak() #math.equation(block: false, alt: "open parenthesis w squared minus 10 w plus 24 close parenthesis")[$( w^(2) − 10 w + 24 )$].

#solutionbox[
#math.equation(block: true, alt: "11 w minus 66")[$11 w − 66$]
]

Find the difference of #linebreak() #math.equation(block: false, alt: "open parenthesis z squared minus 3 z minus 18 close parenthesis")[$( z^(2) − 3 z − 18 )$] and #linebreak() #math.equation(block: false, alt: "open parenthesis z squared plus 5 z minus 20 close parenthesis")[$( z^(2) + 5 z − 20 )$].

Find the difference of #linebreak() #math.equation(block: false, alt: "open parenthesis c squared plus 4 c minus 33 close parenthesis")[$( c^(2) + 4 c − 33 )$] and #linebreak() #math.equation(block: false, alt: "open parenthesis c squared minus 8 c plus 12 close parenthesis")[$( c^(2) − 8 c + 12 )$].

#solutionbox[
#math.equation(block: true, alt: "12 c minus 45")[$12 c − 45$]
]

Find the difference of #linebreak() #math.equation(block: false, alt: "open parenthesis t squared minus 5 t minus 15 close parenthesis")[$( t^(2) − 5 t − 15 )$] and #linebreak() #math.equation(block: false, alt: "open parenthesis t squared plus 4 t minus 17 close parenthesis")[$( t^(2) + 4 t − 17 )$].

#math.equation(block: true, alt: "open parenthesis 7 x squared minus 2 x y plus 6 y squared close parenthesis plus open parenthesis 3 x squared minus 5 x y close parenthesis")[$( 7 x^(2) − 2 x y + 6 y^(2) ) + ( 3 x^(2) − 5 x y )$]

#solutionbox[
#math.equation(block: true, alt: "10 x squared minus 7 x y plus 6 y squared")[$10 x^(2) − 7 x y + 6 y^(2)$]
]

#math.equation(block: true, alt: "open parenthesis −5 x squared minus 4 x y minus 3 y squared close parenthesis plus open parenthesis 2 x squared minus 7 x y close parenthesis")[$( −5 x^(2) − 4 x y − 3 y^(2) ) + ( 2 x^(2) − 7 x y )$]

#math.equation(block: true, alt: "open parenthesis 7 m squared plus m n minus 8 n squared close parenthesis plus open parenthesis 3 m squared plus 2 m n close parenthesis")[$( 7 m^(2) + m n − 8 n^(2) ) + ( 3 m^(2) + 2 m n )$]

#solutionbox[
#math.equation(block: true, alt: "10 m squared plus 3 m n minus 8 n squared")[$10 m^(2) + 3 m n − 8 n^(2)$]
]

#math.equation(block: true, alt: "open parenthesis 2 r squared minus 3 r s minus 2 s squared close parenthesis plus open parenthesis 5 r squared minus 3 r s close parenthesis")[$( 2 r^(2) − 3 r s − 2 s^(2) ) + ( 5 r^(2) − 3 r s )$]

#math.equation(block: true, alt: "open parenthesis a squared minus b squared close parenthesis minus open parenthesis a squared plus 3 a b minus 4 b squared close parenthesis")[$( a^(2) − b^(2) ) − ( a^(2) + 3 a b − 4 b^(2) )$]

#solutionbox[
#math.equation(block: true, alt: "−3 a b plus 3 b squared")[$−3 a b + 3 b^(2)$]
]

#math.equation(block: true, alt: "open parenthesis m squared plus 2 n squared close parenthesis minus open parenthesis m squared minus 8 m n minus n squared close parenthesis")[$( m^(2) + 2 n^(2) ) − ( m^(2) − 8 m n − n^(2) )$]

#math.equation(block: true, alt: "open parenthesis u squared minus v squared close parenthesis minus open parenthesis u squared minus 4 u v minus 3 v squared close parenthesis")[$( u^(2) − v^(2) ) − ( u^(2) − 4 u v − 3 v^(2) )$]

#solutionbox[
#math.equation(block: true, alt: "4 u v plus 2 v squared")[$4 u v + 2 v^(2)$]
]

#math.equation(block: true, alt: "open parenthesis j squared minus k squared close parenthesis minus open parenthesis j squared minus 8 j k minus 5 k squared close parenthesis")[$( j^(2) − k^(2) ) − ( j^(2) − 8 j k − 5 k^(2) )$]

#math.equation(block: true, alt: "open parenthesis p cubed minus 3 p squared q close parenthesis plus open parenthesis 2 p q squared plus 4 q cubed close parenthesis minus open parenthesis 3 p squared q plus p q squared close parenthesis")[$( p^(3) − 3 p^(2) q ) + ( 2 p q^(2) + 4 q^(3) ) − ( 3 p^(2) q + p q^(2) )$]

#solutionbox[
#math.equation(block: true, alt: "p cubed minus 6 p squared q plus p q squared plus 4 q cubed")[$p^(3) − 6 p^(2) q + p q^(2) + 4 q^(3)$]
]

#math.equation(block: true, alt: "open parenthesis a cubed minus 2 a squared b close parenthesis plus open parenthesis a b squared plus b cubed close parenthesis minus open parenthesis 3 a squared b plus 4 a b squared close parenthesis")[$( a^(3) − 2 a^(2) b ) + ( a b^(2) + b^(3) ) − ( 3 a^(2) b + 4 a b^(2) )$]

#math.equation(block: true, alt: "open parenthesis x cubed minus x squared y close parenthesis minus open parenthesis 4 x y squared minus y cubed close parenthesis plus open parenthesis 3 x squared y minus x y squared close parenthesis")[$( x^(3) − x^(2) y ) − ( 4 x y^(2) − y^(3) ) + ( 3 x^(2) y − x y^(2) )$]

#solutionbox[
#math.equation(block: true, alt: "x cubed plus 2 x squared y minus 5 x y squared plus y cubed")[$x^(3) + 2 x^(2) y − 5 x y^(2) + y^(3)$]
]

#math.equation(block: true, alt: "open parenthesis x cubed minus 2 x squared y close parenthesis minus open parenthesis x y squared minus 3 y cubed close parenthesis minus open parenthesis x squared y minus 4 x y squared close parenthesis")[$( x^(3) − 2 x^(2) y ) − ( x y^(2) − 3 y^(3) ) − ( x^(2) y − 4 x y^(2) )$]

#strong[Evaluate a Polynomial for a Given Value]

In the following exercises, evaluate each polynomial for the given value.

Evaluate #math.equation(block: false, alt: "8 y squared minus 3 y plus 2")[$8 y^(2) − 3 y + 2$] when:

ⓐ #math.equation(block: false, alt: "y equals 5")[$y = 5$]
ⓑ #math.equation(block: false, alt: "y equals −2")[$y = −2$]
ⓒ #math.equation(block: false, alt: "y equals 0")[$y = 0$]

#solutionbox[
ⓐ 187 ⓑ 40 ⓒ 2
]

Evaluate #math.equation(block: false, alt: "5 y squared minus y minus 7")[$5 y^(2) − y − 7$] when:

ⓐ #math.equation(block: false, alt: "y equals −4")[$y = −4$]
ⓑ #math.equation(block: false, alt: "y equals 1")[$y = 1$]
ⓒ #math.equation(block: false, alt: "y equals 0")[$y = 0$]

Evaluate #math.equation(block: false, alt: "4 minus 36 x")[$4 − 36 x$] when:

ⓐ #math.equation(block: false, alt: "x equals 3")[$x = 3$]
ⓑ #math.equation(block: false, alt: "x equals 0")[$x = 0$]
ⓒ #math.equation(block: false, alt: "x equals −1")[$x = −1$]

#solutionbox[
ⓐ −104 ⓑ 4 ⓒ 40
]

Evaluate #math.equation(block: false, alt: "16 minus 36 x squared")[$16 − 36 x^(2)$] when:

ⓐ #math.equation(block: false, alt: "x equals −1")[$x = −1$]
ⓑ #math.equation(block: false, alt: "x equals 0")[$x = 0$]
ⓒ #math.equation(block: false, alt: "x equals 2")[$x = 2$]

A painter drops a brush from a platform 75 feet high. The polynomial #math.equation(block: false, alt: "−16 t squared plus 75")[$−16 t^(2) + 75$] gives the height of the brush #math.equation(block: false, alt: "t")[$t$] seconds after it was dropped. Find the height after #math.equation(block: false, alt: "t equals 2")[$t = 2$] seconds.

#solutionbox[
11
]

A girl drops a ball off a cliff into the ocean. The polynomial #math.equation(block: false, alt: "−16 t squared plus 250")[$−16 t^(2) + 250$] gives the height of a ball #math.equation(block: false, alt: "t")[$t$] seconds after it is dropped from a 250-foot tall cliff. Find the height after #math.equation(block: false, alt: "t equals 2")[$t = 2$] seconds.

A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of #emph[p] dollars each is given by the polynomial #math.equation(block: false, alt: "−4 p squared plus 420 p .")[$−4 p^(2) + 420 p .$] Find the revenue received when #math.equation(block: false, alt: "p equals 60")[$p = 60$] dollars.

#solutionbox[
\$10,800
]

A manufacturer of the latest basketball shoes has found that the revenue received from selling the shoes at a cost of #emph[p] dollars each is given by the polynomial #math.equation(block: false, alt: "−4 p squared plus 420 p .")[$−4 p^(2) + 420 p .$] Find the revenue received when #math.equation(block: false, alt: "p equals 90")[$p = 90$] dollars.


==== Everyday Math

#strong[Fuel Efficiency] The fuel efficiency (in miles per gallon) of a car going at a speed of #math.equation(block: false, alt: "x")[$x$] miles per hour is given by the polynomial #math.equation(block: false, alt: "minus the fraction 1 over 150 x squared plus the fraction 1 over 3 x")[$− frac(1, 150) x^(2) + frac(1, 3) x$]. Find the fuel efficiency when #math.equation(block: false, alt: "x equals 30 mph")[$x = 30 #h(0.2em) "mph"$].

#solutionbox[
4
]

#strong[Stopping Distance] The number of feet it takes for a car traveling at #math.equation(block: false, alt: "x")[$x$] miles per hour to stop on dry, level concrete is given by the polynomial #math.equation(block: false, alt: "0.06 x squared plus 1.1 x")[$0.06 x^(2) + 1.1 x$]. Find the stopping distance when #math.equation(block: false, alt: "x equals 40 mph")[$x = 40 #h(0.2em) "mph"$].

#strong[Rental Cost] The cost to rent a rug cleaner for #math.equation(block: false, alt: "d")[$d$] days is given by the polynomial #math.equation(block: false, alt: "5.50 d plus 25")[$5.50 d + 25$]. Find the cost to rent the cleaner for 6 days.

#solutionbox[
\$58
]

#strong[Height of Projectile] The height (in feet) of an object projected upward is given by the polynomial #math.equation(block: false, alt: "−16 t squared plus 60 t plus 90")[$−16 t^(2) + 60 t + 90$] where #math.equation(block: false, alt: "t")[$t$] represents time in seconds. Find the height after #math.equation(block: false, alt: "t equals 2.5")[$t = 2.5$] seconds.

#strong[Temperature Conversion] The temperature in degrees Fahrenheit is given by the polynomial #math.equation(block: false, alt: "the fraction 9 over 5 c plus 32")[$frac(9, 5) c + 32$] where #math.equation(block: false, alt: "c")[$c$] represents the temperature in degrees Celsius. Find the temperature in degrees Fahrenheit when #math.equation(block: false, alt: "c equals 65 ° .")[$c = 65 "°" .$]

#solutionbox[
149
]


==== Writing Exercises

Using your own words, explain the difference between a monomial, a binomial, and a trinomial.

Using your own words, explain the difference between a polynomial with five terms and a polynomial with a degree of 5.

#solutionbox[
Answers will vary.
]

Ariana thinks the sum #math.equation(block: false, alt: "6 y squared plus 5 y to the power 4")[$6 y^(2) + 5 y^(4)$] is #math.equation(block: false, alt: "11 y to the power 6")[$11 y^(6)$]. What is wrong with her reasoning?

Jonathan thinks that #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] and #math.equation(block: false, alt: "the fraction 1 over x")[$frac(1, x)$] are both monomials. What is wrong with his reasoning?

#solutionbox[
Answers will vary.
]


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “identify polynomials, monomials, binomials, and trinomials,” “determine the degree of polynomials,” “add and subtract monomials,” “add and subtract polynomials,” and “evaluate a polynomial for a given value.” The rest of the cells are blank.], alt: "This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “identify polynomials, monomials, binomials, and trinomials,” “determine the degree of polynomials,” “add and subtract monomials,” “add and subtract polynomials,” and “evaluate a polynomial for a given value.” The rest of the cells are blank.", caption: none)

ⓑ If most of your checks were:

#strong[…confidently.] Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

#strong[…with some help.] This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

#strong[…no - I don’t get it!] This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.