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== 10.1#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"))[
Evaluate: #math.equation(block: false, alt: "4 y squared")[$4 y^(2)$] when #math.equation(block: false, alt: "y equals 5")[$y = 5$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "100")[$100$]
]
]


=== Identify Polynomials, Monomials, Binomials, and Trinomials

In Evaluate, Simplify, and Translate Expressions, you learned that a #strong[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. A monomial, or a sum and/or difference of monomials, is called a polynomial.

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

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

#strong[monomial]—A polynomial with exactly one term

#strong[binomial]— A polynomial with exactly two terms

#strong[trinomial]—A polynomial with exactly three terms
]

Notice the roots:

- #emph[poly]- means many
- #emph[mono]- means one
- #emph[bi]- means two
- #emph[tri]- means three

Here are some examples of polynomials:

#figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([#strong[Polynomial]], [#math.equation(block: false, alt: "b plus 1")[$b + 1$]], [#math.equation(block: false, alt: "4 y squared minus 7 y plus 2")[$4 y^(2) − 7 y + 2$]], [#math.equation(block: false, alt: "5 x to the power 5 minus 4 x to the power 4 plus x cubed plus 8 x squared minus 9 x plus 1")[$5 x^(5) − 4 x^(4) + x^(3) + 8 x^(2) − 9 x + 1$]]),
  [#strong[Monomial]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "4 b squared")[$4 b^(2)$]], [#math.equation(block: false, alt: "−9 x cubed")[$−9 x^(3)$]],
  [#strong[Binomial]], [#math.equation(block: false, alt: "3 a minus 7")[$3 a − 7$]], [#math.equation(block: false, alt: "y squared minus 9")[$y^(2) − 9$]], [#math.equation(block: false, alt: "17 x cubed plus 14 x squared")[$17 x^(3) + 14 x^(2)$]],
  [#strong[Trinomial]], [#math.equation(block: false, alt: "x squared minus 5 x plus 6")[$x^(2) − 5 x + 6$]], [#math.equation(block: false, alt: "4 y squared minus 7 y plus 2")[$4 y^(2) − 7 y + 2$]], [#math.equation(block: false, alt: "5 a to the power 4 minus 3 a cubed plus a")[$5 a^(4) − 3 a^(3) + a$]],
))

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

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

+ ⓐ #math.equation(block: false, alt: "8 x squared minus 7 x minus 9")[$#h(0.2em) 8 x^(2) − 7 x − 9$]
+ ⓑ #math.equation(block: false, alt: "−5 a to the power 4")[$#h(0.2em) −5 a^(4)$]
+ ⓒ #math.equation(block: false, alt: "x to the power 4 minus 7 x cubed minus 6 x squared plus 5 x plus 2")[$#h(0.2em) x^(4) − 7 x^(3) − 6 x^(2) + 5 x + 2$]
+ ⓓ #math.equation(block: false, alt: "11 minus 4 y cubed")[$#h(0.2em) 11 − 4 y^(3)$]
+ ⓔ #math.equation(block: false, alt: "n")[$#h(0.2em) n$]

#solutionbox[
#figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([], [Polynomial], [Number of terms], [Type]),
  [ⓐ], [#math.equation(block: false, alt: "8 x squared minus 7 x minus 9")[$8 x^(2) − 7 x − 9$]], [3], [Trinomial],
  [ⓑ], [#math.equation(block: false, alt: "−5 a to the power 4")[$−5 a^(4)$]], [1], [Monomial],
  [ⓒ], [#math.equation(block: false, alt: "x to the power 4 minus 7 x cubed minus 6 x squared plus 5 x plus 2")[$x^(4) − 7 x^(3) − 6 x^(2) + 5 x + 2$]], [5], [Polynomial],
  [ⓓ], [#math.equation(block: false, alt: "11 minus 4 y cubed")[$11 − 4 y^(3)$]], [2], [Binomial],
  [ⓔ], [#math.equation(block: false, alt: "n")[$n$]], [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: "z")[$#h(0.2em) z$]
+ ⓑ #math.equation(block: false, alt: "2 x cubed minus 4 x squared minus x minus 8")[$#h(0.2em) 2 x^(3) − 4 x^(2) − x − 8$]
+ ⓒ #math.equation(block: false, alt: "6 x squared minus 4 x plus 1")[$#h(0.2em) 6 x^(2) − 4 x + 1$]
+ ⓓ #math.equation(block: false, alt: "9 minus 4 y squared")[$#h(0.2em) 9 − 4 y^(2)$]
+ ⓔ #math.equation(block: false, alt: "3 x to the power 7")[$#h(0.2em) 3 x^(7)$]

#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: "y cubed minus 8")[$#h(0.2em) y^(3) − 8$]
+ ⓑ #math.equation(block: false, alt: "9 x cubed minus 5 x squared minus x")[$#h(0.2em) 9 x^(3) − 5 x^(2) − x$]
+ ⓒ #math.equation(block: false, alt: "x to the power 4 minus 3 x squared minus 4 x minus 7")[$#h(0.2em) x^(4) − 3 x^(2) − 4 x − 7$]
+ ⓓ #math.equation(block: false, alt: "− y to the power 4")[$#h(0.2em) "−" y^(4)$]
+ ⓔ #math.equation(block: false, alt: "w")[$#h(0.2em) w$]

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


=== Determine the Degree of Polynomials

In this section, we will work with polynomials that have only one variable in each term. The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.

A monomial that has no variable, just a constant, is a special case. The degree of a constant is #math.equation(block: false, alt: "0")[$0$]—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 exponent of its variable.

The #strong[degree of a constant] is #math.equation(block: false, alt: "0 .")[$0 .$]

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.

Remember: Any base written without an exponent has an implied exponent of #math.equation(block: false, alt: "1 .")[$1 .$]

#figure(figph[A table is shown. The top row is titled “Monomials” and lists the following monomials: 5, 4 b squared, negative 9 x cubed, negative 18. The next row is titled “Degree” and lists, in blue, 0, 2, 3, and 0. The next row is titled “Binomial” and lists the following binomials: b plus 1, 3a minus 7, y squared minus 9, 17 x cubed plus 14 x squared. The next row is titled “Degree of each term,” with “term” written in blue. This row lists 1, 0, 1, 0, 2, 0, 3, 2 in blue. The next row is titled “Degree of polynomial,” with “polynomial” written in red. This row lists 1, 1, 2, 3 in red. The next row is titled “Trinomial” and lists the following trinomials: x squared minus 5x plus 6, 4 y squared minus 7y plus 2, 5 a to the fourth minus 3 a cubed plus a, and x to the fourth plus 2 x squared minus 5. The next row is titled “Degree of each term,” with “term” written in blue. This row lists 2, 1, 0, 2, 1, 0, 4, 3, 1, 4, 2, 0 in blue. The next row is titled “Degree of polynomial,” with “polynomial” written in red. This row lists 2, 2, 4, 4 in red. The next row is titled “Polynomial” and lists the following polynomials: b plus 1, 4 y squared minus 7y plus 2, and 4 x to the fourth plus x cubed plus 8 x squared minus 9x plus 1. The next row is titled “Degree of each term,” with “term” written in blue. This row lists 1, 0, 2, 1, 0, 4, 3, 2, 1, 0 in blue. The next row is titled “Degree of polynomial,” with “polynomial” written in red. This row lists 1, 2, 4 in red.], alt: "A table is shown. The top row is titled “Monomials” and lists the following monomials: 5, 4 b squared, negative 9 x cubed, negative 18. The next row is titled “Degree” and lists, in blue, 0, 2, 3, and 0. The next row is titled “Binomial” and lists the following binomials: b plus 1, 3a minus 7, y squared minus 9, 17 x cubed plus 14 x squared. The next row is titled “Degree of each term,” with “term” written in blue. This row lists 1, 0, 1, 0, 2, 0, 3, 2 in blue. The next row is titled “Degree of polynomial,” with “polynomial” written in red. This row lists 1, 1, 2, 3 in red. The next row is titled “Trinomial” and lists the following trinomials: x squared minus 5x plus 6, 4 y squared minus 7y plus 2, 5 a to the fourth minus 3 a cubed plus a, and x to the fourth plus 2 x squared minus 5. The next row is titled “Degree of each term,” with “term” written in blue. This row lists 2, 1, 0, 2, 1, 0, 4, 3, 1, 4, 2, 0 in blue. The next row is titled “Degree of polynomial,” with “polynomial” written in red. This row lists 2, 2, 4, 4 in red. The next row is titled “Polynomial” and lists the following polynomials: b plus 1, 4 y squared minus 7y plus 2, and 4 x to the fourth plus x cubed plus 8 x squared minus 9x plus 1. The next row is titled “Degree of each term,” with “term” written in blue. This row lists 1, 0, 2, 1, 0, 4, 3, 2, 1, 0 in blue. The next row is titled “Degree of polynomial,” with “polynomial” written in red. This row lists 1, 2, 4 in red.", caption: none)

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

+ ⓐ #math.equation(block: false, alt: "4 x")[$#h(0.2em) 4 x$]
+ ⓑ #math.equation(block: false, alt: "3 x cubed minus 5 x plus 7")[$#h(0.2em) 3 x^(3) − 5 x + 7$]
+ ⓒ #math.equation(block: false, alt: "−11")[$#h(0.2em) −11$]
+ ⓓ #math.equation(block: false, alt: "−6 x squared plus 9 x minus 3")[$#h(0.2em) −6 x^(2) + 9 x − 3$]
+ ⓔ #math.equation(block: false, alt: "8 x plus 2")[$#h(0.2em) 8 x + 2$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], [#math.equation(block: false, alt: "4 x")[$4 x$]]),
  [The exponent of #math.equation(block: false, alt: "x")[$x$] is one. #math.equation(block: false, alt: "x equals x to the power 1")[$x = x^(1)$]], [The degree is 1.],
  [ⓑ], [#math.equation(block: false, alt: "3 x cubed minus 5 x plus 7")[$3 x^(3) − 5 x + 7$]],
  [The highest degree of all the terms is 3.], [The degree is 3],
  [ⓒ], [#math.equation(block: false, alt: "minus 11")[$− 11$]],
  [The degree of a constant is 0.], [The degree is 0.],
  [ⓓ], [#math.equation(block: false, alt: "−6 x squared plus 9 x minus 3")[$−6 x^(2) + 9 x − 3$]],
  [The highest degree of all the terms is 2.], [The degree is 2.],
  [ⓔ], [#math.equation(block: false, alt: "8 x plus 2")[$8 x + 2$]],
  [The highest degree of all the terms is 1.], [The degree is 1.],
))
]
]

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

+ ⓐ #math.equation(block: false, alt: "−6 y")[$#h(0.2em) −6 y$]
+ ⓑ #math.equation(block: false, alt: "4 x minus 1")[$#h(0.2em) 4 x − 1$]
+ ⓒ #math.equation(block: false, alt: "3 x to the power 4 plus 4 x squared minus 8")[$#h(0.2em) 3 x^(4) + 4 x^(2) − 8$]
+ ⓓ #math.equation(block: false, alt: "2 y squared plus 3 y plus 9")[$#h(0.2em) 2 y^(2) + 3 y + 9$]
+ ⓔ #math.equation(block: false, alt: "−18")[$#h(0.2em) −18$]

#solutionbox[
+ ⓐ 1
+ ⓑ 1
+ ⓒ 4
+ ⓓ 2
+ ⓔ 0
]
]

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

+ ⓐ #math.equation(block: false, alt: "47")[$#h(0.2em) 47$]
+ ⓑ #math.equation(block: false, alt: "2 x squared minus 8 x plus 2")[$#h(0.2em) 2 x^(2) − 8 x + 2$]
+ ⓒ #math.equation(block: false, alt: "x to the power 4 minus 16")[$#h(0.2em) x^(4) − 16$]
+ ⓓ #math.equation(block: false, alt: "y to the power 5 minus 5 y cubed plus y")[$#h(0.2em) y^(5) − 5 y^(3) + y$]
+ ⓔ #math.equation(block: false, alt: "9 a cubed")[$#h(0.2em) 9 a^(3)$]

#solutionbox[
+ ⓐ 0
+ ⓑ 2
+ ⓒ 4
+ ⓓ 5
+ ⓔ 3
]
]

Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in #strong[standard form]. Look back at the polynomials in . Notice that they are all written in standard form. Get in the habit of writing the term with the highest degree first.


=== Add and Subtract Monomials

In The Language of Algebra, you simplified expressions by combining like terms. Adding and subtracting monomials is the same as combining like terms. Like terms must have the same variable with the same exponent. Recall that when combining like terms only the coefficients are combined, never the exponents.

#examplebox("Example 3")[][
Add: #math.equation(block: false, alt: "17 x squared plus 6 x squared .")[$17 x^(2) + 6 x^(2) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "17 x squared plus 6 x squared")[$17 x^(2) + 6 x^(2)$]]),
  [Combine like terms.], [#math.equation(block: false, alt: "23 x squared")[$23 x^(2)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "12 x squared plus 5 x squared .")[$12 x^(2) + 5 x^(2) .$]

#solutionbox[
17#emph[x]#super[2]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "−11 y squared plus 8 y squared .")[$−11 y^(2) + 8 y^(2) .$]

#solutionbox[
−3#emph[y]#super[2]
]
]

#examplebox("Example 4")[][
Subtract: #math.equation(block: false, alt: "11 n minus open parenthesis −8 n close parenthesis .")[$11 n − ( −8 n ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "11 n minus open parenthesis −8 n close parenthesis")[$11 n − ( −8 n )$]]),
  [Combine like terms.], [#math.equation(block: false, alt: "19 n")[$19 n$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract: #math.equation(block: false, alt: "9 n minus open parenthesis −5 n close parenthesis .")[$9 n − ( −5 n ) .$]

#solutionbox[
14#emph[n]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract: #math.equation(block: false, alt: "−7 a cubed minus open parenthesis −5 a cubed close parenthesis .")[$−7 a^(3) − ( −5 a^(3) ) .$]

#solutionbox[
−2#emph[a]#super[3]
]
]

#examplebox("Example 5")[][
Simplify: #math.equation(block: false, alt: "a squared plus 4 b squared minus 7 a squared .")[$a^(2) + 4 b^(2) − 7 a^(2) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "a squared plus 4 b squared minus 7 a squared")[$a^(2) + 4 b^(2) − 7 a^(2)$]]),
  [Combine like terms.], [#math.equation(block: false, alt: "−6 a to the power squared plus 4 b squared")[$−6 a^(2) + 4 b^(2)$]],
))

Remember, #math.equation(block: false, alt: "−6 a squared")[$−6 a^(2)$] and #math.equation(block: false, alt: "4 b squared")[$4 b^(2)$] are not like terms. The variables are not the same.
]
]

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

#solutionbox[
−2#emph[x]#super[2] + 3#emph[y]#super[2]
]
]

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

#solutionbox[
−2#emph[a]#super[2] + #emph[b]#super[2]
]
]


=== Add and Subtract Polynomials

Adding and subtracting polynomials can be thought of as just adding and subtracting like terms. Look for like terms—those with the same variables with the same exponent. The #strong[Commutative Property] allows us to rearrange the terms to put like terms together. It may also be helpful to underline, circle, or box like terms.

#examplebox("Example 6")[][
Find the sum: #math.equation(block: false, alt: "open parenthesis 4 x squared minus 5 x plus 1 close parenthesis plus open parenthesis 3 x squared minus 8 x minus 9 close parenthesis .")[$( 4 x^(2) − 5 x + 1 ) + ( 3 x^(2) − 8 x − 9 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing the addition of two quadratic polynomials: (4x^2 - 5x + 1) + (3x^2 - 8x - 9).], alt: "A mathematical expression showing the addition of two quadratic polynomials: (4x^2 - 5x + 1) + (3x^2 - 8x - 9).", caption: none)]),
  [Identify like terms.], [#figure(figph[A mathematical expression shows: 4x^2 - 5x + \[1\] + 3x^2 - 8x - \[9\]. Terms with 'x' and 'x^2' are underlined with single and double lines, respectively, and the constants '1' and '9' are boxed.], alt: "A mathematical expression shows: 4x^2 - 5x + [1] + 3x^2 - 8x - [9]. Terms with 'x' and 'x^2' are underlined with single and double lines, respectively, and the constants '1' and '9' are boxed.", caption: none)],
  [Rearrange to get the like terms together.], [#figure(figph[An algebraic expression demonstrating the combination of like terms: 4x^2 + 3x^2 - 5x - 8x + 1 - 9. Terms with x squared, x, and constants are visually grouped by underlines.], alt: "An algebraic expression demonstrating the combination of like terms: 4x^2 + 3x^2 - 5x - 8x + 1 - 9. Terms with x squared, x, and constants are visually grouped by underlines.", caption: none)],
  [Combine like terms.], [#figure(figph[A mathematical expression displaying a quadratic polynomial: 7x^2 - 13x - 8.], alt: "A mathematical expression displaying a quadratic polynomial: 7x^2 - 13x - 8.", caption: none)],
))
]
]

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

#solutionbox[
4#emph[x]#super[2] − 8#emph[x] + 10
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the sum: #math.equation(block: false, alt: "open parenthesis 7 y squared plus 4 y minus 6 close parenthesis plus open parenthesis 4 y squared plus 5 y plus 1 close parenthesis .")[$( 7 y^(2) + 4 y − 6 ) + ( 4 y^(2) + 5 y + 1 ) .$]

#solutionbox[
11#emph[y]#super[2] + 9#emph[y] − 5
]
]

Parentheses are grouping symbols. When we add polynomials as we did in , we can rewrite the expression without parentheses and then combine like terms. But when we subtract polynomials, we must be very careful with the signs.

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

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing the subtraction of two polynomials: (7u^2 - 5u + 3) - (4u^2 - 2).], alt: "A mathematical expression showing the subtraction of two polynomials: (7u^2 - 5u + 3) - (4u^2 - 2).", caption: none)]),
  [Distribute and identify like terms.], [#figure(figph[An algebraic expression: 7u^2 - 5u + \[3\] - 4u^2 + \[2\]. Terms 7u^2 and 4u^2 are double-underlined, 5u is single-underlined, and 3 and 2 are boxed.], alt: "An algebraic expression: 7u^2 - 5u + [3] - 4u^2 + [2]. Terms 7u^2 and 4u^2 are double-underlined, 5u is single-underlined, and 3 and 2 are boxed.", caption: none)],
  [Rearrange the terms.], [#figure(figph[A mathematical expression is displayed, showing '7u^2 - 4u^2 - 5u + 3 + 2'. The terms '7u^2' and '4u^2' are double-underlined, '5u' is single-underlined, and the final number '2' is enclosed in a box.], alt: "A mathematical expression is displayed, showing '7u^2 - 4u^2 - 5u + 3 + 2'. The terms '7u^2' and '4u^2' are double-underlined, '5u' is single-underlined, and the final number '2' is enclosed in a box.", caption: none)],
  [Combine like terms.], [#figure(figph[A mathematical expression displays three terms: '3u^2 - 5u + 5' on a white background. This quadratic expression features a squared variable, a linear variable, and a constant term.], alt: "A mathematical expression displays three terms: '3u^2 - 5u + 5' on a white background. This quadratic expression features a squared variable, a linear variable, and a constant term.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the difference: #math.equation(block: false, alt: "open parenthesis 6 y squared plus 3 y minus 1 close parenthesis minus open parenthesis 3 y squared minus 4 close parenthesis .")[$( 6 y^(2) + 3 y − 1 ) − ( 3 y^(2) − 4 ) .$]

#solutionbox[
3#emph[y]#super[2] + 3#emph[y] + 3
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the difference: #math.equation(block: false, alt: "open parenthesis 8 u squared minus 7 u minus 2 close parenthesis minus open parenthesis 5 u squared minus 6 u minus 4 close parenthesis .")[$( 8 u^(2) − 7 u − 2 ) − ( 5 u^(2) − 6 u − 4 ) .$]

#solutionbox[
3#emph[u]#super[2] − #emph[u] + 2
]
]

#examplebox("Example 8")[][
Subtract: #math.equation(block: false, alt: "open parenthesis m squared minus 3 m plus 8 close parenthesis")[$( m^(2) − 3 m + 8 )$] from #math.equation(block: false, alt: "open parenthesis 9 m squared minus 7 m plus 4 close parenthesis .")[$( 9 m^(2) − 7 m + 4 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical problem asking to subtract the polynomial (m^2 - 3m + 8) from (9m^2 - 7m + 4).], alt: "A mathematical problem asking to subtract the polynomial (m^2 - 3m + 8) from (9m^2 - 7m + 4).", caption: none)]),
  [Distribute and identify like terms.], [#figure(figph[An algebraic expression is displayed: 9m^2 - 7m + \[4 - m^2 + 3m - 8\]. Some terms are underlined with single or double lines, and constants 4 and 8 are enclosed in square boxes.], alt: "An algebraic expression is displayed: 9m^2 - 7m + [4 - m^2 + 3m - 8]. Some terms are underlined with single or double lines, and constants 4 and 8 are enclosed in square boxes.", caption: none)],
  [Rearrange the terms.], [#figure(figph[A mathematical expression showing 9m^2 - m^2 - 7m + 3m + 4 - 8, featuring like terms for simplification.], alt: "A mathematical expression showing 9m^2 - m^2 - 7m + 3m + 4 - 8, featuring like terms for simplification.", caption: none)],
  [Combine like terms.], [#figure(figph[The image displays the mathematical expression 8m^2 - 4m - 4, which is a quadratic trinomial. The expression involves the variable 'm' raised to the power of two and one, along with constant terms.], alt: "The image displays the mathematical expression 8m^2 - 4m - 4, which is a quadratic trinomial. The expression involves the variable 'm' raised to the power of two and one, along with constant terms.", caption: none)],
))
]
]

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

#solutionbox[
4#emph[n]#super[2] + 12#emph[n]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract: #math.equation(block: false, alt: "open parenthesis a squared minus 4 a minus 9 close parenthesis")[$( a^(2) − 4 a − 9 )$] from #math.equation(block: false, alt: "open parenthesis 6 a squared plus 4 a minus 1 close parenthesis .")[$( 6 a^(2) + 4 a − 1 ) .$]

#solutionbox[
5#emph[a]#super[2] + 8#emph[a] + 8
]
]


=== Evaluate a Polynomial for a Given Value

In The Language of Algebra we evaluated expressions. Since polynomials are expressions, we'll follow the same procedures to evaluate polynomials—substitute the given value for the variable into the polynomial, and then simplify.

#examplebox("Example 9")[][
Evaluate #math.equation(block: false, alt: "3 x squared minus 9 x plus 7")[$3 x^(2) − 9 x + 7$] when

+ ⓐ #math.equation(block: false, alt: "x equals 3")[$#h(0.2em) x = 3$]
+ ⓑ #math.equation(block: false, alt: "x equals −1")[$#h(0.2em) x = −1$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ #math.equation(block: false, alt: "x equals 3")[$x = 3$]], []),
  [], [#math.equation(block: false, alt: "3 x squared minus 9 x plus 7")[$3 x^(2) − 9 x + 7$]],
  [Substitute 3 for #math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "3 open parenthesis 3 close parenthesis squared minus 9 open parenthesis 3 close parenthesis plus 7")[$3 attach(( 3 ), t: 2) − 9 ( 3 ) + 7$]],
  [Simplify the expression with the exponent.], [#math.equation(block: false, alt: "3 times 9 minus 9 open parenthesis 3 close parenthesis plus 7")[$3 · 9 − 9 ( 3 ) + 7$]],
  [Multiply.], [#math.equation(block: false, alt: "27 minus 27 plus 7")[$27 − 27 + 7$]],
  [Simplify.], [#math.equation(block: false, alt: "7")[$7$]],
))

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ #math.equation(block: false, alt: "x equals −1")[$x = −1$]], []),
  [], [#math.equation(block: false, alt: "3 x squared minus 9 x plus 7")[$3 x^(2) − 9 x + 7$]],
  [Substitute −1 for #math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "3 open parenthesis −1 close parenthesis squared minus 9 open parenthesis −1 close parenthesis plus 7")[$3 attach(( −1 ), t: 2) − 9 ( −1 ) + 7$]],
  [Simplify the expression with the exponent.], [#math.equation(block: false, alt: "3 times 1 minus 9 open parenthesis −1 close parenthesis plus 7")[$3 · 1 − 9 ( −1 ) + 7$]],
  [Multiply.], [#math.equation(block: false, alt: "3 plus 9 plus 7")[$3 + 9 + 7$]],
  [Simplify.], [#math.equation(block: false, alt: "19")[$19$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate: #math.equation(block: false, alt: "2 x squared plus 4 x minus 3")[$2 x^(2) + 4 x − 3$] when

+ ⓐ #math.equation(block: false, alt: "x equals 2")[$#h(0.2em) x = 2$]
+ ⓑ #math.equation(block: false, alt: "x equals −3")[$#h(0.2em) x = −3$]

#solutionbox[
+ ⓐ 13
+ ⓑ  3
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate: #math.equation(block: false, alt: "7 y squared minus y minus 2")[$7 y^(2) − y − 2$] when

+ ⓐ #math.equation(block: false, alt: "y equals −4")[$#h(0.2em) y = −4$]
+ ⓑ #math.equation(block: false, alt: "y equals 0")[$#h(0.2em) y = 0$]

#solutionbox[
+ ⓐ 114
+ ⓑ −2
]
]

#examplebox("Example 10")[][
The polynomial #math.equation(block: false, alt: "−16 t squared plus 300")[$−16 t^(2) + 300$] gives the height of an object #math.equation(block: false, alt: "t")[$t$] seconds after it is dropped from a #math.equation(block: false, alt: "300")[$300$] foot tall bridge. Find the height after #math.equation(block: false, alt: "t equals 3")[$t = 3$] seconds.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image shows the mathematical expression -16t^2 + 300, presented in a clear, typeset format against a white background.], alt: "The image shows the mathematical expression -16t^2 + 300, presented in a clear, typeset format against a white background.", caption: none)]),
  [Substitute 3 for #math.equation(block: false, alt: "t")[$t$]], [#figure(figph[The image shows the mathematical expression -16(3)^2 + 300, which evaluates to -16 \* 9 + 300 = -144 + 300 = 156.], alt: "The image shows the mathematical expression -16(3)^2 + 300, which evaluates to -16 * 9 + 300 = -144 + 300 = 156.", caption: none)],
  [Simplify the expression with the exponent.], [#figure(figph[A mathematical expression showing -16 multiplied by 9, then added to 300.], alt: "A mathematical expression showing -16 multiplied by 9, then added to 300.", caption: none)],
  [Multiply.], [#figure(figph[The mathematical expression -144 + 300 is displayed in black font on a white background.], alt: "The mathematical expression -144 + 300 is displayed in black font on a white background.", caption: none)],
  [Simplify.], [#figure(figph[The number 156 is displayed in black text against a plain white background.], alt: "The number 156 is displayed in black text against a plain white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The polynomial #math.equation(block: false, alt: "−8 t squared plus 24 t plus 4")[$−8 t^(2) + 24 t + 4$] gives the height, in feet, of a ball #math.equation(block: false, alt: "t")[$t$] seconds after it is tossed into the air, from an initial height of #math.equation(block: false, alt: "4")[$4$] feet. Find the height after #math.equation(block: false, alt: "t equals 3")[$t = 3$] seconds.

#solutionbox[
4 feet
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The polynomial #math.equation(block: false, alt: "−8 t squared plus 24 t plus 4")[$−8 t^(2) + 24 t + 4$] gives the height, in feet, of a ball #math.equation(block: false, alt: "x")[$x$] seconds after it is tossed into the air, from an initial height of #math.equation(block: false, alt: "4")[$4$] feet. Find the height after #math.equation(block: false, alt: "t equals 2")[$t = 2$] seconds.

#solutionbox[
20 feet
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[ACCESS ADDITIONAL ONLINE RESOURCES]

- #link("https://openstaxcollege.org/l/24addpolynomi")[Adding Polynomials]
- #link("https://openstaxcollege.org/l/24subtractpoly")[Subtracting Polynomials]
]


==== Practice Makes Perfect

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

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

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

#solutionbox[
binomial
]

#math.equation(block: true, alt: "z squared minus 5 z minus 6")[$z^(2) − 5 z − 6$]

#math.equation(block: true, alt: "a squared plus 9 a plus 18")[$a^(2) + 9 a + 18$]

#solutionbox[
trinomial
]

#math.equation(block: true, alt: "−12 p to the power 4")[$−12 p^(4)$]

#math.equation(block: true, alt: "y cubed minus 8 y squared plus 2 y minus 16")[$y^(3) − 8 y^(2) + 2 y − 16$]

#solutionbox[
polynomial
]

#math.equation(block: true, alt: "10 minus 9 x")[$10 − 9 x$]

#math.equation(block: true, alt: "23 y squared")[$23 y^(2)$]

#solutionbox[
monomial
]

#math.equation(block: true, 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[Determine the Degree of Polynomials]

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

#math.equation(block: true, alt: "8 a to the power 5 minus 2 a cubed plus 1")[$8 a^(5) − 2 a^(3) + 1$]

#solutionbox[
5
]

#math.equation(block: true, alt: "5 c cubed plus 11 c squared minus c minus 8")[$5 c^(3) + 11 c^(2) − c − 8$]

#math.equation(block: true, alt: "3 x minus 12")[$3 x − 12$]

#solutionbox[
1
]

#math.equation(block: true, alt: "4 y plus 17")[$4 y + 17$]

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

#solutionbox[
0
]

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

#strong[Add and Subtract Monomials]

In the following exercises, add or subtract the monomials.

#math.equation(block: true, alt: "6x squared plus 9 x squared")[$"6x"^(2) + #h(0.2em) 9 x^(2)$]

#solutionbox[
15#emph[x]#super[2]
]

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

#math.equation(block: true, alt: "−12 u plus 4 u")[$−12 u #h(0.2em) + #h(0.2em) 4 u$]

#solutionbox[
−8#emph[u]
]

#math.equation(block: true, alt: "−3 m plus 9 m")[$−3 m #h(0.2em) + #h(0.2em) 9 m$]

#math.equation(block: true, alt: "5 a plus 7 b")[$5 a #h(0.2em) + #h(0.2em) 7 b$]

#solutionbox[
5#emph[a] + 7#emph[b]
]

#math.equation(block: true, alt: "8 y plus 6 z")[$8 y #h(0.2em) + #h(0.2em) 6 z$]

Add: #math.equation(block: false, alt: "4 a , minus 3 b , minus 8 a")[$4 a #h(0.2em) , #h(0.2em) − 3 b , #h(0.2em) − 8 a$]

#solutionbox[
−4#emph[a] −3#emph[b]
]

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

#math.equation(block: true, alt: "18 x minus 2 x")[$18 x − 2 x$]

#solutionbox[
16#emph[x]
]

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

Subtract #math.equation(block: false, alt: "5 x to the power 6 from minus 12 x to the power 6")[$5 x^(6) #h(0.2em) "from" #h(0.2em) − 12 x^(6)$]

#solutionbox[
−17#emph[x]#super[6]
]

Subtract #math.equation(block: false, alt: "2 p to the power 4 from minus 7 p to the power 4")[$2 p^(4) #h(0.2em) "from" #h(0.2em) − 7 p^(4)$]

#strong[Add and Subtract Polynomials]

In the following exercises, add or subtract the polynomials.

#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")[$( 4 y^(2) + 10 y + 3 ) + ( 8 y^(2) − 6 y + 5 )$]

#solutionbox[
12#emph[y]#super[2] + 4#emph[y] + 8
]

#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 x plus 3 close parenthesis")[$( 7 x^(2) − 9 x + 2 ) + ( 6 x^(2) − 4 x + 3 )$]

#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[
−3#emph[x]#super[2] + 17#emph[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 3 a squared plus 7 close parenthesis plus open parenthesis a squared minus 7 a minus 18 close parenthesis")[$( 3 a^(2) + 7 ) + ( a^(2) − 7 a − 18 )$]

#solutionbox[
4#emph[a]#super[2] − 7#emph[a] − 11
]

#math.equation(block: true, alt: "open parenthesis p squared minus 5 p minus 11 close parenthesis plus open parenthesis 3 p squared plus 9 close parenthesis")[$( p^(2) − 5 p − 11 ) + ( 3 p^(2) + 9 )$]

#math.equation(block: true, alt: "open parenthesis 6 m squared minus 9 m minus 3 close parenthesis minus open parenthesis 2 m squared plus m minus 5 close parenthesis")[$( 6 m^(2) − 9 m − 3 ) − ( 2 m^(2) + m − 5 )$]

#solutionbox[
4#emph[m]#super[2] − 10#emph[m] + 2
]

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

#math.equation(block: true, alt: "open parenthesis z squared plus 8 z plus 9 close parenthesis minus open parenthesis z squared minus 3 z plus 1 close parenthesis")[$( z^(2) + 8 z + 9 ) − ( z^(2) − 3 z + 1 )$]

#solutionbox[
11#emph[z] + 8
]

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

#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[
12#emph[s]#super[2] − 16#emph[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")[$( 10 r^(2) − 20 r ) − ( r − 8 )$]

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[
2#emph[p]#super[3] + #emph[p]#super[2] + 9#emph[p] + 10
]

Find the sum of #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")[$( 7 q^(3) − 3 )$]

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

#solutionbox[
#emph[x]#super[2] + 3#emph[x] + 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 difference of #math.equation(block: false, alt: "open parenthesis w squared plus w minus 42 close parenthesis")[$( w^(2) + w − 42 )$] and #math.equation(block: false, alt: "open parenthesis w squared minus 10 w plus 24 close parenthesis")[$( w^(2) − 10 w + 24 )$]

#solutionbox[
11#emph[w] − 66
]

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

#strong[Evaluate a Polynomial for a Given Value]

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

#math.equation(block: true, alt: "Evaluate 8 y squared minus 3 y plus 2")[$"Evaluate" #h(0.2em) 8 y^(2) − 3 y + 2$]

+ ⓐ #math.equation(block: false, alt: "y equals 5")[$#h(0.2em) y = 5$]
+ ⓑ #math.equation(block: false, alt: "y equals −2")[$#h(0.2em) y = −2$]
+ ⓒ #math.equation(block: false, alt: "y equals 0")[$#h(0.2em) y = 0$]

#solutionbox[
+ ⓐ 187
+ ⓑ 40
+ ⓒ 2
]

#math.equation(block: true, alt: "Evaluate 5 y squared minus y minus 7 when:")[$"Evaluate" #h(0.2em) 5 y^(2) − y − 7 #h(0.2em) "when:"$]

+ ⓐ #math.equation(block: false, alt: "y equals −4")[$#h(0.2em) y = −4 #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "y equals 1")[$#h(0.2em) y = 1$]
+ ⓒ #math.equation(block: false, alt: "y equals 0")[$y = 0$]

#math.equation(block: true, alt: "Evaluate 4 minus 36 x when:")[$"Evaluate" #h(0.2em) 4 − 36 x #h(0.2em) "when:"$]

+ ⓐ #math.equation(block: false, alt: "x equals 3")[$#h(0.2em) x = 3$]
+ ⓑ #math.equation(block: false, alt: "x equals 0")[$#h(0.2em) x = 0$]
+ ⓒ #math.equation(block: false, alt: "x equals −1")[$x = −1$]

#solutionbox[
+ ⓐ −104
+ ⓑ 4
+ ⓒ 40
]

#math.equation(block: true, alt: "Evaluate 16 minus 36 x squared when:")[$"Evaluate" #h(0.2em) 16 − 36 x^(2) #h(0.2em) "when:"$]

+ ⓐ #math.equation(block: false, alt: "x equals −1")[$#h(0.2em) x = −1 #h(0.2em)$]
+ ⓑ #math.equation(block: false, alt: "x equals 0")[$#h(0.2em) x = 0$]
+ ⓒ #math.equation(block: false, alt: "x equals 2")[$x = 2$]

A window washer drops a squeegee from a platform #math.equation(block: false, alt: "275")[$275$] feet high. The polynomial #math.equation(block: false, alt: "−16 t squared plus 275")[$−16 t^(2) + 275$] gives the height of the squeegee #math.equation(block: false, alt: "t")[$t$] seconds after it was dropped. Find the height after #math.equation(block: false, alt: "t equals 4")[$t = 4$] seconds.

#solutionbox[
19 feet
]

A manufacturer of microwave ovens has found that the revenue received from selling microwaves at a cost of #emph[p] dollars each is given by the polynomial #math.equation(block: false, alt: "−5 p squared plus 350 p .")[$−5 p^(2) + 350 p .$] Find the revenue received when #math.equation(block: false, alt: "p equals 50")[$p = 50$] dollars.


==== Everyday Math

#strong[Fuel Efficiency] The fuel efficiency (in miles per gallon) of a bus 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 160 x squared plus the fraction 1 over 2 x .")[$− frac(1, 160) #h(0.1em) x^(2) + frac(1, 2) #h(0.1em) x .$] Find the fuel efficiency when #math.equation(block: false, alt: "x equals 40 mph.")[$x = 40 #h(0.2em) "mph."$]

#solutionbox[
10 mpg
]

#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 60 mph.")[$x = 60 #h(0.2em) "mph."$]


==== Writing Exercises

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

#solutionbox[
Answers will vary.
]

Eloise thinks the sum #math.equation(block: false, alt: "5 x squared plus 3 x to the power 4")[$5 x^(2) + 3 x^(4)$] is #math.equation(block: false, alt: "8 x to the power 6 .")[$8 x^(6) .$] What is wrong with her reasoning?


==== Self Check

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

#figure(figph[Math self-assessment grid for polynomial skills: identifying, determining degree, adding/subtracting monomials & polynomials, and evaluating expressions.], alt: "Math self-assessment grid for polynomial skills: identifying, determining degree, adding/subtracting monomials & polynomials, and evaluating expressions.", caption: none)

ⓑ If most of your checks were:

…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.

…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?

…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.