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== 6.1#h(0.6em)Greatest Common Factor and Factor by Grouping

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

Factor 56 into primes. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "2 times 2 times 2 times 7")[$2 · 2 · 2 · 7$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the least common multiple (LCM) of 18 and 24. #linebreak() 
If you missed this problem, review .

#solutionbox[
72
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "−3 a open parenthesis 7 a plus 8 b close parenthesis .")[$−3 a ( 7 a + 8 b ) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "−21 a squared minus 24 a b")[$−21 a^(2) − 24 a b$]
]
]


=== Find the Greatest Common Factor of Two or More Expressions

Earlier we multiplied factors together to get a #strong[product]. Now, we will reverse this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called #strong[factoring].

#figure(figph[8 times 7 is 56. Here 8 and 7 are factors and 56 is the product. An arrow pointing from 8 times 7 to 56 is labeled multiply. An arrow pointing from 56 to 8 times 7 is labeled factor. 2x open parentheses x plus 3 close parentheses equals 2x squared plus 6x. Here the left side of the equation is labeled factors and the right side is labeled products.], alt: "8 times 7 is 56. Here 8 and 7 are factors and 56 is the product. An arrow pointing from 8 times 7 to 56 is labeled multiply. An arrow pointing from 56 to 8 times 7 is labeled factor. 2x open parentheses x plus 3 close parentheses equals 2x squared plus 6x. Here the left side of the equation is labeled factors and the right side is labeled products.", caption: none)

We have learned how to factor numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the #strong[greatest common factor] of two or more expressions. The method we use is similar to what we used to find the LCM.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Greatest Common Factor]

The #strong[greatest common factor] (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.
]

We summarize the steps we use to find the greatest common factor.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Find the greatest common factor (GCF) of two expressions.]

+ Factor each coefficient into primes. Write all variables with exponents in expanded form.
+ List all factors—matching common factors in a column. In each column, circle the common factors.
+ Bring down the common factors that all expressions share.
+ Multiply the factors.
]

The next example will show us the steps to find the greatest common factor of three expressions.

#examplebox("Example 1")[][
Find the greatest common factor of #math.equation(block: false, alt: "21 x cubed , 9 x squared , 15 x .")[$21 x^(3) , 9 x^(2) , 15 x .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Factor each coefficient into primes and write the #linebreak() variables with exponents in expanded form. #linebreak() Circle the common factors in each column. #linebreak() Bring down the common factors.], [#figure(figph[A mathematical example demonstrating how to find the Greatest Common Factor (GCF) of 21x³, 9x², and 15x by breaking down each expression into its prime factors and identifying common terms.], alt: "A mathematical example demonstrating how to find the Greatest Common Factor (GCF) of 21x³, 9x², and 15x by breaking down each expression into its prime factors and identifying common terms.", caption: none)]),
  [Multiply the factors.], [#figure(figph[The image displays the text 'GCF = 3x' in a bold, gray font against a plain white background, indicating a mathematical equation or a definition of the Greatest Common Factor.], alt: "The image displays the text 'GCF = 3x' in a bold, gray font against a plain white background, indicating a mathematical equation or a definition of the Greatest Common Factor.", caption: none)],
  [], [The GCF of #math.equation(block: false, alt: "21 x cubed")[$21 x^(3)$], #math.equation(block: false, alt: "9 x squared")[$9 x^(2)$] and #math.equation(block: false, alt: "15 x")[$15 x$] is #math.equation(block: false, alt: "3 x")[$3 x$].],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the greatest common factor: #math.equation(block: false, alt: "25 m to the power 4 , 35 m cubed , 20 m squared .")[$25 m^(4) , 35 m^(3) , 20 m^(2) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the greatest common factor: #math.equation(block: false, alt: "14 x cubed , 70 x squared , 105 x .")[$14 x^(3) , 70 x^(2) , 105 x .$]

#solutionbox[
#math.equation(block: true, alt: "7 x")[$7 x$]
]
]


=== Factor the Greatest Common Factor from a Polynomial

It is sometimes useful to represent a number as a product of factors, for example, 12 as #math.equation(block: false, alt: "2 times 6")[$2 · 6$] or #math.equation(block: false, alt: "3 times 4 .")[$3 · 4 .$] In algebra, it can also be useful to represent a polynomial in factored form. We will start with a product, such as #math.equation(block: false, alt: "3 x squared plus 15 x ,")[$3 x^(2) + 15 x ,$] and end with its factors, #math.equation(block: false, alt: "3 x open parenthesis x plus 5 close parenthesis .")[$3 x ( x + 5 ) .$] To do this we apply the Distributive Property “in reverse.”

We state the Distributive Property here just as you saw it in earlier chapters and “in reverse.”

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

If #emph[a], #emph[b], and #emph[c] are real numbers, then

#math.equation(block: true, alt: "a open parenthesis b plus c close parenthesis equals a b plus a c and a b plus a c equals a open parenthesis b plus c close parenthesis")[$a ( b + c ) = a b + a c #h(1em) "and" #h(1em) a b + a c = a ( b + c )$]The form on the left is used to multiply. The form on the right is used to factor.
]

So how do you use the Distributive Property to factor a #strong[polynomial]? You just find the GCF of all the terms and write the polynomial as a product!

#examplebox("Example 2")[How to Use the Distributive Property to factor a polynomial][
Factor: #math.equation(block: false, alt: "8 m cubed minus 12 m squared n plus 20 m n squared .")[$8 m^(3) − 12 m^(2) n + 20 m n^(2) .$]

#solutionbox[
#figure(figph[Step 1 is find the GCF of all the terms in the polynomial. GCF of 8 m cubed, 12 m squared n and 20 mn squared is 4m.], alt: "Step 1 is find the GCF of all the terms in the polynomial. GCF of 8 m cubed, 12 m squared n and 20 mn squared is 4m.", caption: none)

#figure(figph[Step 1 is find the GCF of all the terms in the polynomial. GCF of 8 m cubed, 12 m squared n and 20 mn squared is 4m.], alt: "Step 1 is find the GCF of all the terms in the polynomial. GCF of 8 m cubed, 12 m squared n and 20 mn squared is 4m.", caption: none)

#figure(figph[In step 3, use the reverse Distributive Property to factor the expression as 4m open parentheses 2 m squared minus 3 mn plus 5 n squared close parentheses.], alt: "In step 3, use the reverse Distributive Property to factor the expression as 4m open parentheses 2 m squared minus 3 mn plus 5 n squared close parentheses.", caption: none)

#figure(figph[Step 4 is to check by multiplying the factors. By multiplying the factors, we get the original polynomial.], alt: "Step 4 is to check by multiplying the factors. By multiplying the factors, we get the original polynomial.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "9 x y squared plus 6 x squared y squared plus 21 y cubed .")[$9 x y^(2) + 6 x^(2) y^(2) + 21 y^(3) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "3 p cubed minus 6 p squared q plus 9 p q cubed .")[$3 p^(3) − 6 p^(2) q + 9 p q^(3) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Factor the greatest common factor from a polynomial.]

+ Find the GCF of all the terms of the polynomial.
+ Rewrite each term as a product using the GCF.
+ Use the “reverse” Distributive Property to factor the expression.
+ Check by multiplying the factors.
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Factor as a Noun and a Verb]

We use “factor” as both a noun and a verb:

#math.equation(block: true, alt: "Noun:, 7 is a factor of 14; Verb:, factor 3 from 3 a plus 3")[$"Noun:" &  &  & #h(8em) "7 is a" #h(0.2em) "factor" #h(0.2em) "of 14" \ "Verb:" &  &  & #h(8em) "factor" #h(0.2em) "3 from" #h(0.2em) 3 a + 3$]
]

#examplebox("Example 3")[][
Factor: #math.equation(block: false, alt: "5 x cubed minus 25 x squared .")[$5 x^(3) − 25 x^(2) .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [Find the GCF of #math.equation(block: false, alt: "5 x cubed")[$5 x^(3)$] and #math.equation(block: false, alt: "25 x squared .")[$25 x^(2) .$]], [#figure(figph[This image demonstrates how to find the Greatest Common Factor (GCF) of 5x³ and 25x². It shows the prime factorization of each expression and then identifies the common factors, which are multiplied together to get the GCF.], alt: "This image demonstrates how to find the Greatest Common Factor (GCF) of 5x³ and 25x². It shows the prime factorization of each expression and then identifies the common factors, which are multiplied together to get the GCF.", caption: none)], [],
  [], [], [#figure(figph[The image displays 'GCF = 5x^2' in a gray font on a white background, representing a mathematical equation for the greatest common factor.], alt: "The image displays 'GCF = 5x^2' in a gray font on a white background, representing a mathematical equation for the greatest common factor.", caption: none)],
  [], [], [#figure(figph[The image shows the mathematical expression 5x^3 - 25x^2, which represents a polynomial with two terms, indicating a subtraction operation between 5 times x cubed and 25 times x squared.], alt: "The image shows the mathematical expression 5x^3 - 25x^2, which represents a polynomial with two terms, indicating a subtraction operation between 5 times x cubed and 25 times x squared.", caption: none)],
  [Rewrite each term.], [], [#figure(figph[A mathematical expression 5x^2 \* x - 5x^2 \* 5, which can be factored as 5x^2(x - 5). The numbers and variables are colored in red and black against a white background.], alt: "A mathematical expression 5x^2 * x - 5x^2 * 5, which can be factored as 5x^2(x - 5). The numbers and variables are colored in red and black against a white background.", caption: none)],
  [Factor the GCF.], [], [#figure(figph[A mathematical expression shows 5x squared multiplied by the quantity x minus 5, written as 5x²(x-5).], alt: "A mathematical expression shows 5x squared multiplied by the quantity x minus 5, written as 5x²(x-5).", caption: none)],
  [Check: #linebreak()  #linebreak() #math.equation(block: false, alt: "5 x squared open parenthesis x minus 5 close parenthesis; 5 x squared times x minus 5 x squared times 5; 5 x cubed minus 25 x squared ✓")[$#h(4em) 5 x^(2) ( x − 5 ) \ 5 x^(2) · x − 5 x^(2) · 5 \ 5 x^(3) − 25 x^(2) ✓$]], [], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "2 x cubed plus 12 x squared .")[$2 x^(3) + 12 x^(2) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "6 y cubed minus 15 y squared .")[$6 y^(3) − 15 y^(2) .$]

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

#examplebox("Example 4")[][
Factor: #math.equation(block: false, alt: "8 x cubed y minus 10 x squared y squared plus 12 x y cubed .")[$8 x^(3) y − 10 x^(2) y^(2) + 12 x y^(3) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([The GCF of #math.equation(block: false, alt: "8 x cubed y , −10 x squared y squared , and 12 x y cubed")[$8 x^(3) y , −10 x^(2) y^(2) , #h(0.2em) "and" #h(0.2em) 12 x y^(3)$] #linebreak() is #math.equation(block: false, alt: "2 x y .")[$2 x y .$]], [#figure(figph[This image demonstrates finding the GCF of 8x³y, 10x²y², and 12xy³ by listing their prime factors. Common factors (2, x, y) are circled vertically, and their product gives the GCF: 2xy.], alt: "This image demonstrates finding the GCF of 8x³y, 10x²y², and 12xy³ by listing their prime factors. Common factors (2, x, y) are circled vertically, and their product gives the GCF: 2xy.", caption: none)]),
  [], [#figure(figph[The image displays a mathematical expression 'GCF = 2xy' in a clear, dark gray font against a plain white background.], alt: "The image displays a mathematical expression 'GCF = 2xy' in a clear, dark gray font against a plain white background.", caption: none)],
  [], [#figure(figph[A mathematical expression featuring three terms: 8x^3y - 10x^2y^2 + 12xy^3.], alt: "A mathematical expression featuring three terms: 8x^3y - 10x^2y^2 + 12xy^3.", caption: none)],
  [Rewrite each term using the GCF, #math.equation(block: false, alt: "2 x y .")[$2 x y .$]], [#figure(figph[The image displays the mathematical expression 2xy \* 4x^2 - 2xy \* 5xy + 2xy \* 6y^2, with the term '2xy' highlighted in red in each part, suggesting it's a common factor to be extracted.], alt: "The image displays the mathematical expression 2xy * 4x^2 - 2xy * 5xy + 2xy * 6y^2, with the term '2xy' highlighted in red in each part, suggesting it's a common factor to be extracted.", caption: none)],
  [Factor the GCF.], [#figure(figph[A mathematical expression showing the term 2xy multiplied by the trinomial (4x^2 - 5xy + 6y^2).], alt: "A mathematical expression showing the term 2xy multiplied by the trinomial (4x^2 - 5xy + 6y^2).", caption: none)],
  [Check: #linebreak()  #linebreak() #math.equation(block: false, alt: "2 x y open parenthesis 4 x squared minus 5 x y plus 6 y squared close parenthesis; 2 x y times 4 x squared minus 2 x y times 5 x y plus 2 x y times 6 y squared; 8 x cubed y minus 10 x squared y squared plus 12 x y cubed ✓")[$#h(4em) 2 x y ( 4 x^(2) − 5 x y + 6 y^(2) ) \ 2 x y · 4 x^(2) − 2 x y · 5 x y + 2 x y · 6 y^(2) \ 8 x^(3) y − 10 x^(2) y^(2) + 12 x y^(3) ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "15 x cubed y minus 3 x squared y squared plus 6 x y cubed .")[$15 x^(3) y − 3 x^(2) y^(2) + 6 x y^(3) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "8 a cubed b plus 2 a squared b squared minus 6 a b cubed .")[$8 a^(3) b + 2 a^(2) b^(2) − 6 a b^(3) .$]

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

When the leading coefficient is negative, we factor the negative out as part of the GCF.

#examplebox("Example 5")[][
Factor: #math.equation(block: false, alt: "−4 a cubed plus 36 a squared minus 8 a .")[$−4 a^(3) + 36 a^(2) − 8 a .$]

#solutionbox[
The leading coefficient is negative, so the GCF will be negative.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression displays -4a cubed + 36a squared - 8a.], alt: "A mathematical expression displays -4a cubed + 36a squared - 8a.", caption: none)]),
  [Rewrite each term using the GCF, #math.equation(block: false, alt: "−4 a .")[$−4 a .$]], [#figure(figph[The algebraic expression -4a \* a^2 - (-4a) \* 9a + (-4a) \* 2, illustrating a common factor of -4a in a polynomial.], alt: "The algebraic expression -4a * a^2 - (-4a) * 9a + (-4a) * 2, illustrating a common factor of -4a in a polynomial.", caption: none)],
  [Factor the GCF.], [#figure(figph[The image shows the algebraic expression -4a(a^2 - 9a + 2), which is a monomial multiplied by a trinomial.], alt: "The image shows the algebraic expression -4a(a^2 - 9a + 2), which is a monomial multiplied by a trinomial.", caption: none)],
  [Check: #linebreak()  #linebreak() #math.equation(block: false, alt: "−4 a open parenthesis a squared minus 9 a plus 2 close parenthesis; −4 a times a squared minus open parenthesis −4 a close parenthesis times 9 a plus open parenthesis −4 a close parenthesis times 2; −4 a cubed plus 36 a squared minus 8 a ✓")[$#h(4em) −4 a ( a^(2) − 9 a + 2 ) \ −4 a · a^(2) − ( −4 a ) · 9 a + ( −4 a ) · 2 \ −4 a^(3) + 36 a^(2) − 8 a ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "−4 b cubed plus 16 b squared minus 8 b .")[$−4 b^(3) + 16 b^(2) − 8 b .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "−7 a cubed plus 21 a squared minus 14 a .")[$−7 a^(3) + 21 a^(2) − 14 a .$]

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

So far our greatest common factors have been monomials. In the next example, the greatest common factor is a binomial.

#examplebox("Example 6")[][
Factor: #math.equation(block: false, alt: "3 y open parenthesis y plus 7 close parenthesis minus 4 open parenthesis y plus 7 close parenthesis .")[$3 y ( y + 7 ) − 4 ( y + 7 ) .$]

#solutionbox[
The GCF is the binomial #math.equation(block: false, alt: "y plus 7 .")[$y + 7 .$]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The mathematical expression is 3y(y + 7) - 4(y + 7). It shows a common factor (y+7) being multiplied by 3y and then subtracted from 4 times (y+7).], alt: "The mathematical expression is 3y(y + 7) - 4(y + 7). It shows a common factor (y+7) being multiplied by 3y and then subtracted from 4 times (y+7).", caption: none)]),
  [Factor the GCF, #math.equation(block: false, alt: "open parenthesis y plus 7 close parenthesis .")[$( y + 7 ) .$]], [#figure(figph[An algebraic expression showing the product of two binomials: (y + 7)(3y - 4). The expression is displayed in black text against a white background.], alt: "An algebraic expression showing the product of two binomials: (y + 7)(3y - 4). The expression is displayed in black text against a white background.", caption: none)],
  [Check on your own by multiplying.], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "4 m open parenthesis m plus 3 close parenthesis minus 7 open parenthesis m plus 3 close parenthesis .")[$4 m ( m + 3 ) − 7 ( m + 3 ) .$]

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

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

#solutionbox[
#math.equation(block: true, alt: "open parenthesis n minus 4 close parenthesis open parenthesis 8 n plus 5 close parenthesis")[$( n − 4 ) ( 8 n + 5 )$]
]
]


=== Factor by Grouping

Sometimes there is no common factor of all the terms of a polynomial. When there are four terms we separate the polynomial into two parts with two terms in each part. Then look for the #strong[GCF] in each part. If the polynomial can be factored, you will find a common factor emerges from both parts. Not all polynomials can be factored. Just like some numbers are #strong[prime], some polynomials are prime.

#examplebox("Example 7")[How to Factor a Polynomial by Grouping][
Factor by grouping: #math.equation(block: false, alt: "x y plus 3 y plus 2 x plus 6 .")[$x y + 3 y + 2 x + 6 .$]

#solutionbox[
#figure(figph[Step 1 is to group the terms with common factors. There is no greatest common factor in all the four terms of xy plus 3y plus 2x plus 6. So, separate the first two terms from the second two.], alt: "Step 1 is to group the terms with common factors. There is no greatest common factor in all the four terms of xy plus 3y plus 2x plus 6. So, separate the first two terms from the second two.", caption: none)

#figure(figph[Step 2 is to factor out the common factor in each group. By factoring the GCF from the first 2 terms, we get y open parentheses x plus 3 close parentheses plus 2x plus 6. Factoring the GCF from the second 2 terms, we get y open parentheses x plus 3 close parentheses plus 2 open parentheses x plus 3 close parentheses.], alt: "Step 2 is to factor out the common factor in each group. By factoring the GCF from the first 2 terms, we get y open parentheses x plus 3 close parentheses plus 2x plus 6. Factoring the GCF from the second 2 terms, we get y open parentheses x plus 3 close parentheses plus 2 open parentheses x plus 3 close parentheses.", caption: none)

#figure(figph[Step 3 is to factor the common factor from the expression. Notice that each term has a common factor of x plus 3. By factoring this out, we get open parentheses x plus 3 close parentheses open parentheses y plus 2 close parentheses], alt: "Step 3 is to factor the common factor from the expression. Notice that each term has a common factor of x plus 3. By factoring this out, we get open parentheses x plus 3 close parentheses open parentheses y plus 2 close parentheses", caption: none)

#figure(figph[Step 4 is to check by multiplying the expressions to get the result xy plus 3y plus 2x plus 6.], alt: "Step 4 is to check by multiplying the expressions to get the result xy plus 3y plus 2x plus 6.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor by grouping: #math.equation(block: false, alt: "x y plus 8 y plus 3 x plus 24 .")[$x y + 8 y + 3 x + 24 .$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis x plus 8 close parenthesis open parenthesis y plus 3 close parenthesis")[$( x + 8 ) ( y + 3 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor by grouping: #math.equation(block: false, alt: "a b plus 7 b plus 8 a plus 56 .")[$a b + 7 b + 8 a + 56 .$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis a plus 7 close parenthesis open parenthesis b plus 8 close parenthesis")[$( a + 7 ) ( b + 8 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Factor by grouping.]

+ Group terms with common factors.
+ Factor out the common factor in each group.
+ Factor the common factor from the expression.
+ Check by multiplying the factors.
]

#examplebox("Example 8")[][
Factor by grouping: ⓐ #math.equation(block: false, alt: "x squared plus 3 x minus 2 x minus 6")[$x^(2) + 3 x − 2 x − 6$] ⓑ #math.equation(block: false, alt: "6 x squared minus 3 x minus 4 x plus 2 .")[$6 x^(2) − 3 x − 4 x + 2 .$]

#solutionbox[
ⓐ

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([There is no GCF in all four terms.], [#math.equation(block: false, alt: "x squared plus 3 x minus 2 x minus 6")[$x^(2) + 3 x − 2 x − 6$]]),
  [Separate into two parts.], [#math.equation(block: false, alt: "x squared plus 3 x −2 x minus 6")[$x^(2) + 3 x #h(1.5em) −2 x − 6$]],
  [Factor the GCF from both parts. Be careful with the signs when factoring the GCF from the last two terms.], [#math.equation(block: false, alt: "x open parenthesis x plus 3 close parenthesis minus 2 open parenthesis x plus 3 close parenthesis")[$x ( x + 3 ) − 2 ( x + 3 )$]],
  [Factor out the common factor.], [#math.equation(block: false, alt: "open parenthesis x plus 3 close parenthesis open parenthesis x minus 2 close parenthesis")[$( x + 3 ) ( x − 2 )$]],
  [Check on your own by multiplying.], [],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([There is no GCF in all four terms.], [#math.equation(block: false, alt: "6 x squared minus 3 x minus 4 x plus 2")[$6 x^(2) − 3 x − 4 x + 2$]]),
  [Separate into two parts.], [#math.equation(block: false, alt: "6 x squared minus 3 x −4 x plus 2")[$6 x^(2) − 3 x #h(1.5em) −4 x + 2$]],
  [Factor the GCF from both parts.], [#math.equation(block: false, alt: "3 x open parenthesis 2 x minus 1 close parenthesis minus 2 open parenthesis 2 x minus 1 close parenthesis")[$3 x ( 2 x − 1 ) − 2 ( 2 x − 1 )$]],
  [Factor out the common factor.], [#math.equation(block: false, alt: "open parenthesis 2 x minus 1 close parenthesis open parenthesis 3 x minus 2 close parenthesis")[$( 2 x − 1 ) ( 3 x − 2 )$]],
  [Check on your own by multiplying.], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor by grouping: ⓐ #math.equation(block: false, alt: "x squared plus 2 x minus 5 x minus 10")[$x^(2) + 2 x − 5 x − 10$] ⓑ #math.equation(block: false, alt: "20 x squared minus 16 x minus 15 x plus 12 .")[$20 x^(2) − 16 x − 15 x + 12 .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "open parenthesis x minus 5 close parenthesis open parenthesis x plus 2 close parenthesis")[$( x − 5 ) ( x + 2 )$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis 5 x minus 4 close parenthesis open parenthesis 4 x minus 3 close parenthesis")[$( 5 x − 4 ) ( 4 x − 3 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor by grouping: ⓐ #math.equation(block: false, alt: "y squared plus 4 y minus 7 y minus 28")[$y^(2) + 4 y − 7 y − 28$] ⓑ #math.equation(block: false, alt: "42 m squared minus 18 m minus 35 m plus 15 .")[$42 m^(2) − 18 m − 35 m + 15 .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "open parenthesis y plus 4 close parenthesis open parenthesis y minus 7 close parenthesis")[$( y + 4 ) ( y − 7 )$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis 7 m minus 3 close parenthesis open parenthesis 6 m minus 5 close parenthesis")[$( 7 m − 3 ) ( 6 m − 5 )$]
]
]


=== Key Concepts

- #strong[How to find the greatest common factor (GCF) of two expressions.]
+ Factor each coefficient into primes. Write all variables with exponents in expanded form.
+ List all factors—matching common factors in a column. In each column, circle the common factors.
+ Bring down the common factors that all expressions share.
+ Multiply the factors.
- #strong[Distributive Property:] If #emph[a], #emph[b], and #emph[c] are real numbers, then #linebreak() 
#math.equation(block: true, alt: "a open parenthesis b plus c close parenthesis equals a b plus a c and a b plus a c equals a open parenthesis b plus c close parenthesis")[$a ( b + c ) = a b + a c #h(1em) "and" #h(1em) a b + a c = a ( b + c )$] #linebreak() 
The form on the left is used to multiply. The form on the right is used to factor.
- #strong[How to factor the greatest common factor from a polynomial.]
+ Find the GCF of all the terms of the polynomial.
+ Rewrite each term as a product using the GCF.
+ Use the “reverse” Distributive Property to factor the expression.
+ Check by multiplying the factors.
- #strong[Factor as a Noun and a Verb:] We use “factor” as both a noun and a verb. #linebreak() 
#math.equation(block: true, alt: "Noun:, 7 is a factor of 14; Verb:, factor 3 from 3 a plus 3")[$"Noun:" &  &  & "7 is a" #h(0.2em) "factor" #h(0.2em) "of 14" \ "Verb:" &  &  & "factor" #h(0.2em) "3 from" #h(0.2em) 3 a + 3$]
- #strong[How to factor by grouping.]
+ Group terms with common factors.
+ Factor out the common factor in each group.
+ Factor the common factor from the expression.
+ Check by multiplying the factors.


==== Practice Makes Perfect

#strong[Find the Greatest Common Factor of Two or More Expressions]

In the following exercises, find the greatest common factor.

#math.equation(block: true, alt: "10 p cubed q , 12 p q squared")[$10 p^(3) q , 12 p q^(2)$]

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

#math.equation(block: true, alt: "8 a squared b cubed , 10 a b squared")[$8 a^(2) b^(3) , 10 a b^(2)$]

#math.equation(block: true, alt: "12 m squared n cubed , 30 m to the power 5 n cubed")[$12 m^(2) n^(3) , 30 m^(5) n^(3)$]

#solutionbox[
#math.equation(block: true, alt: "6 m squared n cubed")[$6 m^(2) n^(3)$]
]

#math.equation(block: true, alt: "28 x squared y to the power 4 , 42 x to the power 4 y to the power 4")[$28 x^(2) y^(4) , 42 x^(4) y^(4)$]

#math.equation(block: true, alt: "10 a cubed , 12 a squared , 14 a")[$10 a^(3) , 12 a^(2) , 14 a$]

#solutionbox[
#math.equation(block: true, alt: "2 a")[$2 a$]
]

#math.equation(block: true, alt: "20 y cubed , 28 y squared , 40 y")[$20 y^(3) , 28 y^(2) , 40 y$]

#math.equation(block: true, alt: "35 x cubed y squared , 10 x to the power 4 y , 5 x to the power 5 y cubed")[$35 x^(3) y^(2) , 10 x^(4) y , 5 x^(5) y^(3)$]

#solutionbox[
#math.equation(block: true, alt: "5 x cubed y")[$5 x^(3) y$]
]

#math.equation(block: true, alt: "27 p squared q cubed , 45 p cubed q to the power 4 , 9 p to the power 4 q cubed")[$27 p^(2) q^(3) , 45 p^(3) q^(4) , 9 p^(4) q^(3)$]

#strong[Factor the Greatest Common Factor from a Polynomial]

In the following exercises, factor the greatest common factor from each polynomial.

#math.equation(block: true, alt: "6 m plus 9")[$6 m + 9$]

#solutionbox[
#math.equation(block: true, alt: "3 open parenthesis 2 m plus 3 close parenthesis")[$3 ( 2 m + 3 )$]
]

#math.equation(block: true, alt: "14 p plus 35")[$14 p + 35$]

#math.equation(block: true, alt: "9 n minus 63")[$9 n − 63$]

#solutionbox[
#math.equation(block: true, alt: "9 open parenthesis n minus 7 close parenthesis")[$9 ( n − 7 )$]
]

#math.equation(block: true, alt: "45 b minus 18")[$45 b − 18$]

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

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

#math.equation(block: true, alt: "4 y squared plus 8 y minus 4")[$4 y^(2) + 8 y − 4$]

#math.equation(block: true, alt: "8 p squared plus 4 p plus 2")[$8 p^(2) + 4 p + 2$]

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

#math.equation(block: true, alt: "10 q squared plus 14 q plus 20")[$10 q^(2) + 14 q + 20$]

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

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

#math.equation(block: true, alt: "12 x cubed minus 10 x")[$12 x^(3) − 10 x$]

#math.equation(block: true, alt: "5 x cubed minus 15 x squared plus 20 x")[$5 x^(3) − 15 x^(2) + 20 x$]

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

#math.equation(block: true, alt: "8 m squared minus 40 m plus 16")[$8 m^(2) − 40 m + 16$]

#math.equation(block: true, alt: "24 x cubed minus 12 x squared plus 15 x")[$24 x^(3) − 12 x^(2) + 15 x$]

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

#math.equation(block: true, alt: "24 y cubed minus 18 y squared minus 30 y")[$24 y^(3) − 18 y^(2) − 30 y$]

#math.equation(block: true, alt: "12 x y squared plus 18 x squared y squared minus 30 y cubed")[$12 x y^(2) + 18 x^(2) y^(2) − 30 y^(3)$]

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

#math.equation(block: true, alt: "21 p q squared plus 35 p squared q squared minus 28 q cubed")[$21 p q^(2) + 35 p^(2) q^(2) − 28 q^(3)$]

#math.equation(block: true, alt: "20 x cubed y minus 4 x squared y squared plus 12 x y cubed")[$20 x^(3) y − 4 x^(2) y^(2) + 12 x y^(3)$]

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

#math.equation(block: true, alt: "24 a cubed b plus 6 a squared b squared minus 18 a b cubed")[$24 a^(3) b + 6 a^(2) b^(2) − 18 a b^(3)$]

#math.equation(block: true, alt: "−2 x minus 4")[$−2 x − 4$]

#solutionbox[
#math.equation(block: true, alt: "−2 open parenthesis x plus 2 close parenthesis")[$−2 ( x + 2 )$]
]

#math.equation(block: true, alt: "−3 b plus 12")[$−3 b + 12$]

#math.equation(block: true, alt: "−2 x cubed plus 18 x squared minus 8 x")[$−2 x^(3) + 18 x^(2) − 8 x$]

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

#math.equation(block: true, alt: "−5 y cubed plus 35 y squared minus 15 y")[$−5 y^(3) + 35 y^(2) − 15 y$]

#math.equation(block: true, alt: "−4 p cubed q minus 12 p squared q squared plus 16 p q squared")[$−4 p^(3) q − 12 p^(2) q^(2) + 16 p q^(2)$]

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

#math.equation(block: true, alt: "−6 a cubed b minus 12 a squared b squared plus 18 a b squared")[$−6 a^(3) b − 12 a^(2) b^(2) + 18 a b^(2)$]

#math.equation(block: true, alt: "5 x open parenthesis x plus 1 close parenthesis plus 3 open parenthesis x plus 1 close parenthesis")[$5 x ( x + 1 ) + 3 ( x + 1 )$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis x plus 1 close parenthesis open parenthesis 5 x plus 3 close parenthesis")[$( x + 1 ) ( 5 x + 3 )$]
]

#math.equation(block: true, alt: "2 x open parenthesis x minus 1 close parenthesis plus 9 open parenthesis x minus 1 close parenthesis")[$2 x ( x − 1 ) + #h(0.2em) 9 ( x − 1 )$]

#math.equation(block: true, alt: "3 b open parenthesis b minus 2 close parenthesis minus 13 open parenthesis b minus 2 close parenthesis")[$3 b ( b − 2 ) − 13 ( b − 2 )$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis b minus 2 close parenthesis open parenthesis 3 b minus 13 close parenthesis")[$( b − 2 ) ( 3 b − 13 )$]
]

#math.equation(block: true, alt: "6 m open parenthesis m minus 5 close parenthesis minus 7 open parenthesis m minus 5 close parenthesis")[$6 m ( m − 5 ) − 7 ( m − 5 )$]

#strong[Factor by Grouping]

In the following exercises, factor by grouping.

#math.equation(block: true, alt: "a b plus 5 a plus 3 b plus 15")[$a b + 5 a + 3 b + 15$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis b plus 5 close parenthesis open parenthesis a plus 3 close parenthesis")[$( b + 5 ) ( a + 3 )$]
]

#math.equation(block: true, alt: "c d plus 6 c plus 4 d plus 24")[$c d + 6 c + 4 d + 24$]

#math.equation(block: true, alt: "8 y squared plus y plus 40 y plus 5")[$8 y^(2) + y + 40 y + 5$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis y plus 5 close parenthesis open parenthesis 8 y plus 1 close parenthesis")[$( y + 5 ) ( 8 y + 1 )$]
]

#math.equation(block: true, alt: "6 y squared plus 7 y plus 24 y plus 28")[$6 y^(2) + 7 y + 24 y + 28$]

#math.equation(block: true, alt: "u v minus 9 u plus 2 v minus 18")[$u v − 9 u + 2 v − 18$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis u plus 2 close parenthesis open parenthesis v minus 9 close parenthesis")[$( u + 2 ) ( v − 9 )$]
]

#math.equation(block: true, alt: "p q minus 10 p plus 8 q minus 80")[$p q − 10 p + 8 q − 80$]

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

#solutionbox[
#math.equation(block: true, alt: "open parenthesis u minus 1 close parenthesis open parenthesis u plus 6 close parenthesis")[$( u − 1 ) ( u + 6 )$]
]

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

#math.equation(block: true, alt: "9 p squared plus 12 p minus 15 p minus 20")[$9 p^(2) + 12 p − 15 p − 20$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 3 p minus 5 close parenthesis open parenthesis 3 p plus 4 close parenthesis")[$( 3 p − 5 ) ( 3 p + 4 )$]
]

#math.equation(block: true, alt: "16 q squared plus 20 q minus 28 q minus 35")[$16 q^(2) + 20 q − 28 q − 35$]

#math.equation(block: true, alt: "m n minus 6 m minus 4 n plus 24")[$m n − 6 m − 4 n + 24$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis n minus 6 close parenthesis open parenthesis m minus 4 close parenthesis")[$( n − 6 ) ( m − 4 )$]
]

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

#math.equation(block: true, alt: "2 x squared minus 14 x minus 5 x plus 35")[$2 x^(2) − 14 x − 5 x + 35$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis x minus 7 close parenthesis open parenthesis 2 x minus 5 close parenthesis")[$( x − 7 ) ( 2 x − 5 )$]
]

#math.equation(block: true, alt: "4 x squared minus 36 x minus 3 x plus 27")[$4 x^(2) − 36 x − 3 x + 27$]

#strong[Mixed Practice]

In the following exercises, factor.

#math.equation(block: true, alt: "−18 x y squared minus 27 x squared y")[$−18 x y^(2) − 27 x^(2) y$]

#solutionbox[
#math.equation(block: true, alt: "−9 x y open parenthesis 2 y plus 3 x close parenthesis")[$−9 x y ( 2 y + 3 x )$]
]

#math.equation(block: true, alt: "−4 x cubed y to the power 5 minus x squared y cubed plus 12 x y to the power 4")[$−4 x^(3) y^(5) − x^(2) y^(3) + 12 x y^(4)$]

#math.equation(block: true, alt: "3 x cubed minus 7 x squared plus 6 x minus 14")[$3 x^(3) − 7 x^(2) + 6 x − 14$]

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

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

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

#solutionbox[
#math.equation(block: true, alt: "open parenthesis x plus y close parenthesis open parenthesis x plus 5 close parenthesis")[$( x + y ) ( x + 5 )$]
]

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


==== Writing Exercises

What does it mean to say a polynomial is in factored form?

#solutionbox[
Answers will vary.
]

How do you check result after factoring a polynomial?

The greatest common factor of 36 and 60 is 12. Explain what this means.

#solutionbox[
Answers will vary.
]

What is the GCF of #math.equation(block: false, alt: "y to the power 4 , y to the power 5 ,")[$y^(4) , y^(5) ,$] and #math.equation(block: false, alt: "y to the power 10 ?")[$y^(10) ?$] Write a general rule that tells you how to find the GCF of #math.equation(block: false, alt: "y to the power a , y to the power b ,")[$y^(a) , y^(b) ,$] and #math.equation(block: false, alt: "y to the power c .")[$y^(c) .$]


==== Self Check

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

#figure(figph[This table has 4 columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no I don’t get it. The first column has the following statements: find the greatest common factor of 2 or more expressions, factor the greatest common factor from a polynomial, factor by grouping. The remaining columns are blank.], alt: "This table has 4 columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no I don’t get it. The first column has the following statements: find the greatest common factor of 2 or more expressions, factor the greatest common factor from a polynomial, factor by grouping. The remaining columns are blank.", caption: none)

ⓑ If most of your checks were:

#strong[…confidently.] Congratulations! You have achieved your goals 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 as topics you do not master become potholes in your road to success. Math is sequential - 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 critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.