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== 7.1#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 of 18 and 24. #linebreak() 
If you missed this problem, review .

#solutionbox[
72
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify #math.equation(block: false, alt: "−3 open parenthesis 6 a plus 11 close parenthesis")[$−3 ( 6 a + 11 )$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "minus 18 a minus 33")[$− 18 a − 33$]
]
]


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

Earlier we multiplied factors together to get a product. Now, we will be reversing 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[This figure has two factors being multiplied. They are 8 and 7. Beside this equation there are other factors multiplied. They are 2x and (x+3). The product is given as 2x^2 plus 6x. Above the figure is an arrow towards the right with multiply inside. Below the figure is an arrow to the left with factor inside.], alt: "This figure has two factors being multiplied. They are 8 and 7. Beside this equation there are other factors multiplied. They are 2x and (x+3). The product is given as 2x^2 plus 6x. Above the figure is an arrow towards the right with multiply inside. Below the figure is an arrow to the left with factor inside.", 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.
]

First we’ll find the GCF of two numbers.

#examplebox("Example 1")[How to Find the Greatest Common Factor of Two or More Expressions][
Find the GCF of 54 and 36.

#solutionbox[
#figure(figph[This table has three columns. In the first column are the steps for factoring. The first row has the first step, factor each coefficient into primes and write all variables with exponents in expanded form. The second column in the first row has “factor 54 and 36”. The third column in the first row has 54 and 36 factored with factor trees. The prime factors of 54 are circled and are 3, 3, 2, and3. The prime factors of 36 are circled and are 2,3,2,3.], alt: "This table has three columns. In the first column are the steps for factoring. The first row has the first step, factor each coefficient into primes and write all variables with exponents in expanded form. The second column in the first row has “factor 54 and 36”. The third column in the first row has 54 and 36 factored with factor trees. The prime factors of 54 are circled and are 3, 3, 2, and3. The prime factors of 36 are circled and are 2,3,2,3.", caption: none)

#figure(figph[The second row has the second step of “in each column, circle the common factors. The second column in the second row has the statement “circle the 2, 3 and 3 that are shared by both numbers”. The third column in the second row has the prime factors of 36 and 54 in rows above each other. The common factors of 2, 3, and 3 are circled.], alt: "The second row has the second step of “in each column, circle the common factors. The second column in the second row has the statement “circle the 2, 3 and 3 that are shared by both numbers”. The third column in the second row has the prime factors of 36 and 54 in rows above each other. The common factors of 2, 3, and 3 are circled.", caption: none)

#figure(figph[The third row has the step “bring down the common factors that all expressions share”. The second column in the third row has “bring down the 2,3, and 3 then multiply”. The third column in the third row has “GCF = 2 times 3 times 3”.], alt: "The third row has the step “bring down the common factors that all expressions share”. The second column in the third row has “bring down the 2,3, and 3 then multiply”. The third column in the third row has “GCF = 2 times 3 times 3”.", caption: none)

#figure(figph[The fourth row has the fourth step “multiply the factors”. The second column in the fourth row is blank. The third column in the fourth row has “GCF = 18” and “the GCF of 54 and 36 is 18”.], alt: "The fourth row has the fourth step “multiply the factors”. The second column in the fourth row is blank. The third column in the fourth row has “GCF = 18” and “the GCF of 54 and 36 is 18”.", caption: none)

Notice that, because the GCF is a factor of both numbers, 54 and 36 can be written as multiples of 18.

#math.equation(block: true, alt: "54 equals 18 times 3; 36 equals 18 times 2")[$54 = 18 · 3 \ 36 = 18 · 2$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the GCF of 48 and 80.

#solutionbox[
16
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the GCF of 18 and 40.

#solutionbox[
2
]
]

We summarize the steps we use to find the GCF below.

#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.
]

In the first example, the GCF was a constant. In the next two examples, we will get variables in the greatest common factor.

#examplebox("Example 2")[][
Find the greatest common factor of #math.equation(block: false, alt: "27 x cubed and 18 x to the power 4")[$27 x^(3) #h(0.2em) "and" #h(0.2em) 18 x^(4)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column.], [#figure(figph[Prime factorization of 27x³ and 18x⁴. Common factors (3x3 and x³ ) are circled in purple, illustrating how to find the Greatest Common Factor (GCF) for algebraic expressions.], alt: "Prime factorization of 27x³ and 18x⁴. Common factors (3x3 and x³ ) are circled in purple, illustrating how to find the Greatest Common Factor (GCF) for algebraic expressions.", caption: none)]),
  [Bring down the common factors.], [#figure(figph[The image shows a mathematical equation on a white background, displaying GCF = 3 \* 3 \* X \* X \* X. The text is in a dark gray, sans-serif font.], alt: "The image shows a mathematical equation on a white background, displaying GCF = 3 * 3 * X * X * X. The text is in a dark gray, sans-serif font.", caption: none)],
  [Multiply the factors.], [#figure(figph[The image displays a mathematical equation on a plain white background, stating 'GCF = 9x^3', indicating the Greatest Common Factor equals 9x cubed.], alt: "The image displays a mathematical equation on a plain white background, stating 'GCF = 9x^3', indicating the Greatest Common Factor equals 9x cubed.", caption: none)],
  [], [The GCF of #math.equation(block: false, alt: "27 x cubed")[$27 x^(3)$] and #math.equation(block: false, alt: "18 x to the power 4")[$18 x^(4)$] is #math.equation(block: false, alt: "9 x cubed .")[$9 x^(3) .$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the GCF: #math.equation(block: false, alt: "12 x squared , 18 x cubed")[$12 x^(2) , 18 x^(3)$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the GCF: #math.equation(block: false, alt: "16 y squared , 24 y cubed")[$16 y^(2) , 24 y^(3)$].

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

#examplebox("Example 3")[][
Find the GCF of #math.equation(block: false, alt: "4 x squared y , 6 x y cubed")[$4 x^(2) y , 6 x y^(3)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column.], [#figure(figph[Mathematical expressions 4x^2y and 6xy^3 are factored into their prime and variable components, with common factors 2, x, and y highlighted by magenta ovals.], alt: "Mathematical expressions 4x^2y and 6xy^3 are factored into their prime and variable components, with common factors 2, x, and y highlighted by magenta ovals.", caption: none)]),
  [Bring down the common factors.], [#figure(figph[A mathematical equation showing GCF (Greatest Common Factor) is equal to the product of 2, x, and y, written as GCF = 2  X  Y.], alt: "A mathematical equation showing GCF (Greatest Common Factor) is equal to the product of 2, x, and y, written as GCF = 2 X Y.", caption: none)],
  [Multiply the factors.], [#figure(figph[The image displays a mathematical equation: GCF = 2xy, in black text against a plain white background. GCF stands for Greatest Common Factor, and 2xy is an algebraic expression.], alt: "The image displays a mathematical equation: GCF = 2xy, in black text against a plain white background. GCF stands for Greatest Common Factor, and 2xy is an algebraic expression.", caption: none)],
  [], [The GCF of #math.equation(block: false, alt: "4 x squared y")[$4 x^(2) y$] and #math.equation(block: false, alt: "6 x y cubed")[$6 x y^(3)$] is #math.equation(block: false, alt: "2 xy")[$2 "xy"$].],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the GCF: #math.equation(block: false, alt: "6 a b to the power 4 , 8 a squared b")[$6 a b^(4) , 8 a^(2) b$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the GCF: #math.equation(block: false, alt: "9 m to the power 5 n squared , 12 m cubed n")[$9 m^(5) n^(2) , 12 m^(3) n$].

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

#examplebox("Example 4")[][
Find the GCF 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 variables with exponents in expanded form. Circle the common factors in each column.], [#figure(figph[Prime factorization of 21x^3, 9x^2, and 15x is shown, with common factors '3' and 'x' circled in purple, illustrating a step in finding the Greatest Common Factor (GCF).], alt: "Prime factorization of 21x^3, 9x^2, and 15x is shown, with common factors '3' and 'x' circled in purple, illustrating a step in finding the Greatest Common Factor (GCF).", caption: none)]),
  [Bring down the common factors.], [#figure(figph[The image shows a mathematical expression], alt: "The image shows a mathematical expression", caption: none)],
  [Multiply the factors.], [#figure(figph[The text 'GCF = 3x' is displayed on a white background.], alt: "The text 'GCF = 3x' is displayed on a white background.", 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

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, 12 as #math.equation(block: false, alt: "2 times 6 or 3 times 4 close parenthesis ,")[$2 · 6 #h(0.2em) "or" #h(0.2em) 3 · 4 \) ,$] in algebra, it can be useful to represent a polynomial in factored form. One way to do this is by finding the GCF of all the terms. Remember, we multiply a polynomial by a monomial as follows:

#math.equation(block: true, alt: "2 open parenthesis x plus 7 close parenthesis, factors; 2 times x plus 2 times 7; 2 x plus 14, product")[$2 ( x + 7 ) &  &  & "factors" \ 2 · x + 2 · 7 &  &  &  \ 2 x + 14 &  &  & "product"$]Now we will start with a product, like #math.equation(block: false, alt: "2 x plus 14")[$2 x + 14$], and end with its factors, #math.equation(block: false, alt: "2 open parenthesis x plus 7 close parenthesis")[$2 ( x + 7 )$]. 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 #math.equation(block: false, alt: "a , b , c")[$a , b , c$] 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")[$a ( b + c ) = a b + a c &  &  & "and" &  &  & a b + a c = a ( b + c )$]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 polynomial? You just find the GCF of all the terms and write the polynomial as a product!

#examplebox("Example 5")[How to Factor the Greatest Common Factor from a Polynomial][
Factor: #math.equation(block: false, alt: "4 x plus 12")[$4 x + 12$].

#solutionbox[
#figure(figph[This table has three columns. In the first column are the steps for factoring. The first row has the first step, “Find the G C F of all the terms of the polynomial”. The second column in the first row has “find the G C F of 4 x and 12”. The third column in the first row has 4 x factored as 2 times 2 times x and below it 18 factored as 2 times 2 times 3. Then, below the factors are the statements, “G C F = 2 times 2” and “G C F = 4”.], alt: "This table has three columns. In the first column are the steps for factoring. The first row has the first step, “Find the G C F of all the terms of the polynomial”. The second column in the first row has “find the G C F of 4 x and 12”. The third column in the first row has 4 x factored as 2 times 2 times x and below it 18 factored as 2 times 2 times 3. Then, below the factors are the statements, “G C F = 2 times 2” and “G C F = 4”.", caption: none)

#figure(figph[The second row has the second step “rewrite each term as a product using the G C F”. The second column in the second row has the statement “Rewrite 4 x and 12 as products of their G C F, 4” Then the two equations 4 x = 4 times x and 12 = 4 times 3. The third column in the second row has the expressions 4x + 12 and below this 4 times x + 4 times 3.], alt: "The second row has the second step “rewrite each term as a product using the G C F”. The second column in the second row has the statement “Rewrite 4 x and 12 as products of their G C F, 4” Then the two equations 4 x = 4 times x and 12 = 4 times 3. The third column in the second row has the expressions 4x + 12 and below this 4 times x + 4 times 3.", caption: none)

#figure(figph[The third row has the step “Use the reverse distributive property to factor the expression”. The second column in the third row is blank. The third column in the third row has “4(x + 3)”.], alt: "The third row has the step “Use the reverse distributive property to factor the expression”. The second column in the third row is blank. The third column in the third row has “4(x + 3)”.", caption: none)

#figure(figph[The fourth row has the fourth step “check by multiplying the factors”. The second column in the fourth row is blank. The third column in the fourth row has three expressions. The first is 4(x + 3), the second is 4 times x + 4 times 3. The third is 4 x + 12.], alt: "The fourth row has the fourth step “check by multiplying the factors”. The second column in the fourth row is blank. The third column in the fourth row has three expressions. The first is 4(x + 3), the second is 4 times x + 4 times 3. The third is 4 x + 12.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "6 a plus 24")[$6 a + 24$].

#solutionbox[
#math.equation(block: true, alt: "6 open parenthesis a plus 4 close parenthesis")[$6 ( a + 4 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "2 b plus 14")[$2 b + 14$].

#solutionbox[
#math.equation(block: true, alt: "2 open parenthesis b plus 7 close parenthesis")[$2 ( b + 7 )$]
]
]

#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.

#figure(figph[This figure has two statements. The first statement has “noun”. Beside it the statement “7 is a factor of 14” labeling the word factor as the noun. The second statement has “verb”. Beside this statement is “factor 3 from 3a + 3 labeling factor as the verb.], alt: "This figure has two statements. The first statement has “noun”. Beside it the statement “7 is a factor of 14” labeling the word factor as the noun. The second statement has “verb”. Beside this statement is “factor 3 from 3a + 3 labeling factor as the verb.", caption: none)
]

#examplebox("Example 6")[][
Factor: #math.equation(block: false, alt: "5 a plus 5")[$5 a + 5$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Find the GCF of 5#emph[a] and 5.], [#figure(figph[This image demonstrates finding the Greatest Common Factor (GCF) of 5a and 5. It shows that 5 is the common factor, leading to GCF = 5.], alt: "This image demonstrates finding the Greatest Common Factor (GCF) of 5a and 5. It shows that 5 is the common factor, leading to GCF = 5.", caption: none)]),
  [#linebreak()  #linebreak()], [#figure(figph[The mathematical expression '5a + 5' is displayed, showing the sum of five times a variable 'a' and the number five.], alt: "The mathematical expression '5a + 5' is displayed, showing the sum of five times a variable 'a' and the number five.", caption: none)

 #linebreak()  #linebreak()],
  [Rewrite each term as a product using the GCF.], [#figure(figph[A mathematical expression showing 5 multiplied by 'a' plus 5 multiplied by 1. This illustrates the distributive property, where 5 is a common factor.], alt: "A mathematical expression showing 5 multiplied by 'a' plus 5 multiplied by 1. This illustrates the distributive property, where 5 is a common factor.", caption: none)],
  [Use the Distributive Property "in reverse" to factor the GCF.], [#figure(figph[The mathematical expression '5(a+1)' is displayed on a white background.], alt: "The mathematical expression '5(a+1)' is displayed on a white background.", caption: none)],
  [Check by mulitplying the factors to get the orginal polynomial.], [],
  [#math.equation(block: false, alt: "5 open parenthesis a plus 1 close parenthesis")[$5 ( a + 1 )$]], [],
  [#math.equation(block: false, alt: "5 times a plus 5 times 1")[$5 ⋅ a + 5 ⋅ 1$]], [],
  [#math.equation(block: false, alt: "5 a plus 5 ✓")[$5 a + 5 ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "14 x plus 14")[$14 x + 14$].

#solutionbox[
#math.equation(block: true, alt: "14 open parenthesis x plus 1 close parenthesis")[$14 ( x + 1 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "12 p plus 12")[$12 p + 12$].

#solutionbox[
#math.equation(block: true, alt: "12 open parenthesis p plus 1 close parenthesis")[$12 ( p + 1 )$]
]
]

The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.

#examplebox("Example 7")[][
Factor: #math.equation(block: false, alt: "12 x minus 60")[$12 x − 60$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Find the GCF of 12#emph[x] and 60.], [#figure(figph[Finding the Greatest Common Factor (GCF) of 12x and 60 by breaking down each term into its prime factors and identifying common ones, resulting in a GCF of 12.], alt: "Finding the Greatest Common Factor (GCF) of 12x and 60 by breaking down each term into its prime factors and identifying common ones, resulting in a GCF of 12.", caption: none)]),
  [#linebreak()  #linebreak()], [#figure(figph[A mathematical expression '12x - 60' is displayed in a digital font against a plain white background. It features the numbers 12 and 60, the variable x, and a minus sign.], alt: "A mathematical expression '12x - 60' is displayed in a digital font against a plain white background. It features the numbers 12 and 60, the variable x, and a minus sign.", caption: none)

 #linebreak()  #linebreak()],
  [Rewrite each term as a product using the GCF.], [#figure(figph[An algebraic expression is displayed, featuring 12 multiplied by x, minus 12 multiplied by 5, highlighting the common factor of 12 in a subtraction operation.], alt: "An algebraic expression is displayed, featuring 12 multiplied by x, minus 12 multiplied by 5, highlighting the common factor of 12 in a subtraction operation.", caption: none)],
  [Factor the GCF.], [#figure(figph[The image displays the mathematical expression '12(x-5)' against a plain white background. The numbers and symbols are rendered in a clear, dark font, indicating a algebraic expression likely from a textbook or worksheet.], alt: "The image displays the mathematical expression '12(x-5)' against a plain white background. The numbers and symbols are rendered in a clear, dark font, indicating a algebraic expression likely from a textbook or worksheet.", caption: none)],
  [Check by mulitplying the factors.], [],
  [#math.equation(block: false, alt: "12 open parenthesis x minus 5 close parenthesis")[$12 ( x − 5 )$]], [],
  [#math.equation(block: false, alt: "12 times x minus 12 times 5")[$12 ⋅ x − 12 ⋅ 5$]], [],
  [#math.equation(block: false, alt: "12 x minus 60 ✓")[$12 x − 60 ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "18 u minus 36")[$18 u − 36$].

#solutionbox[
#math.equation(block: true, alt: "18 open parenthesis u minus 2 close parenthesis")[$18 ( u − 2 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "30 y minus 60")[$30 y − 60$].

#solutionbox[
#math.equation(block: true, alt: "30 open parenthesis y minus 2 close parenthesis")[$30 ( y − 2 )$]
]
]

Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.

#examplebox("Example 8")[][
Factor: #math.equation(block: false, alt: "4 y squared plus 24 y plus 28")[$4 y^(2) + 24 y + 28$].

#solutionbox[
We start by finding the GCF of all three terms.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Find the GCF of #math.equation(block: false, alt: "4 y squared")[$4 y^(2)$], #math.equation(block: false, alt: "24 y")[$24 y$] and 28.], [#figure(figph[This image demonstrates finding the GCF of 4y², 24y, and 28 by prime factorization. Common factors (two '2's) are circled, leading to GCF = 2 \* 2 = 4.], alt: "This image demonstrates finding the GCF of 4y², 24y, and 28 by prime factorization. Common factors (two '2's) are circled, leading to GCF = 2 * 2 = 4.", caption: none)]),
  [#linebreak()  #linebreak()], [#figure(figph[The mathematical expression '4y^2 + 24y + 28' is displayed in black text against a white background.], alt: "The mathematical expression '4y^2 + 24y + 28' is displayed in black text against a white background.", caption: none)

 #linebreak()  #linebreak()],
  [Rewrite each term as a product using the GCF.], [#figure(figph[The mathematical expression 4 \* y^2 + 4 \* 6y + 4 \* 7 is presented, with the common factor '4' highlighted in red in each of the three terms.], alt: "The mathematical expression 4 * y^2 + 4 * 6y + 4 * 7 is presented, with the common factor '4' highlighted in red in each of the three terms.", caption: none)],
  [Factor the GCF.], [#figure(figph[The image shows the algebraic expression 4(y^2 + 6y + 7).], alt: "The image shows the algebraic expression 4(y^2 + 6y + 7).", caption: none)],
  [Check by mulitplying.], [],
  [#math.equation(block: false, alt: "4 open parenthesis y squared plus 6 y plus 7 close parenthesis")[$4 ( y^(2) + 6 y + 7 )$]], [],
  [#math.equation(block: false, alt: "4 times y squared plus 4 times 6 y plus 4 times 7")[$4 ⋅ y^(2) + 4 ⋅ 6 y + 4 ⋅ 7$]], [],
  [#math.equation(block: false, alt: "4 y squared plus 24 y plus 28 ✓")[$4 y^(2) + 24 y + 28 ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "5 x squared minus 25 x plus 15")[$5 x^(2) − 25 x + 15$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "3 y squared minus 12 y plus 27")[$3 y^(2) − 12 y + 27$].

#solutionbox[
#math.equation(block: true, alt: "3 open parenthesis y squared minus 4 y plus 9 close parenthesis")[$3 ( y^(2) − 4 y + 9 )$]
]
]

#examplebox("Example 9")[][
Factor: #math.equation(block: false, alt: "5 x cubed minus 25 x squared")[$5 x^(3) − 25 x^(2)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([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[An image demonstrating how to find the Greatest Common Factor (GCF) of 5x^3 and 25x^2 by factoring each term and identifying common factors, resulting in GCF = 5x^2.], alt: "An image demonstrating how to find the Greatest Common Factor (GCF) of 5x^3 and 25x^2 by factoring each term and identifying common factors, resulting in GCF = 5x^2.", caption: none)]),
  [#linebreak()  #linebreak()], [#figure(figph[The image displays the mathematical expression 5x^3 - 25x^2, written in black text on a white background, representing a polynomial expression.], alt: "The image displays the mathematical expression 5x^3 - 25x^2, written in black text on a white background, representing a polynomial expression.", caption: none)

 #linebreak()  #linebreak()],
  [Rewrite each term.], [#figure(figph[A mathematical expression 5x^2 \* x - 5x^2 \* 5 is displayed on a white background, with numbers and the first x in red, and the multiplication dots, second x, minus sign, and second 5 in black.], alt: "A mathematical expression 5x^2 * x - 5x^2 * 5 is displayed on a white background, with numbers and the first x in red, and the multiplication dots, second x, minus sign, and second 5 in black.", caption: none)],
  [Factor the GCF.], [#figure(figph[The mathematical expression 5x^2(x-5) is displayed against a white background.], alt: "The mathematical expression 5x^2(x-5) is displayed against a white background.", caption: none)],
  [Check.], [],
  [#math.equation(block: false, alt: "5 x squared open parenthesis x minus 5 close parenthesis")[$5 x^(2) ( x − 5 )$]], [],
  [#math.equation(block: false, alt: "5 x squared times x minus 5 x squared times 5")[$5 x^(2) ⋅ x − 5 x^(2) ⋅ 5$]], [],
  [#math.equation(block: false, alt: "5 x cubed minus 25 x squared ✓")[$5 x^(3) − 25 x^(2) ✓$]], [],
))
]
]

#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 10")[][
Factor: #math.equation(block: false, alt: "21 x cubed minus 9 x squared plus 15 x")[$21 x^(3) − 9 x^(2) + 15 x$].

#solutionbox[
In a previous example we found the GCF of #math.equation(block: false, alt: "21 x cubed , 9 x squared , 15 x")[$21 x^(3) , 9 x^(2) , 15 x$] to be #math.equation(block: false, alt: "3 x")[$3 x$].

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression reads 21x^3 - 9x^2 + 15x, featuring polynomial terms with coefficients, variables, and exponents.], alt: "A mathematical expression reads 21x^3 - 9x^2 + 15x, featuring polynomial terms with coefficients, variables, and exponents.", caption: none)]),
  [Rewrite each term using the GCF, 3#emph[x].], [#figure(figph[A mathematical expression featuring three terms separated by subtraction and addition, each term having '3x' as a common factor: 3x \* 7x^2 - 3x \* 3x + 3x \* 5.], alt: "A mathematical expression featuring three terms separated by subtraction and addition, each term having '3x' as a common factor: 3x * 7x^2 - 3x * 3x + 3x * 5.", caption: none)],
  [Factor the GCF.], [#figure(figph[A mathematical expression showing 3x multiplied by the quantity 7x squared minus 3x plus 5, indicating a distribution operation in algebra.], alt: "A mathematical expression showing 3x multiplied by the quantity 7x squared minus 3x plus 5, indicating a distribution operation in algebra.", caption: none)],
  [Check.], [],
  [#math.equation(block: false, alt: "3 x open parenthesis 7 x squared minus 3 x plus 5 close parenthesis")[$3 x ( 7 x^(2) − 3 x + 5 )$]], [],
  [#math.equation(block: false, alt: "3 x times 7 x squared minus 3 x times 3 x plus 3 x times 5")[$3 x ⋅ 7 x^(2) − 3 x ⋅ 3 x + 3 x ⋅ 5$]], [],
  [#math.equation(block: false, alt: "21 x cubed minus 9 x squared plus 15 x ✓")[$21 x^(3) − 9 x^(2) + 15 x ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "20 x cubed minus 10 x squared plus 14 x")[$20 x^(3) − 10 x^(2) + 14 x$].

#solutionbox[
#math.equation(block: true, alt: "2 x open parenthesis 10 x squared minus 5 x plus 7 close parenthesis")[$2 x ( 10 x^(2) − 5 x + 7 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "24 y cubed minus 12 y squared minus 20 y")[$24 y^(3) − 12 y^(2) − 20 y$].

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

#examplebox("Example 11")[][
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(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Find the GCF of #math.equation(block: false, alt: "8 m cubed")[$8 m^(3)$], #math.equation(block: false, alt: "12 m squared n")[$12 m^(2) n$], #math.equation(block: false, alt: "20 m n squared")[$20 m n^(2)$].], [#figure(figph[Finding the GCF of 8m^3, 12m^2n, and 20mn^2. The prime factorization method is used to identify common factors (2, 2, m), leading to a GCF of 4m.], alt: "Finding the GCF of 8m^3, 12m^2n, and 20mn^2. The prime factorization method is used to identify common factors (2, 2, m), leading to a GCF of 4m.", caption: none)]),
  [#linebreak()  #linebreak()], [#figure(figph[A mathematical expression reads 8m to the power of 3 minus 12m squared n plus 20mn squared, displayed in a clean, white background.], alt: "A mathematical expression reads 8m to the power of 3 minus 12m squared n plus 20mn squared, displayed in a clean, white background.", caption: none)

 #linebreak()  #linebreak()],
  [Rewrite each term.], [#figure(figph[A mathematical expression featuring the algebraic terms 4m multiplied by 2m^2, minus 4m multiplied by 3mn, plus 4m multiplied by 5n^2, illustrating polynomial expansion.], alt: "A mathematical expression featuring the algebraic terms 4m multiplied by 2m^2, minus 4m multiplied by 3mn, plus 4m multiplied by 5n^2, illustrating polynomial expansion.", caption: none)],
  [Factor the GCF.], [#figure(figph[A mathematical expression is displayed, showing 4m multiplied by the quantity (2m^2 - 3mn + 5n^2). This represents a polynomial multiplication problem.], alt: "A mathematical expression is displayed, showing 4m multiplied by the quantity (2m^2 - 3mn + 5n^2). This represents a polynomial multiplication problem.", caption: none)],
  [Check.], [],
  [#math.equation(block: false, alt: "4 m open parenthesis 2 m squared minus 3 m n plus 5 n squared close parenthesis")[$4 m ( 2 m^(2) − 3 m n + 5 n^(2) )$]], [],
  [#math.equation(block: false, alt: "4 m times 2 m squared minus 4 m times 3 m n plus 4 m times 5 n squared")[$4 m ⋅ 2 m^(2) − 4 m ⋅ 3 m n + 4 m ⋅ 5 n^(2)$]], [],
  [#math.equation(block: false, alt: "8 m cubed minus 12 m squared n plus 20 m n squared ✓")[$8 m^(3) − 12 m^(2) n + 20 m n^(2) ✓$]], [],
))
]
]

#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 cubed close parenthesis")[$3 p ( p^(2) − 2 p q + 3 q^(3) )$]
]
]

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

#examplebox("Example 12")[][
Factor: #math.equation(block: false, alt: "−8 y minus 24")[$−8 y − 24$].

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

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Ignoring the signs of the terms, we first find the GCF of 8#emph[y] and 24 is 8. Since the expression −8#emph[y] − 24 has a negative leading coefficient, we use −8 as the GCF. #linebreak()  #linebreak()], [#figure(figph[Prime factorization method for determining the Greatest Common Factor (GCF) of 8y and 24, resulting in 8.], alt: "Prime factorization method for determining the Greatest Common Factor (GCF) of 8y and 24, resulting in 8.", caption: none)

 #linebreak()  #linebreak()]),
  [Rewrite each term using the GCF.], [#figure(figph[A mathematical expression showing '-8y - 24' in a clean, legible font on a white background, representing a linear algebraic expression.], alt: "A mathematical expression showing '-8y - 24' in a clean, legible font on a white background, representing a linear algebraic expression.", caption: none)

 #linebreak() 
       #figure(figph[A mathematical expression showing -8 multiplied by y, added to -8 multiplied by 3, which is -8 \* y + (-8) \* 3.], alt: "A mathematical expression showing -8 multiplied by y, added to -8 multiplied by 3, which is -8 * y + (-8) * 3.", caption: none)],
  [Factor the GCF.], [#figure(figph[A mathematical expression showing -8 multiplied by the sum of y and 3, written as -8(y + 3).], alt: "A mathematical expression showing -8 multiplied by the sum of y and 3, written as -8(y + 3).", caption: none)],
  [Check.], [],
  [#math.equation(block: false, alt: "−8 open parenthesis y plus 3 close parenthesis")[$−8 ( y + 3 )$]], [],
  [#math.equation(block: false, alt: "−8 times y plus open parenthesis −8 close parenthesis times 3")[$−8 ⋅ y + ( −8 ) ⋅ 3$]], [],
  [#math.equation(block: false, alt: "−8 y minus 24 ✓")[$−8 y − 24 ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "−16 z minus 64")[$−16 z − 64$].

#solutionbox[
#math.equation(block: true, alt: "−16 open parenthesis z plus 4 close parenthesis")[$−16 ( z + 4 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "−9 y minus 27")[$−9 y − 27$].

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

#examplebox("Example 13")[][
Factor: #math.equation(block: false, alt: "−6 a squared plus 36 a")[$−6 a^(2) + 36 a$].

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

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Since the leading coefficient is negative, the GCF is negative, −6#emph[a]. #linebreak()  #linebreak()], [#figure(figph[Calculating the Greatest Common Factor (GCF) of 6a^2 and 36a by prime factorization, highlighting common factors 2, 3, and a, resulting in a GCF of 6a.], alt: "Calculating the Greatest Common Factor (GCF) of 6a^2 and 36a by prime factorization, highlighting common factors 2, 3, and a, resulting in a GCF of 6a.", caption: none)

 #linebreak() #figure(figph[The image displays the mathematical expression -6a^2 + 36a, featuring a quadratic term with a negative coefficient and a linear term with a positive coefficient.], alt: "The image displays the mathematical expression -6a^2 + 36a, featuring a quadratic term with a negative coefficient and a linear term with a positive coefficient.", caption: none)

 #linebreak()  #linebreak()]),
  [Rewrite each term using the GCF.], [#figure(figph[A mathematical expression reads -6a multiplied by a, minus the quantity -6a, multiplied by 6.], alt: "A mathematical expression reads -6a multiplied by a, minus the quantity -6a, multiplied by 6.", caption: none)],
  [Factor the GCF.], [#figure(figph[A mathematical expression shows -6a multiplied by the quantity (a - 6), written as -6a(a-6).], alt: "A mathematical expression shows -6a multiplied by the quantity (a - 6), written as -6a(a-6).", caption: none)],
  [Check.], [],
  [#math.equation(block: false, alt: "−6 a open parenthesis a minus 6 close parenthesis")[$−6 a ( a − 6 )$]], [],
  [#math.equation(block: false, alt: "−6 a times a plus open parenthesis −6 a close parenthesis open parenthesis −6 close parenthesis")[$−6 a ⋅ a + ( −6 a ) ( −6 )$]], [],
  [#math.equation(block: false, alt: "−6 a squared plus 36 a ✓")[$−6 a^(2) + 36 a ✓$]], [],
))
]
]

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

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

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

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

#examplebox("Example 14")[][
Factor: #math.equation(block: false, alt: "5 q open parenthesis q plus 7 close parenthesis minus 6 open parenthesis q plus 7 close parenthesis")[$5 q ( q + 7 ) − 6 ( q + 7 )$].

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

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression displays 5q(q + 7) - 6(q + 7). The 'q + 7' term is highlighted in red, indicating it's a common factor in both parts of the expression, suggesting a factorization step.], alt: "A mathematical expression displays 5q(q + 7) - 6(q + 7). The 'q + 7' term is highlighted in red, indicating it's a common factor in both parts of the expression, suggesting a factorization step.", caption: none)]),
  [Factor the GCF, (#emph[q] + 7).], [#figure(figph[A mathematical expression showing the product of two binomials: (q + 7)(5q - 6).], alt: "A mathematical expression showing the product of two binomials: (q + 7)(5q - 6).", 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

When there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating the polynomial into two parts with two terms in each part. Then look for the 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 prime, some polynomials are prime.)

#examplebox("Example 15")[How to Factor by Grouping][
Factor: #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[This table gives the steps for factoring x y + 3 y + 2 x + 6. In the first row there is the statement, “group terms with common factors”. In the next column, there is the statement of no common factors of all 4 terms. The last column shows the first two terms grouped and the last two terms grouped.], alt: "This table gives the steps for factoring x y + 3 y + 2 x + 6. In the first row there is the statement, “group terms with common factors”. In the next column, there is the statement of no common factors of all 4 terms. The last column shows the first two terms grouped and the last two terms grouped.", caption: none)

#figure(figph[The second row has the statement, “factor out the common factor from each group”. The second column in the second row states to factor out the GCF from the two separate groups. The third column in the second row has the expression y(x + 3) + 2(x + 3).], alt: "The second row has the statement, “factor out the common factor from each group”. The second column in the second row states to factor out the GCF from the two separate groups. The third column in the second row has the expression y(x + 3) + 2(x + 3).", caption: none)

#figure(figph[The third row has the statement, “factor the common factor from the expression”. The second column in this row points out there is a common factor of (x + 3). The third column in the third row shows the factor of (x + 3) factored from the two groups, (x + 3) times (y + 2).], alt: "The third row has the statement, “factor the common factor from the expression”. The second column in this row points out there is a common factor of (x + 3). The third column in the third row shows the factor of (x + 3) factored from the two groups, (x + 3) times (y + 2).", caption: none)

#figure(figph[The last row has the statement, “check”. The second column in this row states to multiply (x + 3)(y + 2). The product is shown in the last column of the original polynomial x y + 3 y + 2 x + 6.], alt: "The last row has the statement, “check”. The second column in this row states to multiply (x + 3)(y + 2). The product is shown in the last column of the original polynomial x y + 3 y + 2 x + 6.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #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: #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 16")[][
Factor: #math.equation(block: false, alt: "x squared plus 3 x minus 2 x minus 6")[$x^(2) + 3 x − 2 x − 6$].

#solutionbox[
#math.equation(block: true, alt: "There is no GCF in all four terms., x squared plus 3 x −2 x minus 6; Separate into two parts., x squared plus 3 x ⎵ −2 x minus 6 ⎵; Factor the GCF from both parts. Be careful; with the signs when factoring the GCF from; the last two terms., x open parenthesis x plus 3 close parenthesis minus 2 open parenthesis x plus 3 close parenthesis; open parenthesis x plus 3 close parenthesis open parenthesis x minus 2 close parenthesis; Check on your own by multiplying.")[$"There is no GCF in all four terms." &  &  & #h(4em) x^(2) + 3 x #h(0.5em) −2 x − 6 \ "Separate into two parts." &  &  & #h(4em) underbrace(x^(2) + 3 x) #h(0.5em) underbrace(−2 x − 6) \  \  \ "Factor the GCF from both parts. Be careful" \ "with the signs when factoring the GCF from" \ "the last two terms." &  &  & #h(4em) x ( x + 3 ) − 2 ( x + 3 ) \ ( x + 3 ) ( x − 2 ) \  \  \ "Check on your own by multiplying." &  &  &$]
]
]

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

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor: #math.equation(block: false, alt: "y squared plus 4 y minus 7 y minus 28")[$y^(2) + 4 y − 7 y − 28$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Access these online resources for additional instruction and practice with greatest common factors (GFCs) and factoring by grouping.

- #link("https://openstax.org/l/25GCF1")[Greatest Common Factor (GCF)]
- #link("https://openstax.org/l/25GCF2")[Factoring Out the GCF of a Binomial]
- #link("https://openstax.org/l/25GCF3")[Greatest Common Factor (GCF) of Polynomials]
]


=== Key Concepts

- #strong[Finding the Greatest Common Factor (GCF):] To find the 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 as in .
- #strong[Factor the Greatest Common Factor from a Polynomial:] To factor a 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 as in .
- #strong[Factor by Grouping:] To factor a polynomial with 4 four or more terms
+ 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 as in .


==== Practice Makes Perfect

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

In the following exercises, find the greatest common factor.

8, 18

#solutionbox[
2
]

24, 40

72, 162

#solutionbox[
18
]

150, 275

10#emph[a], 50

#solutionbox[
10
]

5#emph[b], 30

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

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

#math.equation(block: true, alt: "21 b squared , 14 b")[$21 b^(2) , 14 b$]

#math.equation(block: true, alt: "8 w squared , 24 w cubed")[$8 w^(2) , 24 w^(3)$]

#solutionbox[
#math.equation(block: true, alt: "8 w squared")[$8 w^(2)$]
]

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

#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 , 10 x to the power 4 , 5 x to the power 5")[$35 x^(3) , 10 x^(4) , 5 x^(5)$]

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

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

#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: "4 x plus 20")[$4 x + 20$]

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

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

#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 q plus 9")[$9 q + 9$]

#solutionbox[
#math.equation(block: true, alt: "9 open parenthesis q plus 1 close parenthesis")[$9 ( q + 1 )$]
]

#math.equation(block: true, alt: "7 r plus 7")[$7 r + 7$]

#math.equation(block: true, alt: "8 m minus 8")[$8 m − 8$]

#solutionbox[
#math.equation(block: true, alt: "8 open parenthesis m minus 1 close parenthesis")[$8 ( m − 1 )$]
]

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

#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: "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: "−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: "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")[$2 x ( x − 1 ) + 9 ( x − 1 )$]

#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: "x y plus 2 y plus 3 x plus 6")[$x y + 2 y + 3 x + 6$]

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

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

#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: "b squared plus 5 b minus 4 b minus 20")[$b^(2) + 5 b − 4 b − 20$]

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

#math.equation(block: true, alt: "m squared plus 6 m minus 12 m minus 72")[$m^(2) + 6 m − 12 m − 72$]

#math.equation(block: true, alt: "p squared plus 4 p minus 9 p minus 36")[$p^(2) + 4 p − 9 p − 36$]

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

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

#strong[Mixed Practice]

In the following exercises, factor.

#math.equation(block: true, alt: "−20 x minus 10")[$−20 x − 10$]

#solutionbox[
#math.equation(block: true, alt: "−10 open parenthesis 2 x plus 1 close parenthesis")[$−10 ( 2 x + 1 )$]
]

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

#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 minus x minus 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 plus 3 x squared minus 5 x minus 3")[$5 x^(3) + 3 x^(2) − 5 x − 3$]


==== Everyday Math

#strong[Area of a rectangle] The area of a rectangle with length 6 less than the width is given by the expression #math.equation(block: false, alt: "w squared minus 6 w")[$w^(2) − 6 w$], where #math.equation(block: false, alt: "w equals")[$w =$] width. Factor the greatest common factor from the polynomial.

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

#strong[Height of a baseball] The height of a baseball #emph[t] seconds after it is hit is given by the expression #math.equation(block: false, alt: "−16 t squared plus 80 t plus 4")[$−16 t^(2) + 80 t + 4$]. Factor the greatest common factor from the polynomial.


==== Writing Exercises

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 , and y to the power 10")[$y^(4) , y^(5) , "and" #h(0.2em) y^(10)$]? 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 , and y to the power c")[$y^(a) , y^(b) , "and" y^(c)$].


==== Self Check

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

#figure(figph[This table has the following statements all to be preceded by “I can…”. The first is “find the greatest common factor of two or more expressions”. The second is “factor the greatest common factor from a polynomial”. The third is “factor by grouping”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.], alt: "This table has the following statements all to be preceded by “I can…”. The first is “find the greatest common factor of two or more expressions”. The second is “factor the greatest common factor from a polynomial”. The third is “factor by grouping”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.", 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.