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== 7.5#h(0.6em)General Strategy for Factoring Polynomials

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

Factor #math.equation(block: false, alt: "y squared minus 2 y minus 24")[$y^(2) − 2 y − 24$]. #linebreak() 
If you missed this problem, review .

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor #math.equation(block: false, alt: "3 t squared plus 17 t plus 10")[$3 t^(2) + 17 t + 10$]. #linebreak() 
If you missed this problem, review .

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor #math.equation(block: false, alt: "36 p squared minus 60 p plus 25")[$36 p^(2) − 60 p + 25$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 6 p minus 5 close parenthesis squared")[$attach(( 6 p − 5 ), t: 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor #math.equation(block: false, alt: "5 x squared minus 80")[$5 x^(2) − 80$]. #linebreak() 
If you missed this problem, review .

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


=== Recognize and Use the Appropriate Method to Factor a Polynomial Completely

You have now become acquainted with all the methods of factoring that you will need in this course. (In your next algebra course, more methods will be added to your repertoire.) The figure below summarizes all the factoring methods we have covered.  outlines a strategy you should use when factoring polynomials.

#figure(figph[This figure presents a general strategy for factoring polynomials. First, at the top, there is GCF, which is where factoring starts. Below this, there are three options, binomial, trinomial, and more than three terms. For binomial, there are the difference of two squares, the sum of squares, the sum of cubes, and the difference of cubes. For trinomials, there are two forms, x squared plus bx plus c and ax squared 2 plus b x plus c. There are also the sum and difference of two squares formulas as well as the “a c” method. Finally, for more than three terms, the method is grouping.], alt: "This figure presents a general strategy for factoring polynomials. First, at the top, there is GCF, which is where factoring starts. Below this, there are three options, binomial, trinomial, and more than three terms. For binomial, there are the difference of two squares, the sum of squares, the sum of cubes, and the difference of cubes. For trinomials, there are two forms, x squared plus bx plus c and ax squared 2 plus b x plus c. There are also the sum and difference of two squares formulas as well as the “a c” method. Finally, for more than three terms, the method is grouping.", caption: none)

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

+ Is there a greatest common factor?
- Factor it out.
+ Is the polynomial a binomial, trinomial, or are there more than three terms?
- If it is a binomial: #linebreak() 
Is it a sum?
- Of squares? Sums of squares do not factor.
- Of cubes? Use the sum of cubes pattern.


Is it a difference? #linebreak() 
- Of squares? Factor as the product of conjugates.
- Of cubes? Use the difference of cubes pattern.
- If it is a trinomial: #linebreak() 
Is it of the form #math.equation(block: false, alt: "x squared plus b x plus c")[$x^(2) + b x + c$]? Undo FOIL. #linebreak() 
Is it of the form #math.equation(block: false, alt: "a x squared plus b x plus c")[$a x^(2) + b x + c$]? #linebreak() 
- If #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "c")[$c$] are squares, check if it fits the trinomial square pattern.
- Use the trial and error or “ac” method.
- If it has more than three terms: #linebreak() 
Use the grouping method.
+ Check.
- Is it factored completely?
- Do the factors multiply back to the original polynomial?
]

Remember, a polynomial is completely factored if, other than monomials, its factors are prime!

#examplebox("Example 1")[][
Factor completely: #math.equation(block: false, alt: "4 x to the power 5 plus 12 x to the power 4")[$4 x^(5) + 12 x^(4)$].

#solutionbox[
#math.equation(block: true, alt: "Is there a GCF?, Yes, 4 x to the power 4 ., 4 x to the power 5 plus 12 x to the power 4; Factor out the GCF., 4 x to the power 4 open parenthesis x plus 3 close parenthesis; In the parentheses, is it a binomial, a; trinomial, or are there more than three terms?, Binomial.; Is it a sum?, Yes.; Of squares? Of cubes?, No.; Check.; Is the expression factored completely?, Yes.; Multiply.; 4 x to the power 4 open parenthesis x plus 3 close parenthesis; 4 x to the power 4 times x plus 4 x to the power 4 times 3; 4 x to the power 5 plus 12 x to the power 4 ✓")[$"Is there a GCF?" &  &  & "Yes," #h(0.2em) 4 x^(4) . &  &  & 4 x^(5) + 12 x^(4) \  &  &  & "Factor out the GCF." &  &  & 4 x^(4) ( x + 3 ) \ "In the parentheses, is it a binomial, a" &  &  &  &  &  &  \ "trinomial, or are there more than three terms?" &  &  & "Binomial." &  &  &  \ #h(1em) "Is it a sum?" &  &  &  &  &  & "Yes." \ #h(1em) "Of squares? Of cubes?" &  &  &  &  &  & "No." \ "Check." &  &  &  &  &  &  \  \ #h(1em) "Is the expression factored completely?" &  &  &  &  &  & "Yes." \ #h(1em) "Multiply." &  &  &  &  &  &  \  \  \ #h(2.5em) 4 x^(4) ( x + 3 ) &  &  &  &  &  &  \ #h(2.5em) 4 x^(4) · x + 4 x^(4) · 3 &  &  &  &  &  &  \ #h(2.5em) 4 x^(5) + 12 x^(4) #h(0.2em) ✓ &  &  &  &  &  &$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "3 a to the power 4 plus 18 a cubed")[$3 a^(4) + 18 a^(3)$].

#solutionbox[
#math.equation(block: true, alt: "3 a cubed open parenthesis a plus 6 close parenthesis")[$3 a^(3) ( a + 6 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "45 b to the power 6 plus 27 b to the power 5")[$45 b^(6) + 27 b^(5)$].

#solutionbox[
#math.equation(block: true, alt: "9 b to the power 5 open parenthesis 5 b plus 3 close parenthesis")[$9 b^(5) ( 5 b + 3 )$]
]
]

#examplebox("Example 2")[][
Factor completely: #math.equation(block: false, alt: "12 x squared minus 11 x plus 2")[$12 x^(2) − 11 x + 2$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([], [], [#figure(figph[The image displays the quadratic expression 12x^2 - 11x + 2 in a clear, digital font on a white background. This is a standard polynomial expression.], alt: "The image displays the quadratic expression 12x^2 - 11x + 2 in a clear, digital font on a white background. This is a standard polynomial expression.", caption: none)]),
  [Is there a GCF?], [No.], [],
  [Is it a binomial, trinomial, or are #linebreak() there more than three terms?], [Trinomial.], [],
  [Are #emph[a] and #emph[c] perfect squares?], [No, #emph[a] = 12, #linebreak() not a perfect square.], [],
  [Use trial and error or the “ac” method. #linebreak() We will use trial and error here.], [], [#figure(figph[Factoring the quadratic expression 12x^2 - 11x + 2 by listing potential factors for the first and last terms, a common step in algebraic factorization.], alt: "Factoring the quadratic expression 12x^2 - 11x + 2 by listing potential factors for the first and last terms, a common step in algebraic factorization.", caption: none)],
))

#figure(figph[This table has the heading of 12 x squared minus 11 x plus 2 and gives the possible factors. The first column is labeled possible factors and the second column is labeled product. Four rows have not an option in the product column. This is explained by the text, “if the trinomial has no common factors, then neither factor can contain a common factor”. The last factors, 3 x - 2 in parentheses and 4 x - 1 in parentheses, give the product of 12 x squared minus 11 x plus 2.], alt: "This table has the heading of 12 x squared minus 11 x plus 2 and gives the possible factors. The first column is labeled possible factors and the second column is labeled product. Four rows have not an option in the product column. This is explained by the text, “if the trinomial has no common factors, then neither factor can contain a common factor”. The last factors, 3 x - 2 in parentheses and 4 x - 1 in parentheses, give the product of 12 x squared minus 11 x plus 2.", caption: none)

Check.

#math.equation(block: true, alt: "open parenthesis 3 x minus 2 close parenthesis open parenthesis 4 x minus 1 close parenthesis")[$#h(2.5em) ( 3 x − 2 ) ( 4 x − 1 )$]

#math.equation(block: true, alt: "12 x squared minus 3 x minus 8 x plus 2")[$#h(2.5em) 12 x^(2) − 3 x − 8 x + 2$]

#math.equation(block: true, alt: "12 x squared minus 11 x plus 2 ✓")[$#h(2.5em) 12 x^(2) − 11 x + 2 #h(0.2em) ✓$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "10 a squared minus 17 a plus 6")[$10 a^(2) − 17 a + 6$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 5 a minus 6 close parenthesis open parenthesis 2 a minus 1 close parenthesis")[$( 5 a − 6 ) ( 2 a − 1 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "8 x squared minus 18 x plus 9")[$8 x^(2) − 18 x + 9$].

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

#examplebox("Example 3")[][
Factor completely: #math.equation(block: false, alt: "g cubed plus 25 g")[$g^(3) + 25 g$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Is there a GCF?], [Yes, #emph[g].], [#math.equation(block: false, alt: "g cubed plus 25 g")[$g^(3) + 25 g$]]),
  [Factor out the GCF.], [], [#math.equation(block: false, alt: "g open parenthesis g squared plus 25 close parenthesis")[$g ( g^(2) + 25 )$]],
  [In the parentheses, is it a binomial, trinomial, #linebreak() or are there more than three terms?], [Binomial.], [],
  [Is it a sum ? Of squares?], [Yes.], [Sums of squares are prime.],
  [Check.], [], [],
  [Is the expression factored completely?], [Yes.], [],
  [Multiply. #linebreak() #math.equation(block: false, alt: "g open parenthesis g squared plus 25 close parenthesis; g cubed plus 25 g ✓")[$#h(2.5em) g ( g^(2) + 25 ) \ #h(2.5em) g^(3) + 25 g #h(0.2em) ✓$]], [], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "x cubed plus 36 x")[$x^(3) + 36 x$].

#solutionbox[
#math.equation(block: true, alt: "x open parenthesis x squared plus 36 close parenthesis")[$x ( x^(2) + 36 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "27 y squared plus 48")[$27 y^(2) + 48$].

#solutionbox[
#math.equation(block: true, alt: "3 open parenthesis 9 y squared plus 16 close parenthesis")[$3 ( 9 y^(2) + 16 )$]
]
]

#examplebox("Example 4")[][
Factor completely: #math.equation(block: false, alt: "12 y squared minus 75")[$12 y^(2) − 75$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Is there a GCF?], [Yes, 3.], [#math.equation(block: false, alt: "12 y squared minus 75")[$12 y^(2) − 75$]]),
  [Factor out the GCF.], [], [#math.equation(block: false, alt: "3 open parenthesis 4 y squared minus 25 close parenthesis")[$3 ( 4 y^(2) − 25 )$]],
  [In the parentheses, is it a binomial, trinomial, #linebreak() or are there more than three terms?], [Binomial.], [],
  [Is it a sum?], [No.], [],
  [Is it a difference? Of squares or cubes?], [Yes, squares.], [#math.equation(block: false, alt: "3 open parenthesis open parenthesis 2 y close parenthesis squared minus open parenthesis 5 close parenthesis squared close parenthesis")[$3 ( attach(( 2 y ), t: 2) − attach(( 5 ), t: 2) )$]],
  [Write as a product of conjugates.], [], [#math.equation(block: false, alt: "3 open parenthesis 2 y minus 5 close parenthesis open parenthesis 2 y plus 5 close parenthesis")[$3 ( 2 y − 5 ) ( 2 y + 5 )$]],
  [Check.], [], [],
  [Is the expression factored completely?], [Yes.], [],
  [Neither binomial is a difference of squares.], [], [],
  [Multiply. #linebreak() #math.equation(block: false, alt: "3 open parenthesis 2 y minus 5 close parenthesis open parenthesis 2 y plus 5 close parenthesis; 3 open parenthesis 4 y squared minus 25 close parenthesis; 12 y squared minus 75 ✓")[$#h(2.5em) 3 ( 2 y − 5 ) ( 2 y + 5 ) \ #h(2.5em) 3 ( 4 y^(2) − 25 ) \ #h(2.5em) 12 y^(2) − 75 #h(0.2em) ✓$]], [], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "16 x cubed minus 36 x")[$16 x^(3) − 36 x$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "27 y squared minus 48")[$27 y^(2) − 48$].

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

#examplebox("Example 5")[][
Factor completely: #math.equation(block: false, alt: "4 a squared minus 12 a b plus 9 b squared")[$4 a^(2) − 12 a b + 9 b^(2)$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Is there a GCF?], [No.], [#figure(figph[A mathematical expression showing the quadratic trinomial 4a^2 - 12ab + 9b^2, which is a perfect square equivalent to (2a - 3b)^2.], alt: "A mathematical expression showing the quadratic trinomial 4a^2 - 12ab + 9b^2, which is a perfect square equivalent to (2a - 3b)^2.", caption: none)]),
  [Is it a binomial, trinomial, or are there #linebreak() more terms?], [], [],
  [Trinomial with #math.equation(block: false, alt: "a not equal to 1")[$a ≠ 1$]. But the first term is a #linebreak()   perfect square.], [], [],
  [Is the last term a perfect square?], [Yes.], [#figure(figph[A mathematical expression showing the term (2a) squared, minus 12ab, plus the term (3b) squared.], alt: "A mathematical expression showing the term (2a) squared, minus 12ab, plus the term (3b) squared.", caption: none)],
  [Does it fit the pattern, #math.equation(block: false, alt: "a squared minus 2 a b plus b squared ?")[$a^(2) − 2 a b + b^(2) ?$]], [Yes.], [#figure(figph[The image shows the breakdown of the middle term -12ab in the expression (2a)^2 - 12ab + (3b)^2, revealing it as -2(2a)(3b) for a perfect square trinomial.], alt: "The image shows the breakdown of the middle term -12ab in the expression (2a)^2 - 12ab + (3b)^2, revealing it as -2(2a)(3b) for a perfect square trinomial.", caption: none)],
  [Write it as a square.], [], [#figure(figph[The image shows the mathematical expression (2a - 3b)^2, written in black characters on a white background. This represents a binomial squared, specifically the square of the difference between 2a and 3b.], alt: "The image shows the mathematical expression (2a - 3b)^2, written in black characters on a white background. This represents a binomial squared, specifically the square of the difference between 2a and 3b.", caption: none)],
  [Check your answer.], [], [],
  [Is the expression factored completely?], [], [],
  [Yes.], [], [],
  [The binomial is not a difference of squares.], [], [],
  [Multiply.], [], [],
  [#math.equation(block: false, alt: "open parenthesis 2 a minus 3 b close parenthesis squared")[$attach(( 2 a − 3 b ), t: 2)$]], [], [],
  [#math.equation(block: false, alt: "open parenthesis 2 a close parenthesis squared minus 2 times 2 a times 3 b plus open parenthesis 3 b close parenthesis squared")[$attach(( 2 a ), t: 2) − 2 ⋅ 2 a ⋅ 3 b + attach(( 3 b ), t: 2)$]], [], [],
  [#math.equation(block: false, alt: "4 a squared minus 12 a b plus 9 b squared ✓")[$4 a^(2) − 12 a b + 9 b^(2) ✓$]], [], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "4 x squared plus 20 x y plus 25 y squared")[$4 x^(2) + 20 x y + 25 y^(2)$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "9 m squared plus 42 m n plus 49 n squared")[$9 m^(2) + 42 m n + 49 n^(2)$].

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 3 m plus 7 n close parenthesis squared")[$attach(( 3 m + 7 n ), t: 2)$]
]
]

#examplebox("Example 6")[][
Factor completely: #math.equation(block: false, alt: "6 y squared minus 18 y minus 60")[$6 y^(2) − 18 y − 60$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Is there a GCF?], [Yes, 6.], [#math.equation(block: false, alt: "6 y squared minus 18 y minus 60")[$6 y^(2) − 18 y − 60$]]),
  [Factor out the GCF.], [], [#math.equation(block: false, alt: "6 open parenthesis y squared minus 3 y minus 10 close parenthesis")[$6 ( y^(2) − 3 y − 10 )$]],
  [In the parentheses, is it a binomial, trinomial, #linebreak() or are there more terms?], [Trinomial with leading coefficient 1.], [],
  [“Undo” FOIL.], [#math.equation(block: false, alt: "6 open parenthesis y close parenthesis open parenthesis y close parenthesis")[$6 ( y #h(1em) ) ( y #h(1em) )$]], [#math.equation(block: false, alt: "6 open parenthesis y plus 2 close parenthesis open parenthesis y minus 5 close parenthesis")[$6 ( y + 2 ) ( y − 5 )$]],
  [Check your answer.], [], [],
  [Is the expression factored completely?], [], [Yes.],
  [Neither binomial is a difference of squares.], [], [],
  [Multiply. #linebreak() #math.equation(block: false, alt: "6 open parenthesis y plus 2 close parenthesis open parenthesis y minus 5 close parenthesis; 6 open parenthesis y squared minus 5 y plus 2 y minus 10 close parenthesis; 6 open parenthesis y squared minus 3 y minus 10 close parenthesis; 6 y squared minus 18 y minus 60 ✓")[$#h(2.5em) 6 ( y + 2 ) ( y − 5 ) \ #h(2.5em) 6 ( y^(2) − 5 y + 2 y − 10 ) \ #h(2.5em) 6 ( y^(2) − 3 y − 10 ) \ #h(2.5em) 6 y^(2) − 18 y − 60 #h(0.2em) ✓$]], [], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "8 y squared plus 16 y minus 24")[$8 y^(2) + 16 y − 24$].

#solutionbox[
#math.equation(block: true, alt: "8 open parenthesis y minus 1 close parenthesis open parenthesis y plus 3 close parenthesis")[$8 ( y − 1 ) ( y + 3 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "5 u squared minus 15 u minus 270")[$5 u^(2) − 15 u − 270$]. #linebreak()  #linebreak()

#solutionbox[
#math.equation(block: true, alt: "5 open parenthesis u minus 9 close parenthesis open parenthesis u plus 6 close parenthesis")[$5 ( u − 9 ) ( u + 6 )$]
]
]

#examplebox("Example 7")[][
Factor completely: #math.equation(block: false, alt: "24 x cubed plus 81")[$24 x^(3) + 81$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Is there a GCF?], [Yes, 3.], [#math.equation(block: false, alt: "24 x cubed plus 81")[$24 x^(3) + 81$]]),
  [Factor it out.], [], [#math.equation(block: false, alt: "3 open parenthesis 8 x cubed plus 27 close parenthesis")[$3 ( 8 x^(3) + 27 )$]],
  [In the parentheses, is it a binomial, trinomial, #linebreak() or are there more than three terms?], [Binomial.], [],
  [Is it a sum or difference?], [Sum.], [],
  [Of squares or cubes?], [Sum of cubes.], [#figure(figph[A mathematical expression 3((2x)^3 + (3)^3) is shown, with a red hint 'a^3 + b^3' above the sum of cubes, indicating the formula for the sum of two cubes.], alt: "A mathematical expression 3((2x)^3 + (3)^3) is shown, with a red hint 'a^3 + b^3' above the sum of cubes, indicating the formula for the sum of two cubes.", caption: none)],
  [Write it using the sum of cubes pattern.], [], [#figure(figph[A mathematical expression demonstrating the sum of cubes formula: 3 \* (a + b) \* (a^2 - ab + b^2), where a = 2x and b = 3. This simplifies to 3 \* ((2x)^3 + 3^3).], alt: "A mathematical expression demonstrating the sum of cubes formula: 3 * (a + b) * (a^2 - ab + b^2), where a = 2x and b = 3. This simplifies to 3 * ((2x)^3 + 3^3).", caption: none)],
  [Is the expression factored completely?], [Yes.], [#math.equation(block: false, alt: "3 open parenthesis 2 x plus 3 close parenthesis open parenthesis 4 x squared minus 6 x plus 9 close parenthesis")[$3 ( 2 x + 3 ) ( 4 x^(2) − 6 x + 9 )$]],
  [Check by multiplying.], [], [We leave the check to you.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "250 m cubed plus 432")[$250 m^(3) + 432$].

#solutionbox[
#math.equation(block: true, alt: "2 open parenthesis 5 m plus 6 close parenthesis open parenthesis 25 m squared minus 30 m plus 36 close parenthesis")[$2 ( 5 m + 6 ) ( 25 m^(2) − 30 m + 36 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "81 q cubed plus 192")[$81 q^(3) + 192$].

#solutionbox[
#math.equation(block: true, alt: "3 open parenthesis 3 q plus 4 close parenthesis open parenthesis 9 q squared minus 12 q plus 16 close parenthesis")[$3 ( 3 q + 4 ) ( 9 q^(2) − 12 q + 16 )$]
]
]

#examplebox("Example 8")[][
Factor completely: #math.equation(block: false, alt: "2 x to the power 4 minus 32")[$2 x^(4) − 32$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Is there a GCF?], [Yes, 2.], [#math.equation(block: false, alt: "2 x to the power 4 minus 32")[$2 x^(4) − 32$]]),
  [Factor it out.], [], [#math.equation(block: false, alt: "2 open parenthesis x to the power 4 minus 16 close parenthesis")[$2 ( x^(4) − 16 )$]],
  [In the parentheses, is it a binomial, trinomial, #linebreak() or are there more than three terms?], [Binomial.], [],
  [Is it a sum or difference?], [Yes.], [],
  [Of squares or cubes?], [Difference of squares.], [#math.equation(block: false, alt: "2 open parenthesis open parenthesis x squared close parenthesis squared minus open parenthesis 4 close parenthesis squared close parenthesis")[$2 ( attach(( x^(2) ), t: 2) − attach(( 4 ), t: 2) )$]],
  [Write it as a product of conjugates.], [], [#math.equation(block: false, alt: "2 open parenthesis x squared minus 4 close parenthesis open parenthesis x squared plus 4 close parenthesis")[$2 ( x^(2) − 4 ) ( x^(2) + 4 )$]],
  [The first binomial is again a difference of squares.], [], [#math.equation(block: false, alt: "2 open parenthesis open parenthesis x close parenthesis squared minus open parenthesis 2 close parenthesis squared close parenthesis open parenthesis x squared plus 4 close parenthesis")[$2 ( attach(( x ), t: 2) − attach(( 2 ), t: 2) ) ( x^(2) + 4 )$]],
  [Write it as a product of conjugates.], [], [#math.equation(block: false, alt: "2 open parenthesis x minus 2 close parenthesis open parenthesis x plus 2 close parenthesis open parenthesis x squared plus 4 close parenthesis")[$2 ( x − 2 ) ( x + 2 ) ( x^(2) + 4 )$]],
  [Is the expression factored completely?], [Yes.], [],
  [None of these binomials is a difference of squares.], [], [],
  [Check your answer.], [], [],
  [Multiply. #linebreak() #math.equation(block: false, alt: "2 open parenthesis x minus 2 close parenthesis open parenthesis x plus 2 close parenthesis open parenthesis x squared plus 4 close parenthesis; 2 open parenthesis x squared minus 4 close parenthesis open parenthesis x squared plus 4 close parenthesis; 2 open parenthesis x to the power 4 minus 16 close parenthesis; 2 x to the power 4 minus 32 ✓")[$#h(3em) 2 ( x − 2 ) ( x + 2 ) ( x^(2) + 4 ) \ #h(3em) 2 ( x^(2) − 4 ) ( x^(2) + 4 ) \ #h(3em) 2 ( x^(4) − 16 ) \ #h(3em) 2 x^(4) − 32 ✓$]], [], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "4 a to the power 4 minus 64")[$4 a^(4) − 64$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "7 y to the power 4 minus 7")[$7 y^(4) − 7$].

#solutionbox[
#math.equation(block: true, alt: "7 open parenthesis y squared plus 1 close parenthesis open parenthesis y minus 1 close parenthesis open parenthesis y plus 1 close parenthesis")[$7 ( y^(2) + 1 ) ( y − 1 ) ( y + 1 )$]
]
]

#examplebox("Example 9")[][
Factor completely: #math.equation(block: false, alt: "3 x squared plus 6 b x minus 3 a x minus 6 a b")[$3 x^(2) + 6 b x − 3 a x − 6 a b$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Is there a GCF?], [Yes, 3.], [#math.equation(block: false, alt: "3 x squared plus 6 b x minus 3 a x minus 6 a b")[$3 x^(2) + 6 b x − 3 a x − 6 a b$]]),
  [Factor out the GCF.], [], [#math.equation(block: false, alt: "3 open parenthesis x squared plus 2 b x minus a x minus 2 a b close parenthesis")[$3 ( x^(2) + 2 b x − a x − 2 a b )$]],
  [In the parentheses, is it a binomial, trinomial, #linebreak() or are there more terms?], [More than 3 terms.], [],
  [Use grouping.], [], [#math.equation(block: false, alt: "3 [ x open parenthesis x plus 2 b close parenthesis minus a open parenthesis x plus 2 b close parenthesis ]; 3 open parenthesis x plus 2 b close parenthesis open parenthesis x minus a close parenthesis")[$3 [ x ( x + 2 b ) − a ( x + 2 b ) ] \ 3 ( x + 2 b ) ( x − a )$]],
  [Check your answer.], [], [],
  [Is the expression factored completely? Yes.], [], [],
  [Multiply. #linebreak() #math.equation(block: false, alt: "3 open parenthesis x plus 2 b close parenthesis open parenthesis x minus a close parenthesis; 3 open parenthesis x squared minus a x plus 2 b x minus 2 a b close parenthesis; 3 x squared minus 3 a x plus 6 b x minus 6 a b ✓")[$#h(3em) 3 ( x + 2 b ) ( x − a ) \ #h(3em) 3 ( x^(2) − a x + 2 b x − 2 a b ) \ #h(3em) 3 x^(2) − 3 a x + 6 b x − 6 a b #h(0.2em) ✓$]], [], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "6 x squared minus 12 x c plus 6 b x minus 12 b c")[$6 x^(2) − 12 x c + 6 b x − 12 b c$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "16 x squared plus 24 x y minus 4 x minus 6 y")[$16 x^(2) + 24 x y − 4 x − 6 y$].

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

#examplebox("Example 10")[][
Factor completely: #math.equation(block: false, alt: "10 x squared minus 34 x minus 24")[$10 x^(2) − 34 x − 24$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Is there a GCF?], [Yes, 2.], [#math.equation(block: false, alt: "10 x squared minus 34 x minus 24")[$10 x^(2) − 34 x − 24$]]),
  [Factor out the GCF.], [], [#math.equation(block: false, alt: "2 open parenthesis 5 x squared minus 17 x minus 12 close parenthesis")[$2 ( 5 x^(2) − 17 x − 12 )$]],
  [In the parentheses, is it a binomial, trinomial, #linebreak() or are there more than three terms?], [Trinomial with #math.equation(block: false, alt: "a not equal to 1 .")[$a ≠ 1 .$]], [],
  [Use trial and error or the “ac” method.], [], [#math.equation(block: false, alt: "2 open parenthesis 5 x squared minus 17 x −12 close parenthesis; 2 open parenthesis 5 x plus 3 close parenthesis open parenthesis x minus 4 close parenthesis")[$2 ( 5 x^(2) − 17 x −12 ) \ 2 ( 5 x + 3 ) ( x − 4 )$]],
  [Check your answer. Is the expression factored #linebreak() completely? Yes.], [], [],
  [Multiply. #linebreak() #math.equation(block: false, alt: "2 open parenthesis 5 x plus 3 close parenthesis open parenthesis x minus 4 close parenthesis; 2 open parenthesis 5 x squared minus 20 x plus 3 x minus 12 close parenthesis; 2 open parenthesis 5 x squared minus 17 x minus 12 close parenthesis; 10 x squared minus 34 x minus 24 ✓")[$#h(3em) 2 ( 5 x + 3 ) ( x − 4 ) \ #h(3em) 2 ( 5 x^(2) − 20 x + 3 x − 12 ) \ #h(3em) 2 ( 5 x^(2) − 17 x − 12 ) \ #h(3em) 10 x^(2) − 34 x − 24 #h(0.2em) ✓$]], [], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "4 p squared minus 16 p plus 12")[$4 p^(2) − 16 p + 12$].

#solutionbox[
#math.equation(block: true, alt: "4 open parenthesis p minus 1 close parenthesis open parenthesis p minus 3 close parenthesis")[$4 ( p − 1 ) ( p − 3 )$]
]
]

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

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


=== Key Concepts

- #strong[General Strategy for Factoring Polynomials]
- #strong[How to Factor Polynomials]
+ Is there a greatest common factor? Factor it out.
+ Is the polynomial a binomial, trinomial, or are there more than three terms?
- If it is a binomial: #linebreak() 
Is it a sum?
- Of squares? Sums of squares do not factor.
- Of cubes? Use the sum of cubes pattern.


Is it a difference? #linebreak() 
- Of squares? Factor as the product of conjugates.
- Of cubes? Use the difference of cubes pattern.
- If it is a trinomial: #linebreak() 
Is it of the form #math.equation(block: false, alt: "x squared plus b x plus c")[$x^(2) + b x + c$]? Undo FOIL. #linebreak() 
Is it of the form #math.equation(block: false, alt: "a x squared plus b x plus c")[$a x^(2) + b x + c$]?
- If ‘a’ and ‘c’ are squares, check if it fits the trinomial square pattern.
- Use the trial and error or ‘ac’ method.
- If it has more than three terms: #linebreak() 
Use the grouping method.
+ Check. Is it factored completely? Do the factors multiply back to the original polynomial?


==== Practice Makes Perfect

#strong[Recognize and Use the Appropriate Method to Factor a Polynomial Completely]

In the following exercises, factor completely.

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

#solutionbox[
#math.equation(block: true, alt: "5 x cubed open parenthesis 2 x plus 7 close parenthesis")[$5 x^(3) ( 2 x + 7 )$]
]

#math.equation(block: true, alt: "18 p to the power 6 plus 24 p cubed")[$18 p^(6) + 24 p^(3)$]

#math.equation(block: true, alt: "y squared plus 10 y minus 39")[$y^(2) + 10 y − 39$]

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

#math.equation(block: true, alt: "b squared minus 17 b plus 60")[$b^(2) − 17 b + 60$]

#math.equation(block: true, alt: "2 n squared plus 13 n minus 7")[$2 n^(2) + 13 n − 7$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 2 n minus 1 close parenthesis open parenthesis n plus 7 close parenthesis")[$( 2 n − 1 ) ( n + 7 )$]
]

#math.equation(block: true, alt: "8 x squared minus 9 x minus 3")[$8 x^(2) − 9 x − 3$]

#math.equation(block: true, alt: "a to the power 5 plus 9 a cubed")[$a^(5) + 9 a^(3)$]

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

#math.equation(block: true, alt: "75 m cubed plus 12 m")[$75 m^(3) + 12 m$]

#math.equation(block: true, alt: "121 r squared minus s squared")[$121 r^(2) − s^(2)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 11 r minus s close parenthesis open parenthesis 11 r plus s close parenthesis")[$( 11 r − s ) ( 11 r + s )$]
]

#math.equation(block: true, alt: "49 b squared minus 36 a squared")[$49 b^(2) − 36 a^(2)$]

#math.equation(block: true, alt: "8 m squared minus 32")[$8 m^(2) − 32$]

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

#math.equation(block: true, alt: "36 q squared minus 100")[$36 q^(2) − 100$]

#math.equation(block: true, alt: "25 w squared minus 60 w plus 36")[$25 w^(2) − 60 w + 36$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 5 w minus 6 close parenthesis squared")[$attach(( 5 w − 6 ), t: 2)$]
]

#math.equation(block: true, alt: "49 b squared minus 112 b plus 64")[$49 b^(2) − 112 b + 64$]

#math.equation(block: true, alt: "m squared plus 14 m n plus 49 n squared")[$m^(2) + 14 m n + 49 n^(2)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis m plus 7 n close parenthesis squared")[$attach(( m + 7 n ), t: 2)$]
]

#math.equation(block: true, alt: "64 x squared plus 16 x y plus y squared")[$64 x^(2) + 16 x y + y^(2)$]

#math.equation(block: true, alt: "7 b squared plus 7 b minus 42")[$7 b^(2) + 7 b − 42$]

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

#math.equation(block: true, alt: "3 n squared plus 30 n plus 72")[$3 n^(2) + 30 n + 72$]

#math.equation(block: true, alt: "3 x cubed minus 81")[$3 x^(3) − 81$]

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

#math.equation(block: true, alt: "5 t cubed minus 40")[$5 t^(3) − 40$]

#math.equation(block: true, alt: "k to the power 4 minus 16")[$k^(4) − 16$]

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

#math.equation(block: true, alt: "m to the power 4 minus 81")[$m^(4) − 81$]

#math.equation(block: true, alt: "15 p q minus 15 p plus 12 q minus 12")[$15 p q − 15 p + 12 q − 12$]

#solutionbox[
#math.equation(block: true, alt: "3 open parenthesis 5 p plus 4 close parenthesis open parenthesis q minus 1 close parenthesis")[$3 ( 5 p + 4 ) ( q − 1 )$]
]

#math.equation(block: true, alt: "12 a b minus 6 a plus 10 b minus 5")[$12 a b − 6 a + 10 b − 5$]

#math.equation(block: true, alt: "4 x squared plus 40 x plus 84")[$4 x^(2) + 40 x + 84$]

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

#math.equation(block: true, alt: "5 q squared minus 15 q minus 90")[$5 q^(2) − 15 q − 90$]

#math.equation(block: true, alt: "u to the power 5 plus u squared")[$u^(5) + u^(2)$]

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

#math.equation(block: true, alt: "5 n cubed plus 320")[$5 n^(3) + 320$]

#math.equation(block: true, alt: "4 c squared plus 20 c d plus 81 d squared")[$4 c^(2) + 20 c d + 81 d^(2)$]

#solutionbox[
prime
]

#math.equation(block: true, alt: "25 x squared plus 35 x y plus 49 y squared")[$25 x^(2) + 35 x y + 49 y^(2)$]

#math.equation(block: true, alt: "10 m to the power 4 minus 6250")[$10 m^(4) − 6250$]

#solutionbox[
#math.equation(block: true, alt: "10 open parenthesis m minus 5 close parenthesis open parenthesis m plus 5 close parenthesis open parenthesis m squared plus 25 close parenthesis")[$10 ( m − 5 ) ( m + 5 ) ( m^(2) + 25 )$]
]

#math.equation(block: true, alt: "3 v to the power 4 minus 768")[$3 v^(4) − 768$]


==== Everyday Math

#strong[Watermelon drop] A springtime tradition at the University of California San Diego is the Watermelon Drop, where a watermelon is dropped from the seventh story of Urey Hall.

+ ⓐ The binomial #math.equation(block: false, alt: "−16 t squared plus 80")[$−16 t^(2) + 80$] gives the height of the watermelon #math.equation(block: false, alt: "t")[$t$] seconds after it is dropped. Factor the greatest common factor from this binomial.
+ ⓑ If the watermelon is thrown down with initial velocity 8 feet per second, its height after #math.equation(block: false, alt: "t")[$t$] seconds is given by the trinomial #math.equation(block: false, alt: "−16 t squared minus 8 t plus 80")[$−16 t^(2) − 8 t + 80$]. Completely factor this trinomial.

#solutionbox[
ⓐ #math.equation(block: false, alt: "−16 open parenthesis t squared minus 5 close parenthesis")[$−16 ( t^(2) − 5 )$] ⓑ #math.equation(block: false, alt: "−8 open parenthesis 2 t plus 5 close parenthesis open parenthesis t minus 2 close parenthesis")[$−8 ( 2 t + 5 ) ( t − 2 )$]
]

#strong[Pumpkin drop] A fall tradition at the University of California San Diego is the Pumpkin Drop, where a pumpkin is dropped from the eleventh story of Tioga Hall.

+ ⓐ The binomial #math.equation(block: false, alt: "−16 t squared plus 128")[$−16 t^(2) + 128$] gives the height of the pumpkin #emph[t] seconds after it is dropped. Factor the greatest common factor from this binomial.
+ ⓑ If the pumpkin is thrown down with initial velocity 32 feet per second, its height after #math.equation(block: false, alt: "t")[$t$] seconds is given by the trinomial #math.equation(block: false, alt: "−16 t squared minus 32 t plus 128")[$−16 t^(2) − 32 t + 128$]. Completely factor this trinomial.


==== Writing Exercises

The difference of squares #math.equation(block: false, alt: "y to the power 4 minus 625")[$y^(4) − 625$] can be factored as #math.equation(block: false, alt: "open parenthesis y squared minus 25 close parenthesis open parenthesis y squared plus 25 close parenthesis")[$( y^(2) − 25 ) ( y^(2) + 25 )$]. But it is not #emph[completely] factored. What more must be done to completely factor it?

#solutionbox[
Answer may vary.
]

Of all the factoring methods covered in this chapter (GCF, grouping, undo FOIL, ‘ac’ method, special products) which is the easiest for you? Which is the hardest? Explain your answers.


==== 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 row states “recognize and use the appropriate method to factor a polynomial completely”. 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 row states “recognize and use the appropriate method to factor a polynomial completely”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.", caption: none)

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?