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== 6.4#h(0.6em)General Strategy for Factoring Polynomials


=== 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. The following chart summarizes all the factoring methods we have covered, and outlines a strategy you should use when factoring polynomials.

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

#figure(figph[This chart shows the general strategies for factoring polynomials. It shows ways to find GCF of binomials, trinomials and polynomials with more than 3 terms. For binomials, we have difference of squares: a squared minus b squared equals a minus b, a plus b; sum of squares do not factor; sub of cubes: a cubed plus b cubed equals open parentheses a plus b close parentheses open parentheses a squared minus ab plus b squared close parentheses; difference of cubes: a cubed minus b cubed equals open parentheses a minus b close parentheses open parentheses a squared plus ab plus b squared close parentheses. For trinomials, we have x squared plus bx plus c where we put x as a term in each factor and we have a squared plus bx plus c. Here, if a and c are squares, we have a plus b whole squared equals a squared plus 2 ab plus b squared and a minus b whole squared equals a squared minus 2 ab plus b squared. If a and c are not squares, we use the ac method. For polynomials with more than 3 terms, we use grouping.], alt: "This chart shows the general strategies for factoring polynomials. It shows ways to find GCF of binomials, trinomials and polynomials with more than 3 terms. For binomials, we have difference of squares: a squared minus b squared equals a minus b, a plus b; sum of squares do not factor; sub of cubes: a cubed plus b cubed equals open parentheses a plus b close parentheses open parentheses a squared minus ab plus b squared close parentheses; difference of cubes: a cubed minus b cubed equals open parentheses a minus b close parentheses open parentheses a squared plus ab plus b squared close parentheses. For trinomials, we have x squared plus bx plus c where we put x as a term in each factor and we have a squared plus bx plus c. Here, if a and c are squares, we have a plus b whole squared equals a squared plus 2 ab plus b squared and a minus b whole squared equals a squared minus 2 ab plus b squared. If a and c are not squares, we use the ac method. For polynomials with more than 3 terms, we use grouping.", caption: none)
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Use a general strategy for factoring polynomials.]

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


If it is a trinomial:
- Is it of the form #math.equation(block: false, alt: "x squared plus b x plus c ?")[$x^(2) + b x + c ?$] Undo FOIL.
- 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 #emph[a] and #emph[c] are squares, check if it fits the trinomial square pattern. #linebreak() Use the trial and error or “ac” method.


If it has more than three terms:
- Use the grouping method.
+ Check. #linebreak() Is it factored completely? #linebreak() Do the factors multiply back to the original polynomial?
]

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

#examplebox("Example 1")[][
Factor completely: #math.equation(block: false, alt: "7 x cubed minus 21 x squared minus 70 x .")[$7 x^(3) − 21 x^(2) − 70 x .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "7 x cubed minus 21 x squared minus 70 x")[$7 x^(3) − 21 x^(2) − 70 x$]]),
  [Is there a GCF? Yes, #math.equation(block: false, alt: "7 x")[$7 x$].], [],
  [Factor out the GCF.], [#math.equation(block: false, alt: "7 x open parenthesis x squared minus 3 x minus 10 close parenthesis")[$7 x ( x^(2) − 3 x − 10 )$]],
  [In the parentheses, is it a binomial, trinomial, or are there more terms?], [],
  [Trinomial with leading coefficient 1.], [],
  [“Undo” FOIL.], [#math.equation(block: false, alt: "7 x open parenthesis x close parenthesis open parenthesis x close parenthesis")[$7 x ( x #h(1.6em) ) ( x #h(1.4em) )$]],
  [], [#math.equation(block: false, alt: "7 x open parenthesis x plus 2 close parenthesis open parenthesis x minus 5 close parenthesis")[$7 x ( x + 2 ) ( x − 5 )$]],
  [Is the expression factored completely? Yes.], [],
  [Neither binomial can be factored.], [],
  [Check your answer.], [],
  [Multiply.], [],
  [#math.equation(block: false, alt: "7 x open parenthesis x plus 2 close parenthesis open parenthesis x minus 5 close parenthesis")[$7 x ( x + 2 ) ( x − 5 )$]], [],
  [#math.equation(block: false, alt: "7 x open parenthesis x squared minus 5 x plus 2 x minus 10 close parenthesis")[$7 x ( x^(2) − 5 x + 2 x − 10 )$]], [],
  [#math.equation(block: false, alt: "7 x open parenthesis x squared minus 3 x minus 10 close parenthesis")[$7 x ( x^(2) − 3 x − 10 )$]], [],
  [#math.equation(block: false, alt: "7 x cubed minus 21 x squared minus 70 x ✓")[$7 x^(3) − 21 x^(2) − 70 x ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "8 y cubed plus 16 y squared minus 24 y .")[$8 y^(3) + 16 y^(2) − 24 y .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "5 y cubed minus 15 y squared minus 270 y .")[$5 y^(3) − 15 y^(2) − 270 y .$]

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

Be careful when you are asked to factor a binomial as there are several options!

#examplebox("Example 2")[][
Factor completely: #math.equation(block: false, alt: "24 y squared minus 150 .")[$24 y^(2) − 150 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "24 y squared minus 150")[$24 y^(2) − 150$]]),
  [Is there a GCF? Yes, 6.], [],
  [Factor out the GCF.], [#math.equation(block: false, alt: "6 open parenthesis 4 y squared minus 25 close parenthesis")[$6 ( 4 y^(2) − 25 )$]],
  [In the parentheses, is it a binomial, trinomial 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: "6 open parenthesis open parenthesis 2 y close parenthesis squared minus open parenthesis 5 close parenthesis squared close parenthesis")[$6 ( attach(( 2 y ), t: 2) − attach(( 5 ), t: 2) )$]],
  [Write as a product of conjugates.], [#math.equation(block: false, alt: "6 open parenthesis 2 y minus 5 close parenthesis open parenthesis 2 y plus 5 close parenthesis")[$6 ( 2 y − 5 ) ( 2 y + 5 )$]],
  [#math.equation(block: false, alt: "Is the expression factored completely?")[$#h(2em) "Is the expression factored completely?"$]], [],
  [#math.equation(block: false, alt: "Neither binomial can be factored.")[$#h(2em) "Neither binomial can be factored."$]], [],
  [Check:], [],
  [#math.equation(block: false, alt: "Multiply.")[$#h(2em) "Multiply."$]], [],
  [#math.equation(block: false, alt: "6 open parenthesis 2 y minus 5 close parenthesis open parenthesis 2 y plus 5 close parenthesis")[$6 ( 2 y − 5 ) ( 2 y + 5 )$]], [],
  [#math.equation(block: false, alt: "6 open parenthesis 4 y squared minus 25 close parenthesis")[$6 ( 4 y^(2) − 25 )$]], [],
  [#math.equation(block: false, alt: "24 y squared minus 150 ✓")[$24 y^(2) − 150 ✓$]], [],
))
]
]

#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")[$3 ( 3 y − 4 ) ( 3 y + 4 )$]
]
]

The next example can be factored using several methods. Recognizing the trinomial squares pattern will make your work easier.

#examplebox("Example 3")[][
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: 2, align: left, inset: 6pt,
  [], [#math.equation(block: false, alt: "4 a squared minus 12 a b plus 9 b squared")[$4 a^(2) − 12 a b + 9 b^(2)$]],
  [Is there a GCF? No.], [],
  [Is it a binomial, trinomial, or are there more terms?], [],
  [Trinomial with #math.equation(block: false, alt: "a not equal to 1")[$a ≠ 1$]. But the first term is a perfect square.], [],
  [Is the last term a perfect square? Yes.], [#math.equation(block: false, alt: "open parenthesis 2 a close parenthesis squared minus 12 a b plus open parenthesis 3 b close parenthesis squared")[$attach(( 2 a ), t: 2) − 12 a b + attach(( 3 b ), t: 2)$]],
  [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.], [#math.equation(block: false, alt: "open parenthesis 2 a close parenthesis squared sub ↘ −12 a b plus −2 open parenthesis 2 a close parenthesis open parenthesis 3 b close parenthesis sub ↙ open parenthesis 3 b close parenthesis squared")[$attach(( 2 a ), t: 2) "↘" limits(−12 a b +)_(−2 ( 2 a ) ( 3 b )) "↙" attach(( 3 b ), t: 2)$]],
  [Write it as a square.], [#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: "Is the expression factored completely? Yes.")[$#h(2em) "Is the expression factored completely? Yes."$]], [],
  [#math.equation(block: false, alt: "The binomial cannot be factored.")[$#h(2em) "The binomial cannot be factored."$]], [],
  [Check your answer.], [],
  [#math.equation(block: false, alt: "Multiply.")[$#h(2em) "Multiply."$]], [],
  [#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 x squared minus 24 x y plus 16 y squared .")[$9 x^(2) − 24 x y + 16 y^(2) .$]

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

Remember, sums of squares do not factor, but sums of cubes do!

#examplebox("Example 4")[][
Factor completely #math.equation(block: false, alt: "12 x cubed y squared plus 75 x y squared .")[$12 x^(3) y^(2) + 75 x y^(2) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  [], [#math.equation(block: false, alt: "12 x cubed y squared plus 75 x y squared")[$12 x^(3) y^(2) + 75 x y^(2)$]],
  [Is there a GCF? Yes, #math.equation(block: false, alt: "3 x y squared")[$3 x y^(2)$].], [],
  [Factor out the GCF.], [#math.equation(block: false, alt: "3 x y squared open parenthesis 4 x squared plus 25 close parenthesis")[$3 x y^(2) ( 4 x^(2) + 25 )$]],
  [In the parentheses, is it a binomial, trinomial, or are there more than three terms? Binomial.], [],
  [Is it a sum? Of squares? Yes.], [Sums of squares are prime.],
  [#math.equation(block: false, alt: "Is the expression factored completely? Yes.")[$#h(2em) "Is the expression factored completely? Yes."$]], [],
  [Check:], [],
  [#math.equation(block: false, alt: "Multiply.")[$#h(2em) "Multiply."$]], [],
  [#math.equation(block: false, alt: "3 x y squared open parenthesis 4 x squared plus 25 close parenthesis")[$3 x y^(2) ( 4 x^(2) + 25 )$]], [],
  [#math.equation(block: false, alt: "12 x cubed y squared plus 75 x y squared ✓")[$12 x^(3) y^(2) + 75 x y^(2) ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "50 x cubed y plus 72 x y .")[$50 x^(3) y + 72 x y .$]

#solutionbox[
#math.equation(block: true, alt: "2 x y open parenthesis 25 x squared plus 36 close parenthesis")[$2 x y ( 25 x^(2) + 36 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "27 x y cubed plus 48 x y .")[$27 x y^(3) + 48 x y .$]

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

When using the sum or difference of cubes pattern, being careful with the signs.

#examplebox("Example 5")[][
Factor completely: #math.equation(block: false, alt: "24 x cubed plus 81 y cubed .")[$24 x^(3) + 81 y^(3) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Is there a GCF? Yes, 3.], [#figure(figph[The mathematical expression 24x³ + 81y³ is displayed.], alt: "The mathematical expression 24x³ + 81y³ is displayed.", caption: none)]),
  [Factor it out.], [#figure(figph[A mathematical expression reads 3 multiplied by the sum of 8x cubed and 27y cubed, enclosed in parentheses. The expression is 3(8x^3 + 27y^3) on a white background.], alt: "A mathematical expression reads 3 multiplied by the sum of 8x cubed and 27y cubed, enclosed in parentheses. The expression is 3(8x^3 + 27y^3) on a white background.", caption: none)],
  [In the parentheses, is it a binomial, trinomial, #linebreak() of are there more than three terms? Binomial.], [],
  [Is it a sum or difference? Sum.], [],
  [Of squares or cubes? Sum of cubes.], [#figure(figph[Mathematical expression: 3((2x) ^3 + (3y)^3). Red a^3 and b^3 above the terms suggest the sum of cubes formula, a^3 + b^3.], alt: "Mathematical expression: 3((2x) ^3 + (3y)^3). Red a^3 and b^3 above the terms suggest the sum of cubes formula, a^3 + b^3.", caption: none)],
  [Write it using the sum of cubes pattern.], [#figure(figph[A mathematical expression 3(2x + 3y)((2x)^2 - 2x \* 3y + (3y)^3) is shown. Red annotations 'a', 'b', 'a^2', 'ab', 'b^2' suggest an algebraic identity, but the last term is cubed, not squared.], alt: "A mathematical expression 3(2x + 3y)((2x)^2 - 2x * 3y + (3y)^3) is shown. Red annotations 'a', 'b', 'a^2', 'ab', 'b^2' suggest an algebraic identity, but the last term is cubed, not squared.", caption: none)],
  [Is the expression factored completely? Yes.], [#figure(figph[A mathematical expression showing the factorization 3(2x + 3y)(4x^2 - 6xy + 9y^2), which simplifies to 3(8x^3 + 27y^3).], alt: "A mathematical expression showing the factorization 3(2x + 3y)(4x^2 - 6xy + 9y^2), which simplifies to 3(8x^3 + 27y^3).", caption: none)],
  [Check by multiplying.], [],
))
]
]

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

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "2 p cubed plus 54 q cubed .")[$2 p^(3) + 54 q^(3) .$]

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

#examplebox("Example 6")[][
Factor completely: #math.equation(block: false, alt: "3 x to the power 5 y minus 48 x y .")[$3 x^(5) y − 48 x y .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "3 x to the power 5 y minus 48 x y")[$3 x^(5) y − 48 x y$]]),
  [Is there a GCF? Factor out #math.equation(block: false, alt: "3 x y")[$3 x y$]], [#math.equation(block: false, alt: "3 x y open parenthesis x to the power 4 minus 16 close parenthesis")[$3 x y ( x^(4) − 16 )$]],
  [Is the binomial a sum or difference? Of squares or cubes? #linebreak() 
Write it as a difference of squares.], [#math.equation(block: false, alt: "3 x y open parenthesis open parenthesis x squared close parenthesis squared minus open parenthesis 4 close parenthesis squared close parenthesis")[$3 x y ( attach(( x^(2) ), t: 2) − attach(( 4 ), t: 2) )$]],
  [Factor it as a product of conjugates], [#math.equation(block: false, alt: "3 x y open parenthesis x squared minus 4 close parenthesis open parenthesis x squared plus 4 close parenthesis")[$3 x y ( x^(2) − 4 ) ( x^(2) + 4 )$]],
  [The first binomial is again a difference of squares.], [#math.equation(block: false, alt: "3 x y 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")[$3 x y ( attach(( x ), t: 2) − attach(( 2 ), t: 2) ) ( x^(2) + 4 )$]],
  [Factor it as a product of conjugates.], [#math.equation(block: false, alt: "3 x y open parenthesis x minus 2 close parenthesis open parenthesis x plus 2 close parenthesis open parenthesis x squared plus 4 close parenthesis")[$3 x y ( x − 2 ) ( x + 2 ) ( x^(2) + 4 )$]],
  [Is the expression factored completely? Yes.], [],
  [Check your answer.], [],
  [Multiply.], [],
  [#math.equation(block: false, alt: "3 x y open parenthesis x minus 2 close parenthesis open parenthesis x plus 2 close parenthesis open parenthesis x squared plus 4 close parenthesis")[$3 x y ( x − 2 ) ( x + 2 ) ( x^(2) + 4 )$]], [],
  [#math.equation(block: false, alt: "3 x y open parenthesis x squared minus 4 close parenthesis open parenthesis x squared plus 4 close parenthesis")[$3 x y ( x^(2) − 4 ) ( x^(2) + 4 )$]], [],
  [#math.equation(block: false, alt: "3 x y open parenthesis x to the power 4 minus 16 close parenthesis")[$3 x y ( x^(4) − 16 )$]], [],
  [#math.equation(block: false, alt: "3 x to the power 5 y minus 48 x y ✓")[$3 x^(5) y − 48 x y ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "4 a to the power 5 b minus 64 a b .")[$4 a^(5) b − 64 a b .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "7 x y to the power 5 minus 7 x y .")[$7 x y^(5) − 7 x y .$]

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

#examplebox("Example 7")[][
Factor completely: #math.equation(block: false, alt: "4 x squared plus 8 b x minus 4 a x minus 8 a b .")[$4 x^(2) + 8 b x − 4 a x − 8 a b .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "4 x squared plus 8 b x minus 4 a x minus 8 a b")[$4 x^(2) + 8 b x − 4 a x − 8 a b$]]),
  [Is there a GCF? Factor out the GCF, 4.], [#math.equation(block: false, alt: "4 open parenthesis x squared plus 2 b x minus a x minus 2 a b close parenthesis")[$4 ( x^(2) + 2 b x − a x − 2 a b )$]],
  [There are four terms. Use grouping.], [#math.equation(block: false, alt: "4 [ x open parenthesis x plus 2 b close parenthesis minus a open parenthesis x plus 2 b close parenthesis ]; 4 open parenthesis x plus 2 b close parenthesis open parenthesis x minus a close parenthesis")[$4 [ x ( x + 2 b ) − a ( x + 2 b ) ] \ 4 ( x + 2 b ) ( x − a )$]],
  [Is the expression factored completely? Yes.], [],
  [Check your answer. #linebreak() Multiply. #linebreak() #math.equation(block: false, alt: "4 open parenthesis x plus 2 b close parenthesis open parenthesis x minus a close parenthesis; 4 open parenthesis x squared minus a x plus 2 b x minus 2 a b close parenthesis; 4 x squared plus 8 b x minus 4 a x minus 8 a b ✓")[$#h(4em) 4 ( x + 2 b ) ( x − a ) \ 4 ( x^(2) − a x + 2 b x − 2 a b ) \ 4 x^(2) + 8 b x − 4 a x − 8 a b ✓$]], [],
))
]
]

#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")[$2 ( 4 x − 1 ) ( 2 x + 3 y )$]
]
]

Taking out the complete GCF in the first step will always make your work easier.

#examplebox("Example 8")[][
Factor completely: #math.equation(block: false, alt: "40 x squared y plus 44 x y minus 24 y .")[$40 x^(2) y + 44 x y − 24 y .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "40 x squared y plus 44 x y minus 24 y")[$40 x^(2) y + 44 x y − 24 y$]]),
  [Is there a GCF? Factor out the GCF, #math.equation(block: false, alt: "4 y")[$4 y$].], [#math.equation(block: false, alt: "4 y open parenthesis 10 x squared plus 11 x minus 6 close parenthesis")[$4 y ( 10 x^(2) + 11 x − 6 )$]],
  [Factor the trinomial with #math.equation(block: false, alt: "a not equal to 1")[$a ≠ 1$].], [#math.equation(block: false, alt: "4 y open parenthesis 10 x squared plus 11 x minus 6 close parenthesis")[$4 y ( 10 x^(2) + 11 x − 6 )$]],
  [], [#math.equation(block: false, alt: "4 y open parenthesis 5 x minus 2 close parenthesis open parenthesis 2 x plus 3 close parenthesis")[$4 y ( 5 x − 2 ) ( 2 x + 3 )$]],
  [Is the expression factored completely? Yes.], [],
  [Check your answer.], [],
  [Multiply.], [],
  [#math.equation(block: false, alt: "4 y open parenthesis 5 x minus 2 close parenthesis open parenthesis 2 x plus 3 close parenthesis")[$4 y ( 5 x − 2 ) ( 2 x + 3 )$]], [],
  [#math.equation(block: false, alt: "4 y open parenthesis 10 x squared plus 11 x minus 6 close parenthesis")[$4 y ( 10 x^(2) + 11 x − 6 )$]], [],
  [#math.equation(block: false, alt: "40 x squared y plus 44 x y minus 24 y ✓")[$40 x^(2) y + 44 x y − 24 y ✓$]], [],
))
]
]

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

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

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

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

When we have factored a polynomial with four terms, most often we separated it into two groups of two terms. Remember that we can also separate it into a trinomial and then one term.

#examplebox("Example 9")[][
Factor completely: #math.equation(block: false, alt: "9 x squared minus 12 x y plus 4 y squared minus 49 .")[$9 x^(2) − 12 x y + 4 y^(2) − 49 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "9 x squared minus 12 x y plus 4 y squared minus 49")[$9 x^(2) − 12 x y + 4 y^(2) − 49$]]),
  [Is there a GCF? No.], [],
  [With more than 3 terms, use grouping. Last 2 terms have no GCF. Try grouping first 3 terms.], [#math.equation(block: false, alt: "9 x squared minus 12 x y plus 4 y squared minus 49")[$9 x^(2) − 12 x y + 4 y^(2) − 49$]],
  [Factor the trinomial with #math.equation(block: false, alt: "a not equal to 1")[$a ≠ 1$]. But the first term is a perfect square.], [],
  [Is the last term of the trinomial a perfect square? Yes.], [#math.equation(block: false, alt: "open parenthesis 3 x close parenthesis squared minus 12 x y plus open parenthesis 2 y close parenthesis squared minus 49")[$attach(( 3 x ), t: 2) − 12 x y + attach(( 2 y ), t: 2) − 49$]],
  [Does the trinomial 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.], [#math.equation(block: false, alt: "open parenthesis 3 x close parenthesis squared sub ↘ −12 x y plus −2 open parenthesis 3 x close parenthesis open parenthesis 2 y close parenthesis sub ↙ open parenthesis 2 y close parenthesis squared minus 49")[$attach(( 3 x ), t: 2) "↘" limits(−12 x y +)_(−2 ( 3 x ) ( 2 y )) "↙" attach(( 2 y ), t: 2) − 49$]],
  [Write the trinomial as a square.], [#math.equation(block: false, alt: "open parenthesis 3 x minus 2 y close parenthesis squared minus 49")[$attach(( 3 x − 2 y ), t: 2) − 49$]],
  [Is this binomial a sum or difference? Of squares or cubes? Write it as a difference of squares.], [#math.equation(block: false, alt: "open parenthesis 3 x minus 2 y close parenthesis squared minus 7 squared")[$attach(( 3 x − 2 y ), t: 2) − 7^(2)$]],
  [Write it as a product of conjugates.], [#math.equation(block: false, alt: "open parenthesis open parenthesis 3 x minus 2 y close parenthesis minus 7 close parenthesis open parenthesis open parenthesis 3 x minus 2 y close parenthesis plus 7 close parenthesis")[$( ( 3 x − 2 y ) − 7 ) ( ( 3 x − 2 y ) + 7 )$]],
  [], [#math.equation(block: false, alt: "open parenthesis 3 x minus 2 y minus 7 close parenthesis open parenthesis 3 x minus 2 y plus 7 close parenthesis")[$( 3 x − 2 y − 7 ) ( 3 x − 2 y + 7 )$]],
  [Is the expression factored completely? Yes.], [],
  [Check your answer.], [],
  [Multiply.], [],
  [#math.equation(block: false, alt: "open parenthesis 3 x minus 2 y minus 7 close parenthesis open parenthesis 3 x minus 2 y plus 7 close parenthesis")[$( 3 x − 2 y − 7 ) ( 3 x − 2 y + 7 )$]], [],
  [#math.equation(block: false, alt: "9 x squared minus 6 x y minus 21 x minus 6 x y plus 4 y squared plus 14 y plus 21 x minus 14 y minus 49")[$9 x^(2) − 6 x y − 21 x − 6 x y + 4 y^(2) + 14 y + 21 x − 14 y − 49$]], [],
  [#math.equation(block: false, alt: "9 x squared minus 12 x y plus 4 y squared minus 49 ✓")[$9 x^(2) − 12 x y + 4 y^(2) − 49 ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "4 x squared minus 12 x y plus 9 y squared minus 25 .")[$4 x^(2) − 12 x y + 9 y^(2) − 25 .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "16 x squared minus 24 x y plus 9 y squared minus 64 .")[$16 x^(2) − 24 x y + 9 y^(2) − 64 .$]

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


=== Key Concepts

#figure(figph[This chart shows the general strategies for factoring polynomials. It shows ways to find GCF of binomials, trinomials and polynomials with more than 3 terms. For binomials, we have difference of squares: a squared minus b squared equals a minus b, a plus b; sum of squares do not factor; sub of cubes: a cubed plus b cubed equals open parentheses a plus b close parentheses open parentheses a squared minus ab plus b squared close parentheses; difference of cubes: a cubed minus b cubed equals open parentheses a minus b close parentheses open parentheses a squared plus ab plus b squared close parentheses. For trinomials, we have x squared plus bx plus c where we put x as a term in each factor and we have a squared plus bx plus c. Here, if a and c are squares, we have a plus b whole squared equals a squared plus 2 ab plus b squared and a minus b whole squared equals a squared minus 2 ab plus b squared. If a and c are not squares, we use the ac method. For polynomials with more than 3 terms, we use grouping.], alt: "This chart shows the general strategies for factoring polynomials. It shows ways to find GCF of binomials, trinomials and polynomials with more than 3 terms. For binomials, we have difference of squares: a squared minus b squared equals a minus b, a plus b; sum of squares do not factor; sub of cubes: a cubed plus b cubed equals open parentheses a plus b close parentheses open parentheses a squared minus ab plus b squared close parentheses; difference of cubes: a cubed minus b cubed equals open parentheses a minus b close parentheses open parentheses a squared plus ab plus b squared close parentheses. For trinomials, we have x squared plus bx plus c where we put x as a term in each factor and we have a squared plus bx plus c. Here, if a and c are squares, we have a plus b whole squared equals a squared plus 2 ab plus b squared and a minus b whole squared equals a squared minus 2 ab plus b squared. If a and c are not squares, we use the ac method. For polynomials with more than 3 terms, we use grouping.", caption: none)

- #strong[How to use a general strategy for factoring polynomials.]
+ Is there a greatest common factor? #linebreak() Factor it out.
+ Is the polynomial a binomial, trinomial, or are there more than three terms? #linebreak() If it is a binomial: #linebreak() Is it a sum? #linebreak() Of squares? Sums of squares do not factor. #linebreak() Of cubes? Use the sum of cubes pattern. #linebreak() Is it a difference? #linebreak() Of squares? Factor as the product of conjugates. #linebreak() Of cubes? Use the difference of cubes pattern. #linebreak() 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 #emph[a] and #emph[c] are squares, check if it fits the trinomial square pattern. #linebreak() Use the trial and error or “ac” method. #linebreak() If it has more than three terms: #linebreak() Use the grouping method.
+ Check. #linebreak() Is it factored completely? #linebreak() 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: "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: "30 n squared plus 30 n plus 72")[$30 n^(2) + 30 n + 72$]

#math.equation(block: true, alt: "3 x to the power 4 y minus 81 x y")[$3 x^(4) y − 81 x y$]

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

#math.equation(block: true, alt: "4 x to the power 5 y minus 32 x squared y")[$4 x^(5) y − 32 x^(2) y$]

#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: "5 x to the power 5 y squared minus 80 x y squared")[$5 x^(5) y^(2) − 80 x y^(2)$]

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

#math.equation(block: true, alt: "48 x to the power 5 y squared minus 243 x y squared")[$48 x^(5) y^(2) − 243 x y^(2)$]

#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: "4 u to the power 5 plus 4 u squared v cubed")[$4 u^(5) + 4 u^(2) v^(3)$]

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

#math.equation(block: true, alt: "5 m to the power 4 n plus 320 m n to the power 4")[$5 m^(4) n + 320 m n^(4)$]

#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")[$3 v^(4) − 768$]

#math.equation(block: true, alt: "36 x squared y plus 15 x y minus 6 y")[$36 x^(2) y + 15 x y − 6 y$]

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

#math.equation(block: true, alt: "60 x squared y minus 75 x y plus 30 y")[$60 x^(2) y − 75 x y + 30 y$]

#math.equation(block: true, alt: "8 x cubed minus 27 y cubed")[$8 x^(3) − 27 y^(3)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 2 x minus 3 y close parenthesis open parenthesis 4 x squared plus 6 x y plus 9 y squared close parenthesis")[$( 2 x − 3 y ) ( 4 x^(2) + 6 x y + 9 y^(2) )$]
]

#math.equation(block: true, alt: "64 x cubed plus 125 y cubed")[$64 x^(3) + 125 y^(3)$]

#math.equation(block: true, alt: "y to the power 6 minus 1")[$y^(6) − 1$]

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

#math.equation(block: true, alt: "y to the power 6 plus 1")[$y^(6) + 1$]

#math.equation(block: true, alt: "9 x squared minus 6 x y plus y squared minus 49")[$9 x^(2) − 6 x y + y^(2) − 49$]

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

#math.equation(block: true, alt: "16 x squared minus 24 x y plus 9 y squared minus 64")[$16 x^(2) − 24 x y + 9 y^(2) − 64$]

#math.equation(block: true, alt: "open parenthesis 3 x plus 1 close parenthesis squared minus 6 open parenthesis 3 x plus 1 close parenthesis plus 9")[$attach(( 3 x + 1 ), t: 2) − 6 ( 3 x + 1 ) + 9$]

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

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


==== Writing Exercises

Explain what it mean to factor a polynomial completely.

#solutionbox[
Answers will vary.
]

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 completely factored. What more must be done to completely factor.

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.

#solutionbox[
Answers will vary.
]

Create three factoring problems that would be good test questions to measure your knowledge of factoring. Show the solutions.


==== Self Check

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

#figure(figph[This table has 4 columns, 1 row and a header row. The header row labels each column: I can, confidently, with some help and no, I don’t get it. The first column has the following statement: recognize and use the appropriate method to factor a polynomial completely. The remaining columns are blank.], alt: "This table has 4 columns, 1 row and a header row. The header row labels each column: I can, confidently, with some help and no, I don’t get it. The first column has the following statement: recognize and use the appropriate method to factor a polynomial completely. The remaining columns are blank.", caption: none)

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?