Loading [MathJax]/extensions/TeX/boldsymbol.js
Skip to main content
permalink

Section 10.6 Factoring Strategies

permalink
permalink
Figure 10.6.1. Alternative Video Lesson
permalink

Subsection 10.6.1 Factoring Strategies

permalinkDeciding which method to use when factoring a random polynomial can seem like a daunting task. Understanding all of the techniques that we have learned and how they fit together can be done using a decision tree.

permalink
Figure 10.6.2. Factoring Decision Tree

permalinkUsing the decision tree can guide us when we are given an expression to factor.

permalink
Example 10.6.3.

Factor the expression 4k^2+12k-40 completely.

Explanation

Start by noting that the GCF is \(4\text{.}\) Factoring this out, we get

\begin{equation*} 4k^2+12k-40=4\left(k^2+3k-10\right)\text{.} \end{equation*}

Following the decision tree, we now have a trinomial where the leading coefficient is \(1\) and we need to look for factors of \(-10\) that add to \(3\text{.}\) We find that \(-2\) and \(5\) work. So, the full factorization is:

\begin{align*} 4k^2+12k-40\amp=4\left(k^2+3k-10\right)\\ \amp=4(k-2)(k+5) \end{align*}
permalink
Example 10.6.4.

Factor the expression 64d^2+144d+81 completely.

Explanation

Start by noting that the GCF is \(1\text{,}\) and there is no GCF to factor out. We continue along the decision tree for a trinomial. Notice that both \(64\) and \(81\) are perfect squares and that this expression might factor using the pattern \(A^2+2AB+B^2=(A+B)^2\text{.}\) To find \(A\) and \(B\text{,}\) take the square roots of the first and last terms, so \(A=8d\) and \(B=9\text{.}\) We have to check that the middle term is correct: since \(2AB=2(8d)(9)=144d\) matches our middle term, the expression must factor as

\begin{equation*} 64d^2+144d+81=(8d+9)^2\text{.} \end{equation*}
permalink
Example 10.6.5.

Factor the expression 10x^2y-12xy^2 completely.

Explanation

Start by noting that the GCF is \(2xy\text{.}\) Factoring this out, we get

\begin{equation*} 10x^2y-12xy^2=2xy(5x-6y)\text{.} \end{equation*}

Since we have a binomial inside the parentheses, the only options on the decision tree for a binomial involve squares or cubes. Since there are none, we conclude that \(2xy(5x-6y)\) is the complete factorization.

permalink
Example 10.6.6.

Factor the expression 9b^2-25y^2 completely.

Explanation

Start by noting that the GCF is \(1\text{,}\) and there is no GCF to factor out. We continue along the decision tree for a binomial and notice that we now have a difference of squares, \(A^2-B^2=(A-B)(A+B)\text{.}\) To find the values for \(A\) and \(B\) that fit the patterns, just take the square roots. So \(A=3b\) since \((3b)^2=9b^2\) and \(B=5y\) since \((5y)^2=25y^2\text{.}\) So, the expression must factor as

\begin{equation*} 9b^2-25y^2=(3b-5y)(3b+5y)\text{.} \end{equation*}
permalink
Example 10.6.7.

Factor the expression 24w^3+6w^2-9w completely.

Explanation

Start by noting that the GCF is \(3w\text{.}\) Factoring this out, we get

\begin{equation*} 24w^3+6w^2-9w=3w\left(8w^2+2w-3\right)\text{.} \end{equation*}

Following the decision tree, we now have a trinomial inside the parentheses where \(a\neq1\text{.}\) We should try the AC method because neither \(8\) nor \(-3\) are perfect squares. In this case, \(ac=-24\) and we must find two factors of \(-24\) that add to be \(2\text{.}\) The numbers \(6\) and \(-4\) work in this case. The rest of the factoring process is:

\begin{align*} 24w^3+6w^2-9w\amp=3w\left(8w^2\overbrace{{}+2w}-3\right)\\ \amp=3w\left(8w^2\overbrace{{}+6w-4w}-3\right)\\ \amp=3w\left(\left(8w^2+6w\right)+\left(-4w-3\right)\right)\\ \amp=3w\left(2w\highlight{(4w+3)}-1\highlight{(4w+3)}\right)\\ \amp=3w\highlight{(4w+3)}(2w-1) \end{align*}
permalink
Example 10.6.8.

Factor the expression -6xy+9y+2x-3 completely.

Explanation

Start by noting that the GCF is \(1\text{,}\) and there is no GCF to factor out. We continue along the decision tree. Since we have a four-term polynomial, we should try to factor by grouping. The full process is:

\begin{align*} -6xy+9y+2x-3\amp=(-6xy+9y)+(2x-3)\\ \amp=-3y\highlight{(2x-3)}+1\highlight{(2x-3)}\\ \amp=\highlight{(2x-3)}(-3y+1)\\ \end{align*}

Note that the negative sign in front of the \(3y\) can be factored out if you wish. That would look like:

\begin{align*} \amp=-(2x-3)(3y-1) \end{align*}
permalink
Example 10.6.9.

Factor the expression 4w^3-20w^2+24w completely.

Explanation

Start by noting that the GCF is \(4w\text{.}\) Factoring this out, we get

\begin{equation*} 4w^3-20w^2+24w=4w\left(w^2-5w+6\right)\text{.} \end{equation*}

Following the decision tree, we now have a trinomial with \(a=1\) inside the parentheses. So, we can look for factors of \(6\) that add up to \(-5\text{.}\) Since \(-3\) and \(-2\) fit the requirements, the full factorization is:

\begin{align*} 4w^3-20w^2+24w\amp=4w\left(w^2-5w+6\right)\\ \amp=4w(w-3)(w-2) \end{align*}
permalink
Example 10.6.10.

Factor the expression 9-24y+16y^2 completely.

Explanation

Start by noting that the GCF is \(1\text{,}\) and there is no GCF to factor out. Continue along the decision tree. We now have a trinomial where both the first term, \(9\text{,}\) and last term, \(16y^2\text{,}\) look like perfect squares. To use the perfect squares difference pattern, \(A^2-2AB+B^2=(A-B)^2\text{,}\) recall that we need to mentally take the square roots of these two terms to find \(A\) and \(B\text{.}\) So, \(A=3\) since \(3^2=9\text{,}\) and \(B=4y\) since \((4y)^2=16y^2\text{.}\) Now we have to check that \(2AB\) matches \(24y\text{:}\)

\begin{equation*} 2AB=2(3)(4y)=24y\text{.} \end{equation*}

So the full factorization is:

\begin{equation*} 9-24y+16y^2=(3-4y)^2\text{.} \end{equation*}
permalink
Example 10.6.11.

Factor the expression 9-25y+16y^2 completely.

Explanation

Start by noting that the GCF is \(1\text{,}\) and there is no GCF to factor out. Since we now have a trinomial where both the first term and last term are perfect squares in exactly the same way as in Example 10. However, we cannot apply the perfect squares method to this problem because it worked when \(2AB=24y\text{.}\) Since our middle term is \(25y\text{,}\) we can be certain that it won't be a perfect square.

Continuing on with the decision tree, our next option is to use the AC method. You might be tempted to rearrange the order of the terms, but that is unnecessary. In this case, \(ac=144\) and we need to come up with two factors of \(144\) that add to be \(-25\text{.}\) After a brief search, we conclude that those values are \(-16\) and \(-9\text{.}\) The remainder of the factorization is:

\begin{align*} 9\overbrace{{}-25y}+16y^2\amp=9\overbrace{{}-16y-9y}+16y^2\\ \amp=\left(9-16y\right)+\left(-9y+16y^2\right)\\ \amp=1\highlight{\left(9-16y\right)}-y\highlight{\left(9+16y\right)}\\ \amp=\highlight{\left(9-16y\right)}(1-y) \end{align*}
permalink
Example 10.6.12.

Factor the expression 20x^4+13x^3-21x^2 completely.

Explanation

Start by noting that the GCF is \(x^2\text{.}\) Factoring this out, we get

\begin{equation*} 20x^4+13x^3-21x^2=x^2\left(20x^2+13x-21\right)\text{.} \end{equation*}

Following the decision tree, we now have a trinomial inside the parentheses where \(a\neq1\) and we should try the AC method. In this case, \(ac=-420\) and we need factors of \(-420\) that add to \(13\text{.}\)

Factor Pair Sum
\(1\cdot-420\) \(-419\)
\(2\cdot-210\) \(-208\)
\(3\cdot-140\) \(-137\)
\(4\cdot-105\) \(-101\)
Factor Pair Sum
\(5\cdot-84\) \(-79\)
\(6\cdot-70\) \(-64\)
\(7\cdot-60\) \(-53\)
\(10\cdot-42\) \(-32\)
Factor Pair Sum
\(12\cdot-35\) \(-23\)
\(14\cdot-30\) \(-16\)
\(15\cdot-28\) \(-13\)
\(20\cdot-21\) \(-1\)

In the table of the factor pairs of \(-420\) we find \(15+(-28)=-13\text{,}\) the opposite of what we want, so we want the opposite numbers: \(-15\) and \(28\text{.}\) The rest of the factoring process is shown:

\begin{align*} 20x^4+13x^3-21x^2\amp=x^2\left(20x^2\overbrace{{}+13x}-21\right)\\ \amp=x^2\left(20x^2\overbrace{-15x+28x}-21\right)\\ \amp=x^2\left(\left(20x^2-15x\right)+\left(28x-21\right)\right)\\ \amp=x^2\left(5x\highlight{(4x-3)}+7\highlight{(4x-3)}\right)\\ \amp=x^2\highlight{(4x-3)}(5x+7) \end{align*}
permalink

Reading Questions 10.6.2 Reading Questions

permalink
1.

Do you find a factoring flowchart helpful?

permalink

Exercises 10.6.3 Exercises

Strategies
permalink
1.

Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.

  • Factoring out a GCF

  • Factoring by grouping

  • Finding two numbers that multiply to the constant term and sum to the linear coefficient

  • The AC Method

  • Difference of Squares

  • Difference of Cubes

  • Sum of Cubes

  • Perfect Square Trinomial

  • None of the above

{64B^{2}-216Bb+140b^{2}}

permalink
2.

Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.

  • Factoring out a GCF

  • Factoring by grouping

  • Finding two numbers that multiply to the constant term and sum to the linear coefficient

  • The AC Method

  • Difference of Squares

  • Difference of Cubes

  • Sum of Cubes

  • Perfect Square Trinomial

  • None of the above

{6m^{6}+384m^{3}x^{3}}

permalink
3.

Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.

  • Factoring out a GCF

  • Factoring by grouping

  • Finding two numbers that multiply to the constant term and sum to the linear coefficient

  • The AC Method

  • Difference of Squares

  • Difference of Cubes

  • Sum of Cubes

  • Perfect Square Trinomial

  • None of the above

{49n^{3}-35n^{2}-28n+20}

permalink
4.

Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.

  • Factoring out a GCF

  • Factoring by grouping

  • Finding two numbers that multiply to the constant term and sum to the linear coefficient

  • The AC Method

  • Difference of Squares

  • Difference of Cubes

  • Sum of Cubes

  • Perfect Square Trinomial

  • None of the above

{4q^{4}-2916q}

permalink
5.

Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.

  • Factoring out a GCF

  • Factoring by grouping

  • Finding two numbers that multiply to the constant term and sum to the linear coefficient

  • The AC Method

  • Difference of Squares

  • Difference of Cubes

  • Sum of Cubes

  • Perfect Square Trinomial

  • None of the above

{9y^{2}-24y+16}

permalink
6.

Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.

  • Factoring out a GCF

  • Factoring by grouping

  • Finding two numbers that multiply to the constant term and sum to the linear coefficient

  • The AC Method

  • Difference of Squares

  • Difference of Cubes

  • Sum of Cubes

  • Perfect Square Trinomial

  • None of the above

{3r^{4}-39r^{3}+120r^{2}}

permalink
7.

Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.

  • Factoring out a GCF

  • Factoring by grouping

  • Finding two numbers that multiply to the constant term and sum to the linear coefficient

  • The AC Method

  • Difference of Squares

  • Difference of Cubes

  • Sum of Cubes

  • Perfect Square Trinomial

  • None of the above

{40x^{2}-408xa+432a^{2}}

permalink
8.

Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.

  • Factoring out a GCF

  • Factoring by grouping

  • Finding two numbers that multiply to the constant term and sum to the linear coefficient

  • The AC Method

  • Difference of Squares

  • Difference of Cubes

  • Sum of Cubes

  • Perfect Square Trinomial

  • None of the above

{b^{3}-b^{2}-2b+2}

permalink
9.

Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.

  • Factoring out a GCF

  • Factoring by grouping

  • Finding two numbers that multiply to the constant term and sum to the linear coefficient

  • The AC Method

  • Difference of Squares

  • Difference of Cubes

  • Sum of Cubes

  • Perfect Square Trinomial

  • None of the above

{54A^{6}-686A^{3}B^{3}}

permalink
10.

Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.

  • Factoring out a GCF

  • Factoring by grouping

  • Finding two numbers that multiply to the constant term and sum to the linear coefficient

  • The AC Method

  • Difference of Squares

  • Difference of Cubes

  • Sum of Cubes

  • Perfect Square Trinomial

  • None of the above

{63n-81}

permalink
11.

Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.

  • Factoring out a GCF

  • Factoring by grouping

  • Finding two numbers that multiply to the constant term and sum to the linear coefficient

  • The AC Method

  • Difference of Squares

  • Difference of Cubes

  • Sum of Cubes

  • Perfect Square Trinomial

  • None of the above

{32m^{3}-50mp^{2}}

permalink
12.

Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.

  • Factoring out a GCF

  • Factoring by grouping

  • Finding two numbers that multiply to the constant term and sum to the linear coefficient

  • The AC Method

  • Difference of Squares

  • Difference of Cubes

  • Sum of Cubes

  • Perfect Square Trinomial

  • None of the above

{n^{3}+64A^{3}}

permalink
13.

Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.

  • Factoring out a GCF

  • Factoring by grouping

  • Finding two numbers that multiply to the constant term and sum to the linear coefficient

  • The AC Method

  • Difference of Squares

  • Difference of Cubes

  • Sum of Cubes

  • Perfect Square Trinomial

  • None of the above

{210q^{3}t-168q^{3}-270q^{2}t+216q^{2}}

permalink
14.

Which factoring techniques/tools will be useful for factoring the polynomial below? Check all that apply.

  • Factoring out a GCF

  • Factoring by grouping

  • Finding two numbers that multiply to the constant term and sum to the linear coefficient

  • The AC Method

  • Difference of Squares

  • Difference of Cubes

  • Sum of Cubes

  • Perfect Square Trinomial

  • None of the above

{4y^{5}-2916y^{2}p^{3}}

Factoring
permalink
15.

Factor the given polynomial.

6x+6=

permalink
16.

Factor the given polynomial.

-3y-3=

permalink
17.

Factor the given polynomial.

36y^2+27=

permalink
18.

Factor the given polynomial.

{30r^{4}-12r^{3}+42r^{2}}=

permalink
19.

Factor the given polynomial.

{6r+12r^{2}+15r^{3}}=

permalink
20.

Factor the given polynomial.

{8xy+8y}=

permalink
21.

Factor the given polynomial.

{54x^{5}y^{8}-6x^{4}y^{8}+42x^{3}y^{8}}=

permalink
22.

Factor the given polynomial.

{t^{2}-2t+3t-6}=

permalink
23.

Factor the given polynomial.

{xy+2x+8y+16}=

permalink
24.

Factor the given polynomial.

{x^{3}-3+2x^{3}y-6y}=

permalink
25.

Factor the given polynomial.

{y^{2}-3y-4}=

permalink
26.

Factor the given polynomial.

{5y^{2}-2y-7}=

permalink
27.

Factor the given polynomial.

{2r^{2}y^{2}+3ry-9}=

permalink
28.

Factor the given polynomial.

{2r^{2}-6r+5}=

permalink
29.

Factor the given polynomial.

{4r^{2}-7r-2}=

permalink
30.

Factor the given polynomial.

{8t^{2}+22t+15}=

permalink
31.

Factor the given polynomial.

{12t^{2}-23t+10}=

permalink
32.

Factor the given polynomial.

{3x^{2}+10xr+3r^{2}}=

permalink
33.

Factor the given polynomial.

{3x^{2}-13xy+12y^{2}}=

permalink
34.

Factor the given polynomial.

{4y^{2}+3yt-7t^{2}}=

permalink
35.

Factor the given polynomial.

{8y^{2}+22yr+9r^{2}}=

permalink
36.

Factor the given polynomial.

{8r^{2}-18rx+9x^{2}}=

permalink
37.

Factor the given polynomial.

{12r^{2}-8r-4}=

permalink
38.

Factor the given polynomial.

{15r^{2}t^{2}-3rt-12}=

permalink
39.

Factor the given polynomial.

{10t^{9}+25t^{8}+15t^{7}}=

permalink
40.

Factor the given polynomial.

{6t^{10}-9t^{9}+3t^{8}}=

permalink
41.

Factor the given polynomial.

{6x^{2}+20xy+14y^{2}}=

permalink
42.

Factor the given polynomial.

{10x^{2}-34xy+12y^{2}}=

permalink
43.

Factor the given polynomial.

{y^{2}+9y+8}=

permalink
44.

Factor the given polynomial.

{y^{2}-6y+5}=

permalink
45.

Factor the given polynomial.

{r^{2}+10rx+16x^{2}}=

permalink
46.

Factor the given polynomial.

{r^{2}y^{2}-4ry-12}=

permalink
47.

Factor the given polynomial.

{r^{2}-10ry+24y^{2}}=

permalink
48.

Factor the given polynomial.

{4t^{2}x^{2}+12tx+8}=

permalink
49.

Factor the given polynomial.

{2t^{2}+8t-10}=

permalink
50.

Factor the given polynomial.

{3x^{4}+18x^{3}+24x^{2}}=

permalink
51.

Factor the given polynomial.

{7x^{9}-28x^{8}+21x^{7}}=

permalink
52.

Factor the given polynomial.

{2x^{2}y+6xy+4y}=

permalink
53.

Factor the given polynomial.

{2x^{2}y-18xy+28y}=

permalink
54.

Factor the given polynomial.

{2x^{2}y^{3}-10xy^{2}+8y}=

permalink
55.

Factor the given polynomial.

{x^{2}y^{2}+6x^{2}yz-7x^{2}z^{2}}=

permalink
56.

Factor the given polynomial.

{r^{2}+0.9r+0.2}=

permalink
57.

Factor the given polynomial.

{t^{2}-144}=

permalink
58.

Factor the given polynomial.

{t^{2}r^{2}-16}=

permalink
59.

Factor the given polynomial.

{9-x^{2}}=

permalink
60.

Factor the given polynomial.

{x^{4}-121}=

permalink
61.

Factor the given polynomial.

{y^{12}-49}=

permalink
62.

Factor the given polynomial.

{x^{6}-36y^{14}}=

permalink
63.

Factor the given polynomial.

{81y^{4}-16}=

permalink
64.

Factor the given polynomial.

{3r^{3}-147r}=

permalink
65.

Factor the given polynomial.

{r^{2}+4}=

permalink
66.

Factor the given polynomial.

{32-8t^{2}}=

permalink
67.

Factor the given polynomial.

{t^{2}+12t+36}=

permalink
68.

Factor the given polynomial.

{x^{2}-12xy+36y^{2}}=

permalink
69.

Factor the given polynomial.

{x^{2}-18x+81}=

permalink
70.

Factor the given polynomial.

{36y^{2}-12y+1}=

permalink
71.

Factor the given polynomial.

{y^{2}+18yx+81x^{2}}=

permalink
72.

Factor the given polynomial.

{9y^{2}+30yr+25r^{2}}=

permalink
73.

Factor the given polynomial.

{98r^{2}y^{2}+28ry+2}=

permalink
74.

Factor the given polynomial.

{4r^{10}+4r^{9}+r^{8}}=

permalink
75.

Factor the given polynomial.

{98t^{8}+28t^{7}+2t^{6}}=

permalink
76.

Factor the given polynomial.

{0.16t-t^{3}}=

permalink
77.

Factor the given polynomial.

{2x^{4}-162}=

permalink
78.

Factor the given polynomial.

{x^{2}-14x+49-64y^{2}}=