Learning Objectives
By the end of this section, you will be able to:
 Recognize and use the appropriate method to factor a polynomial completely
Be Prepared 7.15
Before you get started, take this readiness quiz.
Factor ${y}^{2}2y24$.
If you missed this problem, review Example 7.23.
Be Prepared 7.16
Factor $3{t}^{2}+17t+10$.
If you missed this problem, review Example 7.38.
Be Prepared 7.17
Factor $36{p}^{2}60p+25$.
If you missed this problem, review Example 7.42.
Be Prepared 7.18
Factor $5{x}^{2}80$.
If you missed this problem, review Example 7.52.
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. Factor polynomials. outlines a strategy you should use when factoring polynomials.
How To
Factor polynomials.
 Step 1.
Is there a greatest common factor?
 Factor it out.
 Step 2.
Is the polynomial a binomial, trinomial, or are there more than three terms?
 If it is a binomial:
Is it a sum? Of squares? Sums of squares do not factor.
 Of cubes? Use the sum of cubes pattern.
 Of squares? Factor as the product of conjugates.
 Of cubes? Use the difference of cubes pattern.
 If it is a trinomial:
Is it of the form ${x}^{2}+bx+c$? Undo FOIL.
Is it of the form $a{x}^{2}+bx+c$?
 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:
Use the grouping method.
 If it is a binomial:
 Step 3.
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!
Example 7.59
Factor completely: $4{x}^{5}+12{x}^{4}$.
Solution
$\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes,}\phantom{\rule{0.2em}{0ex}}4{x}^{4}.\hfill & & & \hfill 4{x}^{5}+12{x}^{4}\hfill \\ & & & \text{Factor out the GCF.}\hfill & & & \hfill 4{x}^{4}\left(x+3\right)\hfill \\ \text{In the parentheses, is it a binomial, a}\hfill & & & & & & \\ \text{trinomial, or are there more than three terms?}\hfill & & & \text{Binomial.}\hfill & & & \\ \phantom{\rule{1em}{0ex}}\text{Is it a sum?}\hfill & & & & & & \text{Yes.}\hfill \\ \phantom{\rule{1em}{0ex}}\text{Of squares? Of cubes?}\hfill & & & & & & \text{No.}\hfill \\ \text{Check.}\hfill & & & & & & \\ \\ \phantom{\rule{1em}{0ex}}\text{Is the expression factored completely?}\hfill & & & & & & \text{Yes.}\hfill \\ \phantom{\rule{1em}{0ex}}\text{Multiply.}\hfill & & & & & & \\ \\ \\ \phantom{\rule{2.5em}{0ex}}4{x}^{4}\left(x+3\right)\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}4{x}^{4}\xb7x+4{x}^{4}\xb73\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}4{x}^{5}+12{x}^{4}\phantom{\rule{0.2em}{0ex}}\u2713\hfill & & & & & & \end{array}$
Try It 7.117
Factor completely: $3{a}^{4}+18{a}^{3}$.
Try It 7.118
Factor completely: $45{b}^{6}+27{b}^{5}$.
Example 7.60
Factor completely: $12{x}^{2}11x+2$.
Solution
Is there a GCF?  No.  
Is it a binomial, trinomial, or are there more than three terms? 
Trinomial.  
Are a and c perfect squares?  No, a = 12, not a perfect square. 

Use trial and error or the “ac” method. We will use trial and error here. 
Check.
$\phantom{\rule{2.5em}{0ex}}\left(3x2\right)\left(4x1\right)$
$\phantom{\rule{2.5em}{0ex}}12{x}^{2}3x8x+2$
$\phantom{\rule{2.5em}{0ex}}12{x}^{2}11x+2\phantom{\rule{0.2em}{0ex}}\u2713$
Try It 7.119
Factor completely: $10{a}^{2}17a+6$.
Try It 7.120
Factor completely: $8{x}^{2}18x+9$.
Example 7.61
Factor completely: ${g}^{3}+25g$.
Solution
Is there a GCF?  Yes, g.  ${g}^{3}+25g$ 
Factor out the GCF.  $g\left({g}^{2}+25\right)$  
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. 
Check.  
Is the expression factored completely?  Yes.  
Multiply. $\begin{array}{}\\ \\ \\ \phantom{\rule{2.5em}{0ex}}g\left({g}^{2}+25\right)\hfill \\ \phantom{\rule{2.5em}{0ex}}{g}^{3}+25g\phantom{\rule{0.2em}{0ex}}\u2713\hfill \end{array}$ 
Try It 7.121
Factor completely: ${x}^{3}+36x$.
Try It 7.122
Factor completely: $27{y}^{2}+48$.
Example 7.62
Factor completely: $12{y}^{2}75$.
Solution
Is there a GCF?  Yes, 3.  $12{y}^{2}75$ 
Factor out the GCF.  $3\left(4{y}^{2}25\right)$  
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.  $3\left({\left(2y\right)}^{2}{\left(5\right)}^{2}\right)$ 
Write as a product of conjugates.  $3\left(2y5\right)\left(2y+5\right)$  
Check.  
Is the expression factored completely?  Yes.  
Neither binomial is a difference of squares.  
Multiply. $\begin{array}{}\\ \\ \phantom{\rule{2.5em}{0ex}}3\left(2y5\right)\left(2y+5\right)\hfill \\ \phantom{\rule{2.5em}{0ex}}3\left(4{y}^{2}25\right)\hfill \\ \phantom{\rule{2.5em}{0ex}}12{y}^{2}75\phantom{\rule{0.2em}{0ex}}\u2713\hfill \end{array}$ 
Try It 7.123
Factor completely: $16{x}^{3}36x$.
Try It 7.124
Factor completely: $27{y}^{2}48$.
Example 7.63
Factor completely: $4{a}^{2}12ab+9{b}^{2}$.
Solution
Is there a GCF?  No.  
Is it a binomial, trinomial, or are there more terms? 

Trinomial with $a\ne 1$. But the first term is a perfect square. 

Is the last term a perfect square?  Yes.  
Does it fit the pattern, ${a}^{2}2ab+{b}^{2}?$  Yes.  
Write it as a square.  
Check your answer.  
Is the expression factored completely?  
Yes.  
The binomial is not a difference of squares.  
Multiply.  
${(2a3b)}^{2}$  
${\left(2a\right)}^{2}2\cdot 2a\cdot 3b+{\left(3b\right)}^{2}$  
$4{a}^{2}12ab+9{b}^{2}\u2713$ 
Try It 7.125
Factor completely: $4{x}^{2}+20xy+25{y}^{2}$.
Try It 7.126
Factor completely: $9{m}^{2}+42mn+49{n}^{2}$.
Example 7.64
Factor completely: $6{y}^{2}18y60$.
Solution
Is there a GCF?  Yes, 6.  $6{y}^{2}18y60$ 
Factor out the GCF.  $6\left({y}^{2}3y10\right)$  
In the parentheses, is it a binomial, trinomial, or are there more terms? 
Trinomial with leading coefficient 1.  
“Undo” FOIL.  $6\left(y\phantom{\rule{1em}{0ex}}\right)\left(y\phantom{\rule{1em}{0ex}}\right)$  $6\left(y+2\right)\left(y5\right)$ 
Check your answer.  
Is the expression factored completely?  Yes.  
Neither binomial is a difference of squares.  
Multiply. $\begin{array}{}\\ \\ \phantom{\rule{2.5em}{0ex}}6\left(y+2\right)\left(y5\right)\hfill \\ \phantom{\rule{2.5em}{0ex}}6\left({y}^{2}5y+2y10\right)\hfill \\ \phantom{\rule{2.5em}{0ex}}6\left({y}^{2}3y10\right)\hfill \\ \phantom{\rule{2.5em}{0ex}}6{y}^{2}18y60\phantom{\rule{0.2em}{0ex}}\u2713\hfill \end{array}$ 
Try It 7.127
Factor completely: $8{y}^{2}+16y24$.
Try It 7.128
Factor completely: $5{u}^{2}15u270$.
Example 7.65
Factor completely: $24{x}^{3}+81$.
Solution
Is there a GCF?  Yes, 3.  $24{x}^{3}+81$ 
Factor it out.  $3(8{x}^{3}+27)$  
In the parentheses, is it a binomial, trinomial, or are there more than three terms? 
Binomial.  
Is it a sum or difference?  Sum.  
Of squares or cubes?  Sum of cubes.  
Write it using the sum of cubes pattern.  
Is the expression factored completely?  Yes.  $3(2x+3)(4{x}^{2}6x+9)$ 
Check by multiplying.  We leave the check to you. 
Try It 7.129
Factor completely: $250{m}^{3}+432$.
Try It 7.130
Factor completely: $81{q}^{3}+192$.
Example 7.66
Factor completely: $2{x}^{4}32$.
Solution
Is there a GCF?  Yes, 2.  $2{x}^{4}32$ 
Factor it out.  $2\left({x}^{4}16\right)$  
In the parentheses, is it a binomial, trinomial, or are there more than three terms? 
Binomial.  
Is it a sum or difference?  Yes.  
Of squares or cubes?  Difference of squares.  $2\left({\left({x}^{2}\right)}^{2}{\left(4\right)}^{2}\right)$ 
Write it as a product of conjugates.  $2\left({x}^{2}4\right)\left({x}^{2}+4\right)$  
The first binomial is again a difference of squares.  $2\left({\left(x\right)}^{2}{\left(2\right)}^{2}\right)\left({x}^{2}+4\right)$  
Write it as a product of conjugates.  $2\left(x2\right)\left(x+2\right)\left({x}^{2}+4\right)$  
Is the expression factored completely?  Yes.  
None of these binomials is a difference of squares.  
Check your answer.  
Multiply. $\begin{array}{}\\ \\ \\ \\ \phantom{\rule{3em}{0ex}}2(x2)(x+2)({x}^{2}+4)\hfill \\ \phantom{\rule{3em}{0ex}}2({x}^{2}4)({x}^{2}+4)\hfill \\ \phantom{\rule{3em}{0ex}}2({x}^{4}16)\hfill \\ \phantom{\rule{3em}{0ex}}2{x}^{4}32\u2713\hfill \end{array}$ 
Try It 7.131
Factor completely: $4{a}^{4}64$.
Try It 7.132
Factor completely: $7{y}^{4}7$.
Example 7.67
Factor completely: $3{x}^{2}+6bx3ax6ab$.
Solution
Is there a GCF?  Yes, 3.  $3{x}^{2}+6bx3ax6ab$ 
Factor out the GCF.  $3\left({x}^{2}+2bxax2ab\right)$  
In the parentheses, is it a binomial, trinomial, or are there more terms? 
More than 3 terms.  
Use grouping.  $\begin{array}{}\\ \hfill 3\left[x\left(x+2b\right)a\left(x+2b\right)\right]\hfill \\ \hfill 3\left(x+2b\right)\left(xa\right)\hfill \end{array}$  
Check your answer.  
Is the expression factored completely? Yes.  
Multiply. $\begin{array}{}\\ \\ \phantom{\rule{3em}{0ex}}3\left(x+2b\right)\left(xa\right)\hfill \\ \phantom{\rule{3em}{0ex}}3\left({x}^{2}ax+2bx2ab\right)\hfill \\ \phantom{\rule{3em}{0ex}}3{x}^{2}3ax+6bx6ab\phantom{\rule{0.2em}{0ex}}\u2713\hfill \end{array}$ 
Try It 7.133
Factor completely: $6{x}^{2}12xc+6bx12bc$.
Try It 7.134
Factor completely: $16{x}^{2}+24xy4x6y$.
Example 7.68
Factor completely: $10{x}^{2}34x24$.
Solution
Is there a GCF?  Yes, 2.  $10{x}^{2}34x24$ 
Factor out the GCF.  $2\left(5{x}^{2}17x12\right)$  
In the parentheses, is it a binomial, trinomial, or are there more than three terms? 
Trinomial with $a\ne 1.$  
Use trial and error or the “ac” method.  $\begin{array}{}\\ \hfill 2(5{x}^{2}17x\mathrm{12})\hfill \\ \hfill 2\left(5x+3\right)\left(x4\right)\hfill \end{array}$  
Check your answer. Is the expression factored completely? Yes. 

Multiply. $\begin{array}{}\\ \\ \phantom{\rule{3em}{0ex}}2\left(5x+3\right)\left(x4\right)\hfill \\ \phantom{\rule{3em}{0ex}}2\left(5{x}^{2}20x+3x12\right)\hfill \\ \phantom{\rule{3em}{0ex}}2\left(5{x}^{2}17x12\right)\hfill \\ \phantom{\rule{3em}{0ex}}10{x}^{2}34x24\phantom{\rule{0.2em}{0ex}}\u2713\hfill \end{array}$ 
Try It 7.135
Factor completely: $4{p}^{2}16p+12$.
Try It 7.136
Factor completely: $6{q}^{2}9q6$.
Section 7.5 Exercises
Practice Makes Perfect
Recognize and Use the Appropriate Method to Factor a Polynomial Completely
In the following exercises, factor completely.
$18{p}^{6}+24{p}^{3}$
${b}^{2}17b+60$
$8{x}^{2}9x3$
$75{m}^{3}+12m$
$49{b}^{2}36{a}^{2}$
$36{q}^{2}100$
$49{b}^{2}112b+64$
$64{x}^{2}+16xy+{y}^{2}$
$3{n}^{2}+30n+72$
$5{t}^{3}40$
${m}^{4}81$
$12ab6a+10b5$
$5{q}^{2}15q90$
$5{n}^{3}+320$
$25{x}^{2}+35xy+49{y}^{2}$
$3{v}^{4}768$
Everyday Math
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 $\mathrm{16}{t}^{2}+80$ gives the height of the watermelon $t$ 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 $t$ seconds is given by the trinomial $\mathrm{16}{t}^{2}8t+80$. Completely factor this trinomial.
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 $\mathrm{16}{t}^{2}+128$ gives the height of the pumpkin 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 $t$ seconds is given by the trinomial $\mathrm{16}{t}^{2}32t+128$. Completely factor this trinomial.
Writing Exercises
The difference of squares ${y}^{4}625$ can be factored as $\left({y}^{2}25\right)\left({y}^{2}+25\right)$. But it is not completely factored. What more must be done to completely factor it?
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.
ⓑ Overall, after looking at the checklist, do you think you are wellprepared for the next section? Why or why not?