General Strategy for Factoring Polynomials

7.5 General Strategy for Factoring Polynomials

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 $y^{2} - 2 y - 24$ .
If you missed this problem, review Example 7.23.

Be Prepared 7.16

Factor $3 t^{2} + 17 t + 10$ .
If you missed this problem, review Example 7.38.

Be Prepared 7.17

Factor $36 p^{2} - 60 p + 25$ .
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.

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.

Figure 7.3

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.
  Is it a difference?
  - 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 x2+bx+cx2+bx+c? Undo FOIL.
  Is it of the form ax2+bx+cax2+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.
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

$\begin{matrix} Is there a GCF? & Yes, 4 x^{4} . & 4 x^{5} + 12 x^{4} \\ Factor out the GCF. & 4 x^{4} (x + 3) \\ In the parentheses, is it a binomial, a \\ trinomial, or are there more than three terms? & Binomial. \\ Is it a sum? & Yes. \\ Of squares? Of cubes? & No. \\ Check. \\ Is the expression factored completely? & Yes. \\ Multiply. \\ 4 x^{4} (x + 3) \\ 4 x^{4} \cdot x + 4 x^{4} \cdot 3 \\ 4 x^{5} + 12 x^{4} ✓ \end{matrix}$

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: $12 x^{2} - 11 x + 2$ .

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.

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.

Check.

$(3 x - 2) (4 x - 1)$

$12 x^{2} - 3 x - 8 x + 2$

$12 x^{2} - 11 x + 2 ✓$

Try It 7.119

Factor completely: $10 a^{2} - 17 a + 6$ .

Try It 7.120

Factor completely: $8 x^{2} - 18 x + 9$ .

Example 7.61

Factor completely: $g^{3} + 25 g$ .

Solution

Is there a GCF?	Yes, g.	$g^{3} + 25 g$
Factor out the GCF.		$g (g^{2} + 25)$
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.
Check.
Is the expression factored completely?	Yes.
Multiply. $\begin{matrix} g (g^{2} + 25) \\ g^{3} + 25 g ✓ \end{matrix}$

A step-by-step guide to factoring an algebraic expression, illustrating the process with questions, outcomes, and mathematical forms.

Try It 7.121

Factor completely: $x^{3} + 36 x$ .

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.		$3 (4 y^{2} - 25)$
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.	$3 ({(2 y)}^{2} - {(5)}^{2})$
Write as a product of conjugates.		$3 (2 y - 5) (2 y + 5)$
Check.
Is the expression factored completely?	Yes.
Neither binomial is a difference of squares.
Multiply. $\begin{matrix} 3 (2 y - 5) (2 y + 5) \\ 3 (4 y^{2} - 25) \\ 12 y^{2} - 75 ✓ \end{matrix}$

Detailed steps for factoring the algebraic expression 12y^2 - 75, showing identification of GCF and difference of squares.

Try It 7.123

Factor completely: $16 x^{3} - 36 x$ .

Try It 7.124

Factor completely: $27 y^{2} - 48$ .

Example 7.63

Factor completely: $4 a^{2} - 12 a b + 9 b^{2}$ .

Solution

Is there a GCF?	No.
Is it a binomial, trinomial, or are there more terms?
Trinomial with $a \neq 1$ . But the first term is a perfect square.
Is the last term a perfect square?	Yes.
Does it fit the pattern, $a^{2} - 2 a b + b^{2} ?$	Yes.
Write it as a square.
Check your answer.
Is the expression factored completely?
Yes.
The binomial is not a difference of squares.
Multiply.
${(2 a - 3 b)}^{2}$
${(2 a)}^{2} - 2 \cdot 2 a \cdot 3 b + {(3 b)}^{2}$
$4 a^{2} - 12 a b + 9 b^{2} ✓$

Try It 7.125

Factor completely: $4 x^{2} + 20 x y + 25 y^{2}$ .

Try It 7.126

Factor completely: $9 m^{2} + 42 m n + 49 n^{2}$ .

Example 7.64

Factor completely: $6 y^{2} - 18 y - 60$ .

Solution

Is there a GCF?	Yes, 6.	$6 y^{2} - 18 y - 60$
Factor out the GCF.		$6 (y^{2} - 3 y - 10)$
In the parentheses, is it a binomial, trinomial, or are there more terms?	Trinomial with leading coefficient 1.
“Undo” FOIL.	$6 (y) (y)$	$6 (y + 2) (y - 5)$
Check your answer.
Is the expression factored completely?		Yes.
Neither binomial is a difference of squares.
Multiply. $\begin{matrix} 6 (y + 2) (y - 5) \\ 6 (y^{2} - 5 y + 2 y - 10) \\ 6 (y^{2} - 3 y - 10) \\ 6 y^{2} - 18 y - 60 ✓ \end{matrix}$

Step-by-step guide demonstrating how to factor a trinomial polynomial, from identifying the GCF to checking the final factored form.

Try It 7.127

Factor completely: $8 y^{2} + 16 y - 24$ .

Try It 7.128

Factor completely: $5 u^{2} - 15 u - 270$ .

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.	$3 (2 x + 3) (4 x^{2} - 6 x + 9)$
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.		$2 (x^{4} - 16)$
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.	$2 ({(x^{2})}^{2} - {(4)}^{2})$
Write it as a product of conjugates.		$2 (x^{2} - 4) (x^{2} + 4)$
The first binomial is again a difference of squares.		$2 ({(x)}^{2} - {(2)}^{2}) (x^{2} + 4)$
Write it as a product of conjugates.		$2 (x - 2) (x + 2) (x^{2} + 4)$
Is the expression factored completely?	Yes.
None of these binomials is a difference of squares.
Check your answer.
Multiply. $\begin{matrix} 2 (x - 2) (x + 2) (x^{2} + 4) \\ 2 (x^{2} - 4) (x^{2} + 4) \\ 2 (x^{4} - 16) \\ 2 x^{4} - 32 ✓ \end{matrix}$

A step-by-step guide demonstrating the process of factoring the polynomial 2x^4 - 32, including questions and corresponding expressions.

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: $3 x^{2} + 6 b x - 3 a x - 6 a b$ .

Solution

Is there a GCF?	Yes, 3.	$3 x^{2} + 6 b x - 3 a x - 6 a b$
Factor out the GCF.		$3 (x^{2} + 2 b x - a x - 2 a b)$
In the parentheses, is it a binomial, trinomial, or are there more terms?	More than 3 terms.
Use grouping.		$\begin{matrix} 3 [x (x + 2 b) - a (x + 2 b)] \\ 3 (x + 2 b) (x - a) \end{matrix}$
Check your answer.
Is the expression factored completely? Yes.
Multiply. $\begin{matrix} 3 (x + 2 b) (x - a) \\ 3 (x^{2} - a x + 2 b x - 2 a b) \\ 3 x^{2} - 3 a x + 6 b x - 6 a b ✓ \end{matrix}$

This table demonstrates the step-by-step process of factoring a polynomial expression by grouping, including GCF identification and checking the solution.

Try It 7.133

Factor completely: $6 x^{2} - 12 x c + 6 b x - 12 b c$ .

Try It 7.134

Factor completely: $16 x^{2} + 24 x y - 4 x - 6 y$ .

Example 7.68

Factor completely: $10 x^{2} - 34 x - 24$ .

Solution

Is there a GCF?	Yes, 2.	$10 x^{2} - 34 x - 24$
Factor out the GCF.		$2 (5 x^{2} - 17 x - 12)$
In the parentheses, is it a binomial, trinomial, or are there more than three terms?	Trinomial with $a \neq 1 .$
Use trial and error or the “ac” method.		$\begin{matrix} 2 (5 x^{2} - 17 x −12) \\ 2 (5 x + 3) (x - 4) \end{matrix}$
Check your answer. Is the expression factored completely? Yes.
Multiply. $\begin{matrix} 2 (5 x + 3) (x - 4) \\ 2 (5 x^{2} - 20 x + 3 x - 12) \\ 2 (5 x^{2} - 17 x - 12) \\ 10 x^{2} - 34 x - 24 ✓ \end{matrix}$

Step-by-step method for factoring a quadratic trinomial, illustrating GCF, classification, trial/error, and verification.

Try It 7.135

Factor completely: $4 p^{2} - 16 p + 12$ .

Try It 7.136

Factor completely: $6 q^{2} - 9 q - 6$ .

Section 7.5 Exercises

Practice Makes Perfect

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

279.

$10 x^{4} + 35 x^{3}$

280.

$18 p^{6} + 24 p^{3}$

281.

$y^{2} + 10 y - 39$

282.

$b^{2} - 17 b + 60$

283.

$2 n^{2} + 13 n - 7$

284.

$8 x^{2} - 9 x - 3$

285.

$a^{5} + 9 a^{3}$

286.

$75 m^{3} + 12 m$

287.

$121 r^{2} - s^{2}$

288.

$49 b^{2} - 36 a^{2}$

289.

$8 m^{2} - 32$

290.

$36 q^{2} - 100$

291.

$25 w^{2} - 60 w + 36$

292.

$49 b^{2} - 112 b + 64$

293.

$m^{2} + 14 m n + 49 n^{2}$

294.

$64 x^{2} + 16 x y + y^{2}$

295.

$7 b^{2} + 7 b - 42$

296.

$3 n^{2} + 30 n + 72$

297.

$3 x^{3} - 81$

298.

$5 t^{3} - 40$

299.

$k^{4} - 16$

300.

$m^{4} - 81$

301.

$15 p q - 15 p + 12 q - 12$

302.

$12 a b - 6 a + 10 b - 5$

303.

$4 x^{2} + 40 x + 84$

304.

$5 q^{2} - 15 q - 90$

305.

$u^{5} + u^{2}$

306.

$5 n^{3} + 320$

307.

$4 c^{2} + 20 c d + 81 d^{2}$

308.

$25 x^{2} + 35 x y + 49 y^{2}$

309.

$10 m^{4} - 6250$

310.

$3 v^{4} - 768$

Everyday Math

311.

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 $−16 t^{2} + 80$ 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 $−16 t^{2} - 8 t + 80$ . Completely factor this trinomial.

312.

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 $−16 t^{2} + 128$ 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 $−16 t^{2} - 32 t + 128$ . Completely factor this trinomial.

Writing Exercises

313.

The difference of squares $y^{4} - 625$ can be factored as $(y^{2} - 25) (y^{2} + 25)$ . But it is not completely factored. What more must be done to completely factor it?

314.

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.

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!”.

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