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.

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

Factor completely: $7 x^{3} - 21 x^{2} - 70 x .$

Solution

This table demonstrates the step-by-step process of factoring a polynomial completely, including the verification of the factored form.
	$7 x^{3} - 21 x^{2} - 70 x$
Is there a GCF? Yes, $7 x$ .
Factor out the GCF.	$7 x (x^{2} - 3 x - 10)$
In the parentheses, is it a binomial, trinomial, or are there more terms?
Trinomial with leading coefficient 1.
“Undo” FOIL.	$7 x (x) (x)$
	$7 x (x + 2) (x - 5)$
Is the expression factored completely? Yes.
Neither binomial can be factored.
Check your answer.
Multiply.
$7 x (x + 2) (x - 5)$
$7 x (x^{2} - 5 x + 2 x - 10)$
$7 x (x^{2} - 3 x - 10)$
$7 x^{3} - 21 x^{2} - 70 x ✓$

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

Factor completely: $24 y^{2} - 150 .$

Solution

This table details the step-by-step factorization of the polynomial 24y^2 - 150, showcasing GCF extraction, difference of squares, and result verification.
	$24 y^{2} - 150$
Is there a GCF? Yes, 6.
Factor out the GCF.	$6 (4 y^{2} - 25)$
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.	$6 ({(2 y)}^{2} - {(5)}^{2})$
Write as a product of conjugates.	$6 (2 y - 5) (2 y + 5)$
$Is the expression factored completely?$
$Neither binomial can be factored.$
Check:
$Multiply.$
$6 (2 y - 5) (2 y + 5)$
$6 (4 y^{2} - 25)$
$24 y^{2} - 150 ✓$

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

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

Solution

Step-by-step guide to factoring the perfect square trinomial 4a^2 - 12ab + 9b^2, including identification, factorization, and verification.
	$4 a^{2} - 12 a b + 9 b^{2}$
Is there a GCF? No.
Is it a binomial, trinomial, or are there more terms?
Trinomial with $a \neq 1$ . But the first term is a perfect square.
Is the last term a perfect square? Yes.	${(2 a)}^{2} - 12 a b + {(3 b)}^{2}$
Does it fit the pattern, $a^{2} - 2 a b + b^{2}$ ? Yes.	${(2 a)}^{2}_{↘} \underset{−2 (2 a) (3 b)}{−12 a b +}_{↙} {(3 b)}^{2}$
Write it as a square.	${(2 a - 3 b)}^{2}$
$Is the expression factored completely? Yes.$
$The binomial cannot be factored.$
Check your answer.
$Multiply.$
${(2 a - 3 b)}^{2}$
${(2 a)}^{2} - 2 \cdot 2 a \cdot 3 b + {(3 b)}^{2}$
$4 a^{2} - 12 a b + 9 b^{2} ✓$

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

Factor completely $12 x^{3} y^{2} + 75 x y^{2} .$

Solution

Step-by-step process for factoring the algebraic expression 12x^3y^2 + 75xy^2, including finding the GCF and verifying the result.
	$12 x^{3} y^{2} + 75 x y^{2}$
Is there a GCF? Yes, $3 x y^{2}$ .
Factor out the GCF.	$3 x y^{2} (4 x^{2} + 25)$
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.
$Is the expression factored completely? Yes.$
Check:
$Multiply.$
$3 x y^{2} (4 x^{2} + 25)$
$12 x^{3} y^{2} + 75 x y^{2} ✓$

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

Factor completely: $24 x^{3} + 81 y^{3} .$

Solution

Is there a GCF? Yes, 3.
Factor it out.
In the parentheses, is it a binomial, trinomial, of 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.
Check by multiplying.

Factor completely: $3 x^{5} y - 48 x y .$

Solution

This table demonstrates the step-by-step factorization of the polynomial 3x^5y - 48xy, concluding with a verification of the result.
	$3 x^{5} y - 48 x y$
Is there a GCF? Factor out $3 x y$	$3 x y (x^{4} - 16)$
Is the binomial a sum or difference? Of squares or cubes? Write it as a difference of squares.	$3 x y ({(x^{2})}^{2} - {(4)}^{2})$
Factor it as a product of conjugates	$3 x y (x^{2} - 4) (x^{2} + 4)$
The first binomial is again a difference of squares.	$3 x y ({(x)}^{2} - {(2)}^{2}) (x^{2} + 4)$
Factor it as a product of conjugates.	$3 x y (x - 2) (x + 2) (x^{2} + 4)$
Is the expression factored completely? Yes.
Check your answer.
Multiply.
$3 x y (x - 2) (x + 2) (x^{2} + 4)$
$3 x y (x^{2} - 4) (x^{2} + 4)$
$3 x y (x^{4} - 16)$
$3 x^{5} y - 48 x y ✓$

Factor completely: $4 x^{2} + 8 b x - 4 a x - 8 a b .$

Solution

Step-by-step factorization of the polynomial 4x^2 + 8bx - 4ax - 8ab, demonstrating GCF extraction, grouping, and solution verification.
	$4 x^{2} + 8 b x - 4 a x - 8 a b$
Is there a GCF? Factor out the GCF, 4.	$4 (x^{2} + 2 b x - a x - 2 a b)$
There are four terms. Use grouping.	$\begin{matrix} 4 [x (x + 2 b) - a (x + 2 b)] \\ 4 (x + 2 b) (x - a) \end{matrix}$
Is the expression factored completely? Yes.
Check your answer. Multiply. $\begin{matrix} 4 (x + 2 b) (x - a) \\ 4 (x^{2} - a x + 2 b x - 2 a b) \\ 4 x^{2} + 8 b x - 4 a x - 8 a b ✓ \end{matrix}$

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

Factor completely: $40 x^{2} y + 44 x y - 24 y .$

Solution

Step-by-step guide to factoring the polynomial 40x^2y + 44xy - 24y, showing GCF and trinomial factorization, and verification.
	$40 x^{2} y + 44 x y - 24 y$
Is there a GCF? Factor out the GCF, $4 y$ .	$4 y (10 x^{2} + 11 x - 6)$
Factor the trinomial with $a \neq 1$ .	$4 y (10 x^{2} + 11 x - 6)$
	$4 y (5 x - 2) (2 x + 3)$
Is the expression factored completely? Yes.
Check your answer.
Multiply.
$4 y (5 x - 2) (2 x + 3)$
$4 y (10 x^{2} + 11 x - 6)$
$40 x^{2} y + 44 x y - 24 y ✓$

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.

Factor completely: $9 x^{2} - 12 x y + 4 y^{2} - 49 .$

Solution

Detailed steps for factoring the algebraic expression 9x^2 - 12xy + 4y^2 - 49, identifying a perfect square trinomial and difference of squares, including verification.
	$9 x^{2} - 12 x y + 4 y^{2} - 49$
Is there a GCF? No.
With more than 3 terms, use grouping. Last 2 terms have no GCF. Try grouping first 3 terms.	$9 x^{2} - 12 x y + 4 y^{2} - 49$
Factor the trinomial with $a \neq 1$ . But the first term is a perfect square.
Is the last term of the trinomial a perfect square? Yes.	${(3 x)}^{2} - 12 x y + {(2 y)}^{2} - 49$
Does the trinomial fit the pattern, $a^{2} - 2 a b + b^{2}$ ? Yes.	${(3 x)}^{2}_{↘} \underset{−2 (3 x) (2 y)}{−12 x y +}_{↙} {(2 y)}^{2} - 49$
Write the trinomial as a square.	${(3 x - 2 y)}^{2} - 49$
Is this binomial a sum or difference? Of squares or cubes? Write it as a difference of squares.	${(3 x - 2 y)}^{2} - 7^{2}$
Write it as a product of conjugates.	$((3 x - 2 y) - 7) ((3 x - 2 y) + 7)$
	$(3 x - 2 y - 7) (3 x - 2 y + 7)$
Is the expression factored completely? Yes.
Check your answer.
Multiply.
$(3 x - 2 y - 7) (3 x - 2 y + 7)$
$9 x^{2} - 6 x y - 21 x - 6 x y + 4 y^{2} + 14 y + 21 x - 14 y - 49$
$9 x^{2} - 12 x y + 4 y^{2} - 49 ✓$

Key Concepts

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.

How to use a general strategy for factoring polynomials.
1. Is there a greatest common factor?
  Factor it out.
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 $x^{2} + b x + c ?$ Undo FOIL.
  Is it of the form $a x^{2} + b x + c ?$
  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:
  Use the grouping method.
3. Check.
  Is it factored completely?
  Do the factors multiply back to the original polynomial?

Practice Makes Perfect

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

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

Solution

$(2 n - 1) (n + 7)$

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

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

Solution

$a^{3} (a^{2} + 9)$

$75 m^{3} + 12 m$

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

Solution

$(11 r - s) (11 r + s)$

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

$8 m^{2} - 32$

Solution

$8 (m - 2) (m + 2)$

$36 q^{2} - 100$

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

Solution

${(5 w - 6)}^{2}$

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

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

Solution

${(m + 7 n)}^{2}$

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

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

Solution

$7 (b + 3) (b - 2)$

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

$3 x^{4} y - 81 x y$

Solution

$3 x y (x - 3) (x^{2} + 3 x + 9)$

$4 x^{5} y - 32 x^{2} y$

$k^{4} - 16$

Solution

$(k - 2) (k + 2) (k^{2} + 4)$

$m^{4} - 81$

$5 x^{5} y^{2} - 80 x y^{2}$

Solution

$5 x y^{2} (x^{2} + 4) (x + 2) (x - 2)$

$48 x^{5} y^{2} - 243 x y^{2}$

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

Solution

$3 (5 p + 4) (q - 1)$

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

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

Solution

$4 (x + 3) (x + 7)$

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

$4 u^{5} + 4 u^{2} v^{3}$

Solution

$4 u^{2} (u + v) (u^{2} - u v + v^{2})$

$5 m^{4} n + 320 m n^{4}$

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

Solution

prime

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

$10 m^{4} - 6250$

Solution

$10 (m - 5) (m + 5) (m^{2} + 25)$

$3 v^{4} - 768$

$36 x^{2} y + 15 x y - 6 y$

Solution

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

$60 x^{2} y - 75 x y + 30 y$

$8 x^{3} - 27 y^{3}$

Solution

$(2 x - 3 y) (4 x^{2} + 6 x y + 9 y^{2})$

$64 x^{3} + 125 y^{3}$

$y^{6} - 1$

Solution

$(y + 1) (y - 1) (y^{2} - y + 1) (y^{2} + y + 1)$

$y^{6} + 1$

$9 x^{2} - 6 x y + y^{2} - 49$

Solution

$(3 x - y + 7) (3 x - y - 7)$

$16 x^{2} - 24 x y + 9 y^{2} - 64$

${(3 x + 1)}^{2} - 6 (3 x + 1) + 9$

Solution

$(3 x - 2)^{2}$

${(4 x - 5)}^{2} - 7 (4 x - 5) + 12$

Writing Exercises

Explain what it mean to factor a polynomial completely.

Solution

Answers will vary.

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.

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.

Solution

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.

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.

ⓑ 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?