2.01 Review: Factoring techniques

Factorization is an application of the distributive property.  Recall the five methods of factoring a polynomial discussed in Algebra I: greatest common factor (GCF), grouping with 4 terms, perfect square trinomials, factoring trinomials, and difference of squares.

 

Greatest common factor (GCF)

Find the greatest common factor (GCF) of all terms of the polynomial and then apply the distributive property.

Worked example

Question 1

Factor: $24a^3b+8a$24a3b+8a

Think: The first method to try is GCF. 

Do: Determine the GCF of $24a^3b$24a3b and $8a$8a.

$24a^3b+8a$24a3b+8a
$2\times2\times2\times3aaab+2\times2\times2a$2×2×2×3aaab+2×2×2a	From the prime factorization we can see that GCF: $2\times2\times2a=8a$2×2×2a=8a
$8a\left(3a^2b+1\right)$8a(3a2b+1)

 

Grouping with 4 terms

Group pairs of terms with grouping symbol that contain a common factor, then factor out the GCF from each separate binomial. Lastly, factor out the common binomial.

Worked example

question 2

Factor: $x^3+7x^2+2x+14$x3+7x2+2x+14

Think: The first method to try is GCF. Once we have tried GCF, then notice that the expression has $4$4terms, therefore try using the grouping with $4$4 terms method. 

Do: Since the expression does not have a GCF, group terms in pairs that have at least one GCF.

$x^3+7x^2+2x+14$x3+7x2+2x+14

$x^3+7x^2+2x+14$x3+7x2+2x+14

GCF:  $x^2$x2                       GCF: $2$2

$x^2\left(x+7\right)+2\left(x+7\right)$x2(x+7)+2(x+7)

$\left(x^2+2\right)\left(x+7\right)$(x2+2)(x+7)

 

Perfect square trinomials

Look for three terms where the first and third terms are perfect squares, and the middle term is twice the product of their square roots. 

Worked examples

question 3

Factor: $x^2+6x+9$x2+6x+9

Think: Always, check for GCF first. Since the expression has $3$3 terms, determine whether the expression meets the criteria for perfect squares trinomials method of factoring.  

Do: When written in order, the first and last terms perfect squares $x^2=x^2$x2=x2 and $9=3^2$9=32. The expression must equal the perfect square trinomial formula:  $\left(ax\right)^2+2abx+b^2$(ax)2+2abx+b2

$x^2+6x+9$x2+6x+9
$x^2+2\times3x+3^2$x2+2×3x+32	Does the quadratic fit the perfect squares formula? $\left(ax\right)^2+2abx+b^2$(ax)2+2abx+b2
$\left(x+3\right)^2$(x+3)2

 

Question 4

Factor: $x^2-6x+9$x2−6x+9

Think: Always, check for GCF first. Since the expression has $3$3 terms, determine whether the expression meets the criteria for perfect squares trinomials method of factoring.  

Do: When written in order, the first and last terms perfect squares $x^2=x^2$x2=x2 and $9=3^2$9=32. The expression must equal the perfect square trinomial formula:  $\left(ax\right)^2-2abx+b^2$(ax)2−2abx+b2

$x^2-6x+9$x2−6x+9
$x^2-2\times3x+3^2$x2−2×3x+32	Does the quadratic fit the perfect squares formula? $\left(ax\right)^2-2abx+b^2$(ax)2−2abx+b2
$\left(x+3\right)^2$(x+3)2

 

Factoring trinomials

Look for three terms of the form $ax^2+bx+c$ax2+bx+c , find factors of $ac$ac that add to equal $b$b. Replace $b$b with the factors then continue by factoring using the grouping method. There are several organizing tools to assist with this process (i.e., box method and X method). 

 

Worked examples

Question 5

Use the Box method to factor: $x^2+7x+10$x2+7x+10

Think: The Box method requires that we understand the area model (length x width = $lw$lw)

Do: Fill in the pieces we know: $x^2$x2, $10$10. The $7x$7x will be the sum of the other $2$2 boxes. Now, we can fill in the length and width for the $x^2$x2 box ($x$x and $x$x). The length and width for the $10$10 box has to be $5$5 and $2$2 because the sum of the boxes they create equals $7x$7x.

The factorization of $x^2+7x+10$x2+7x+10 is $\left(x+5\right)\left(x+2\right)$(x+5)(x+2).

 

Difference of squares

Look for the difference of two terms which are both perfect squares. 

Worked examples

Question 6

Factor: $9x^2-4y^2$9x2−4y2

Think: Check for GCF first. After the GCF has been checked, then determine whether the $2$2 terms are perfect squares.

Do: There is no GCF between the $2$2 terms. However, the terms are perfect squares: $9x^2=\left(3x\right)^2$9x2=(3x)2 and $4y^2=\left(2y\right)^2$4y2=(2y)2. Because the terms are being subtracted, then we will use the difference of squares method.

$9x^2-4y^2$9x2−4y2	$=$=	$\left(3x\right)^2-\left(2y\right)^2$(3x)2−(2y)2
	$=$=	$\left(3x-2y\right)\left(3x+2y\right)$(3x−2y)(3x+2y)

The factorization of $9x^2-4y^2$9x2−4y2 is $\left(3x-2y\right)\left(3x+2y\right)$(3x−2y)(3x+2y)

 

Practice questions

Question 7

Question 8

Question 9

Question 10

Question 11