 # 2.01 Review: Factoring techniques

Lesson

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: $x^2-6x+9$x26x+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)22abx+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$9x24y2

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$9x24y2 is $\left(3x-2y\right)\left(3x+2y\right)$(3x2y)(3x+2y)

#### Practice questions

##### Question 7

Find the greatest common factor of $x^2y^4+x^5y^6$x2y4+x5y6.

##### Question 8

Factor $x^2-5x+10x-50$x25x+10x50 by grouping in pairs.

##### Question 9

Factor $x^2-2x-8$x22x8.

##### Question 10

Factor $k^2-81$k281.

##### Question 11

Factor $x^2+12x+36$x2+12x+36.