Factor Trinomials: Mini-Lesson Review

Mini Lesson Question

Question #2: Factor Trinomials

Factor:

y^2-y-42

$(y-7)(y-6)$
$(y+7)(y+6)$
$(y+7)(y-6)$
$(y-7)(y+6)$

Factor Trinomials

Example 1

To factor the trinomial $x^{2} + 8 x + 12$ , you can reverse the process of FOIL.

The trinomial factors to two binomials of the form $(x + m) (x + n)$ .

Using FOIL: $(x + m) (x + n) = x^{2} + n x + m x + m n$ .

Using the distributive property: $x^{2} + (n + m) x + m n$ .

Take a look at the trinomial $x^{2} + 8 x + 12$ :

Image showing x squared plus the quantity of n plus m times x plus m times n. On the next line, the trinomial is rewritten as x squared plus 8 times x plus 12 and arrows point from the quantity of n plus m in the trinomial above to the 8, and an arrow from m times n in the trinomial above to 12.

To factor the trinomial, we need factors of 12, $m$ and $n$ , so that the sum, $n + m$ , is 8.

Use a table to find all the different combinations of factors of 12. Then, find the sum of the factors.

Factors of 12	Sum of factors
$1, 12$	$1 + 12 = 13$
$2, 6$	$2 + 6 = 8 ✓$
$3, 4$	$3 + 4 = 7$

The numbers 2 and 6 have a product of 12 and a sum of 8. They are the factors we need:

$x^{2} + 8 x + 12 = (x + 6) (x + 2)$

Example 2

Factor the trinomial $x^{2} - 5 x - 24$ .

To factor the trinomial, we need factors of -24, $m$ and $n$ , so that the sum, $n + m$ is -5.

The product of the factors is negative, so the factors must have opposite signs.
The sum is negative, so the negative sign will go with the factor with the greater absolute value.

Factors of 12	Sum of factors
$1, - 24$ $2, - 12$ $3, - 8$ $4, - 6$	$1 + (- 24) = - 23$ $2 + (- 12) = - 10$ $3 + (- 8) = - 5 ✓$ $4 + (- 6) = - 2$

Factors of 12

Sum of factors

$1, - 24$

$2, - 12$

$3, - 8$

$4, - 6$

$1 + (- 24) = - 23$

$2 + (- 12) = - 10$

$3 + (- 8) = - 5 ✓$

$4 + (- 6) = - 2$

The numbers 3 and -8 have a product of -24 and a sum of -5.

$x^{2} - 5 x - 24 = (x + 3) (x - 8)$

Example 3

Factor the trinomial $x^{2} - 9 x + 18$ .

To factor the trinomial, we need factors of 18, $m$ and $n$ , so that the sum, $n + m$ , is -9.

The product of the factors is positive, so the factors must have the same signs.
The sum is negative, so both factors are negative.

Factors of 18	Sum of factors
$- 1, - 18$ $- 2, - 9$ $- 3, - 6$	$- 1 + (- 18) = - 19$ $- 2 + (- 9) = - 11$ $- 3 + (- 6) = - 9 ✓$

Factors of 18

Sum of factors

$- 1, - 18$

$- 2, - 9$

$- 3, - 6$

$- 1 + (- 18) = - 19$

$- 2 + (- 9) = - 11$

$- 3 + (- 6) = - 9 ✓$

The numbers -3 and -6 have a product of 18 and a sum of -9.

$x^{2} - 9 x + 18 = (x - 3) (x - 6)$

Try it

Factoring Trinomials

Factor the trinomial $x^{2} + 2 x - 24$ .

Here is how to factor the trinomial $x^{2} + 2 x - 24$ .

To factor the trinomial, we need factors of -24, $m$ and $n$ , so that the sum, $n + m$ , is 2.

The product of the factors is negative, so the factors must have opposite signs.
The sum is positive, so the greater factor is positive.

Factors of 24	Sum of factors
$- 1, 24$ $- 2, 12$ $- 3, 8$ $- 4, 6$	$- 1 + 24 = 23$ $- 2 + 12 = 10$ $- 3 + 8 = 5$ $- 4 + 6 = 2 ✓$

Factors of 24

Sum of factors

$- 1, 24$

$- 2, 12$

$- 3, 8$

$- 4, 6$

$- 1 + 24 = 23$

$- 2 + 12 = 10$

$- 3 + 8 = 5$

$- 4 + 6 = 2 ✓$

The numbers 6 and -4 have a product of -24 and a sum of 2.

$x^{2} + 2 x - 24 = (x + 6) (x - 4)$

Check Your Understanding

Factor: $y^{2} - 13 y + 36$ .

$(y - 4) (y - 9)$
$(y - 2) (y + 18)$
$(y + 4) (y + 9)$
$(y - 6) (y - 6)$

$(y - 4) (y - 9)$

Check yourself: To factor the trinomial, we need factors of 36 that sum to – 13.

Our factor pairs could be: 1 & 36, 2 & 18, 3 & 12, 4 & 9, 6 & 6
Because 36 is positive, we know the factors will have the same sign.
Since the sum is negative, both of the factors must be negative.

The factor pair that sums to 13 are 4 & 9, so our binomials are $(y - 4) (y - 9)$

Video: Factoring Trinomials

Watch the following video to learn more about factoring trinomials.

Khan Academy: Factoring Quadratics

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