4.8 Factoring Quadratic Trinomials where a ≠ 1

Leanne Thompson

4.8 Factoring Quadratic Trinomials where a ≠ 1

To factor quadratic trinomials of the form ax²+ bx + c, we can use a method called decomposition. Decomposition can be used when a = 1, but in this lesson, we will focus on using this method to factor quadratic trinomials where a ≠ 1. In the process of decomposition, we will use factoring by grouping, which was covered in a previous lesson.

Table 4.8.1
To factor trinomials of the form $a{x}^{2}+bx+c$ where a ≠ 1:
Step 1: Find the product ( $a \bullet {c}$ ). Step 2: Find two numbers, m and n, that multiply to ac, $(m \bullet n=ac)$ , and add them to b, $(m+n=b)$ . Step 3: Split the middle term using m and n: (ax²+ bx + c = ax²+ mx + nx + c). Step 4: Factor by grouping. Step 5: Check by multiplying the factors.

Example 1

Factor: $6{x}^{2}+7x+2$

Table 4.8.2

Steps

Solution

Find the product $a \bullet {c}$ .

$6 \bullet 2=12$

Find two numbers, m and n, that multiply to ac, $(m \bullet n=ac)$ , and add them to b, $(m+n=b)$ .

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

The numbers that multiply to 12 and add to 7 are 3 and 4.

Split the middle term using m and n: (ax²+ bx + c = ax²+ mx + nx + c).

$6{x}^{2}+7x+2$
$6{x}^{2}+3x+4x+2$

Factor by grouping.

$(6{x}^{2}+3x$ ) $+ (4x+2)$
$3x(2x+1)+2(2x+1)$
$(2x+1)(3x+2)$

Check by multiplying the factors.

$(2x+1)(3x+2)$
$6{x}^{2}+4x+3x+2$
$6{x}^{2}+7x+2$

Practice 1

Factor.

a) $3x^{2} + 8x + 4$

b) $4y^{2} + 12y + 9$

c) $2p^{2} - 11p + 9$

d) $7m^{2} + 40m - 12$

e) $7n^{2} - n - 6$

f) $-3x^{2} + 10x - 8$

If the terms in the trinomial you are factoring are not written in the same order as the form ax²+ bx + c, it is a good idea to rearrange the terms to be in this order. This will make using decomposition easier.

Practice 2

Factor.

a) $9c +5c^2 + 4$

b) $4 - 12x + 9x^2$

As mentioned in the previous lesson, when given a polynomial to factor, first try to factor out a greatest common factor, if possible. In the questions below, you will need to factor out the greatest common factor, then factor using decomposition.

Practice 3

Factor.

a) $5t^{2} + 15t + 10$

b) $14x^{2} + 7x - 21$

c) $2b^{3} + 12b^{2} + 18b$

d) $-2x^{2} + 7x - 3$

Homework

Factor.

a) $2x^{2} + 7x + 6$

b) $3x^{2} - 11x + 6$

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

d) $5x^{2} + 18x + 9$

e) $6x^{2} + 11x + 4$

f) $5x^{2} - 13x + 6$

g) $-4x^{2} - 5x - 6$

h) $x^{2} + 5x - 14$

i) $2x^{2} + x - 3$

j) $6x^{2} - 7x - 3$

k) $-3x^{2} + 11x - 6$

l) $6x^{2} + 11x - 35$
Factor.

a) $8c + 4 - 5c^2$

b) $-12x + 9x^2 + 4$

c) $7y + 13 - 6y^2$

d) $12 + 23p + 5p^2$

e) $-2m^2 + m + 3$

f) $3z^2 - 14z + 8$

g) $4a^2 + 23a + 15$

h) $-3b + 2b^2 - 5$
Factor.

a) $6x^2 + 27x + 12$

b) $8y^2 - 4y - 24$

c) $15z^2 - 35z + 20$

d) $4a^2 + 10a - 6$

e) $6b^2 - 21b + 18$

f) $18m^2 - 30m + 12$

g) $12x^2 - 20x + 8$

h) $10y^2 + 35y - 20$
A backyard has an area of $2y^2 - 3y - 5$ square metres.
a) Write expressions to represent the length and width of the backyard.
b) If x = 8, find the length and width of the backyard.
A box in the shape of a rectangular prism has a volume of $2x^3 + 11x^2 + 15x$ cubic inches.
a) Write expressions to represent the length, width, and height of the box.
b) If x = 12, find the length, width, height, and volume of the box
Which of the following is not a factor of $6x^2 + 22x + 12$ ?

a) x + 3
b) 3x + 1
c) 2
d) 3x + 2
Which of the following trinomials is (m – 4) not a factor of?

a) $5m^2 + 16m + 3$
b) $2m^2 - 7m - 4$
c) $3m^2 - 18m + 24$
d) $4m^2 - 14m - 8$
Which of the following trinomials can be factored? Circle all that apply.

a) $2x^2 - 5x - 3$
b) $x^2 + 7x + 12$
c) $3x^2 + 8x - 3$
d) $2x^2 + 3x + 7$

Answers

1.

a) $(2x + 3)(x + 2)$	b) $(3x - 2)(x - 3)$	c) $(2x - 3)(x + 1)$
d) $(5x + 3)(x + 3)$	e) $(2x + 1)(3x + 4)$	f) $(5x - 3)(x - 2)$
g) Cannot be factored	h) $(x + 7)(x - 2)$	i) $(2x + 3)(x - 1)$
j) $(3x + 1)(2x - 3)$	k) $-(3x - 2)(x - 3)$	l) $(2x + 7)(3x - 5)$

2.

a) $-(5c + 2)(c - 2)$	b) $(3x - 2)^2$	c) $-(6y - 13)(y + 1)$	d) $(5p + 3)(p + 4)$
e) $-(2m - 3)(m + 1)$	f) $(3z - 2)(z - 4)$	g) $(4a + 3)(a + 5)$	h) $(b + 1)(2b - 5)$

3.

a) $3(2x + 1)(x + 4)$	b) $4(2y + 3)(y - 2)$	c) $5(3z - 4)(z - 1)$	d) $2(2a - 1)(a + 3)$
e) $3(2b - 3)(b - 2)$	f) $6(3m - 2)(m - 1)$	g) $4(3x - 2)(x - 1)$	h) $5(2y - 1)(y + 4)$