4.6 Factoring Quadratic Trinomials where a = 1

Leanne Thompson

4.6 Factoring Quadratic Trinomials where a = 1

In earlier lessons in this unit, we multiplied factors together to get a trinomial product. This process can be seen in the left column of the table below. Now, we will cover how this process can be reversed by factoring a trinomial. This process can be seen in the right column of the table below. In this lesson, we will specifically go over how to factor quadratic trinomials with a leading coefficient of 1. The process that we use is called factoring by inspection.

Table 4.6.1
Multiplying	Factoring
$\left(x+2\right)\left(x+3\right)$ ${x}^{2}+3x+2x+6$ ${x}^{2}+5x+6$	${x}^{2}+5x+6$ ${x}^{2}+3x+2x+6$ $\left(x+2\right)\left(x+3\right)$

To explore how to factor quadratic trinomials with a leading coefficient of 1, expand the following binomials.

$\left(x+4\right)\left(x+3\right)$ =

$\left(x+1\right)\left(x+5\right)$ =

$\left(x-3\right)\left(x+5\right)$ =

$\left(x-4\right)\left(x-2\right)$ =

Based on the examples above, (x+m)(x+n) = x²+ bx + c. Consider the following questions to figure out how we can factor quadratic trinomials with a leading coefficient of 1.

What is the relationship between the value of c and the values of m and n?	c =
What is the relationship between the value of b and the values of m and n?	b =

To factor x²+ bx + c by inspection, we need to find two numbers that have a product of and a sum of .

Table 4.6.2
To factor trinomials of the form $a{x}^{2}+bx+c$ where a = 1:
Step 1: Write the factors as two binomials with first terms x: $(x)(x)$ . Step 2: Find two numbers, m and n, that multiply to c, $(m \bullet n=c)$ , and add them to b, $(m+n=b)$ . Step 3: Use m and n as the last terms of the factors: $\left(x+m\right)\left(x+n\right)$ . Step 4: Check by multiplying the factors.

Example 1

Factor: ${x}^{2}+7x+12$

Table 4.6.3

Steps

Solution

Write the factors as two binomials with first terms x: $(x)(x)$ .

${x}^{2}+7x+12$
(x )(x )

Find two numbers, m and n, that multiply to c, $m \bullet n=c$ , 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.

Use m and n as the last terms of the factors: $\left(x+m\right)\left(x+n\right)$ .

(x + 3)(x + 4)

Check by multiplying the factors.

(x + 3)(x + 4)
${x}^{2}+4x+3x+12$
${x}^{2}+7x+12$

Practice 1

Factor.

a) ${x}^{2}+6x+8$

b) ${u}^{2}+11u+24$

c) ${t}^{2}-11t+28$

d) ${u}^{2}-9u+18$

e) ${z}^{2}+4z-5$

f) ${h}^{2}+4h-12$

g) ${z}^{2}-4z-5$

h) ${q}^{2}-2q-15$

Practice 2

Factor where possible.

a) $2x+{x}^{2}-48$

b) $-11-10x+{x}^{2}$

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

d) $-6 - 9x + x^{2}$

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 inspection.

Practice 3

Factor.

a) $6x^2 + 18x + 12$

b) $4y^2 - 20y - 24$

c) $9z^3 + 18z^2 + 9z$

d) $5a^3 - 25a^2 - 30a$

Homework

Factor.

a) ${y}^{2}+8y+15$

b) ${q}^{2}+10q+24$

c) ${t}^{2}+14t+24$

d) ${y}^{2}+17y+60$

e) ${x}^{2}+19x+60$

f) ${v}^{2}+23v+60$
Factor.

a) ${y}^{2}-16y+63$

b) ${k}^{2}+k-20$

c) ${x}^{2}-4x-12$

d) ${y}^{2}-y-20$

e) ${r}^{2}-3r-40$

f) ${s}^{2}-3s-10$
Factor.

a) $9m+{m}^{2}+18$

b) $-7n+12+{n}^{2}$

c) $8-6x+{x}^{2}$

d) ${x}^{2}-12-11x$

e) $6 - 4x + x^2$

f) $y^2 - 14 - 5y$

g) $6x^2 + 30x + 36$

h) $8y^2 - 64y + 120$

i) $10z^2 + 50z + 60$

j) $12a^2 - 36a + 24$

k) $12x^3 + 24x^2 + 12x$

l) $8y^3 - 32y^2 - 40y$
Factor where possible.

a) ${y}^{2}-6y+15$

b) $y^2 + 12y + 36$

c) $2a^2 - 8a + 6$

d) $y^2 - 7y - 30$

e) $q^2 + 14q + 49$

f) $s^2 + 4s - 21$

g) $3x^2 + 6x - 72$

h) $k^2 + 5k - 14$

i) $z^2 - 15z + 50$

j) $x^2 - 10x + 21$

k) ${m}^{2}+4m+18$

l) ${n}^{2}-10n+12$
A backyard has an area of ${x}^{2}+22x+112$ square feet.
a) Write expressions to represent the length and width of the backyard.
b) If x = 4, find the length and width of the backyard.
Which of the following is not a factor of $15p^3 - 45p^2 - 60p$ ?

a) p
b) p – 4
c) 15
d) p – 1
Which of the following trinomials is (a – 3) not a factor of?

a) $a^2 - 4a + 3$
b) $a^2 - 6a + 9$
c) $a^2 + 6a + 8$
d) $a^2 - 5a + 6$
Which of the following trinomials can be factored?

a) $n^2 - 8n + 15$
b) $x^2 + 3x + 3$
c) $y^2 + 5y + 7$
d) $z^2 + 2z + 9$

Answers

1.

a) $\left(y+3\right)\left(y+5\right)$	b) $\left(q+4\right)\left(q+6\right)$	c) $\left(t+2\right)\left(t+12\right)$
d) $\left(y+5\right)\left(y+12\right)$	e) $\left(x+4\right)\left(x+15\right)$	f) $\left(v+3\right)\left(v+20\right)$

2.

a) $\left(y-7\right)\left(y-9\right)$	b) $\left(k-4\right)\left(k+5\right)$	c) $\left(x+2\right)\left(x-6\right)$
d) $\left(y+4\right)\left(y-5\right)$	e) $\left(r+5\right)\left(r-8\right)$	f) $\left(s+2\right)\left(s-5\right)$

3.

a) $\left(m+3\right)\left(m+6\right)$	b) $\left(n-3\right)\left(n-4\right)$	c) $\left(x-4\right)\left(x-2\right)$
d) $\left(x-12\right)\left(x+1\right)$	e) Cannot be factored	f) $(y - 7)(y + 2)$
g) $6(x + 2)(x + 3)$	h) $8(y - 3)(y - 5)$	i) $10(z + 3)(z + 2)$
j) $12(a - 2)(a - 1)$	k) $12x(x + 1)(x + 1)$	l) $8y(y - 5)(y + 1)$

4.

a) Cannot be factored	b) $(y + 6)(y + 6)$	c) $2(a - 1)(a - 3)$
d) $(y - 10)(y + 3)$	e) $(q + 7)(q + 7)$	f) $(s + 7)(s - 3)$
g) $3(x + 6)(x - 4)$	h) $(k + 7)(k - 2)$	i) $(z - 10)(z - 5)$
j) $(x - 3)(x - 7)$	k) Cannot be factored	l) Cannot be factored