4.7 Difference of Squares

Leanne Thompson

4.7 Difference of Squares

We will now look at how to factor binomials in the form a² – b². We can factor these binomials using the difference of squares method. To explore how to factor binomials of the form a² – b², expand the following binomials.

$\left(x+6\right)\left(x-6\right)$ =

$\left(x-7\right)\left(x+7\right)$ =

$\left(2x+1\right)\left(2x-1\right)$ =

$\left(3x-2\right)\left(3x+2\right)$ =

Consider the following questions to figure out how we can factor binomials in the form a² – b².

1. What do you notice about the constant terms in the final product and the constant terms in the binomial products?

2. What do you notice about the x² terms in the final product and the x terms in the binomial products?

3. What do you notice about the sign in the final product and the signs in the binomial products?

4. To factor a² – b² by the difference of squares method, find the square root of and , and write your factors in the form ( + )( – ).

Table 4.7.1
To factor binomials of the form a² – b²:
Step 1: Find the square root of a² and b². Step 2: Use the square roots, a and b, to write your factors in the form: $\left(a+b\right)\left(a-b\right)$ . Step 3: Check by multiplying the factors. In short, $a^2-b^2=\left(a+b\right)\left(a-b\right)$ .

Example 1

Factor: ${x}^{2}-36$

Table 4.7.2
Steps	Solution
Find the square root of a² and b².	$\sqrt{x^2}=x$ $\sqrt{36}=6$
Use the square roots, a and b, to write your factors in the form: $\left(a+b\right)\left(a-b\right)$ .	$\left(x+6\right)\left(x-6\right)$
Check by multiplying the factors.	$\left(x+6\right)\left(x-6\right)$ ${x}^{2}+6x-6x-36$ ${x}^{2}-36$

Practice 1

Factor where possible.

a) ${a}^{2}-25$

b) ${x}^{2}-49$

c) ${p}^{2}-1$

d) ${m}^{2}+81$

e) ${y}^{2}-121$

f) ${q}^{2}-{p}^{2}$

Practice 2

Factor where possible.

a) $4{x}^{2}-49$

b) $9{p}^{2}-121$

c) $81-4{p}^{2}{q}^{2}$

d) $1-{r}^{2}$

e) $49{m}^{2}-{n}^{2}$

f) ${x}^{2}{y}^{2}-9$

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 difference of squares.

Practice 3

Factor where possible.

a) ${2x}^{2}-18$

b) ${3y}^{2}-48$

c) ${2a}^{2}{b}^{2}-{2a}^{2}$

d) ${4a}^{2}-100$

e) ${7m}^{2}+28$

f) ${m}^{3}-25m$

Homework

Factor where possible.

a) $r^2-16$

b) $x^2-9$

c) $v^2-25$

d) $x^2-1$

e) $p^2-4$

f) $f^2 - 100$

g) $25-c^2$

h) $1+t^2$

i) $49-4x^2$

j) $441a^2-16b^2$

k) $1-p^2$

l) $100x^2-49y^2$
Factor where possible.

a) $4v^2-1$

b) $9w^2 - 16$

c) $p^2q^2 - 4$

d) $25x^2 - 36$

e) $36a^2 - 49b^2$

f) $49y^2 - 64$

g) $25m^2 - 64n^2$

h) $k^2 + 25m^2$

i) $121p^2 - 144q^2$

j) $64z^2 - 81$

k) $81r^2 - 16s^2$

l) $4a^2 - 9b^2$
Factor where possible.

a) $3x^2-27$

b) $5n^2-20$

c) $16x^2-36$

d) $125x^2+45y^2$

e) ${x}^{2}{y}^{2}-9{x}^{2}$

f) $4x^2-100y^2$

g) $2a^2 - 18$

h) $7m^2 - 28$

i) $9x^2 - 25$

j) $36p^2 + 64q^2$

k) $4y^2z^2 - 81y^2$

l) $8r^2 - 200s^2$
The area of a rectangular bedroom can be expressed with the polynomial 121 – 9a²b² square feet.
a) Write expressions to represent the length and width of the bedroom.
b) If the length of the bedroom is 17 ft and b = 2, what is the value of a?
c) Find the width and area of the bedroom.
d) Find the perimeter of the bedroom.
The area of a triangular mat can be represented by the expression $\frac{4x^2-25}{2}$ in^2.
a) Write expressions to represent the base and height of the triangle.
b) Given that the base is larger than the height and x = 5.5, what are the base, height, and area of the mat. Rounded to the nearest tenth where necessary.
Which of the following is not a factor of $18p^2 - 18$ ?

a) p
b) p + 1
c) 18
d) p – 1
Which of the following binomials is (a + 3) not a factor of?

a) $2x^2 + 6$
b) $a^2 - 9$
c) $a^2 + 9$
d) $3a^2 - 27$
Which of the following trinomials can be factored? Circle all that apply.

a) $q^2 + 100$
b) $4a^2 + 64$
c) $2b^2 - 49$
d) $6c^2 - 24$

Answers

1.

a) $(r-4)(r+4)$	b) $(x-3)(x+3)$	c) $(v-5)(v+5)$
d) $(x-1)(x+1)$	e) $(p-2)(p+2)$	f) $(f+10)(f-10)$
g) $(5+c)(5-c)$	h) Cannot be factored	i) $(7+2x)(7-2x)$
j) $(21a+4b)(21a-4b)$	k) $(1+p)(1-p)$	l) $(10x+7y)(10x-7y)$

2.

a) $(2v-1)(2v+1)$	b) $(3w - 4)(3w + 4)$	c) $(pq + 2)(pq - 2)$
d) $(5x - 6)(5x + 6)$	e) $(6a - 7b)(6a + 7b)$	f) $(7y - 8)(7y + 8)$
g) $(5m - 8n)(5m + 8n)$	h) Cannot be factored	i) $(11p - 12q)(11p + 12q)$
j) $(8z - 9)(8z + 9)$	k) $(9r - 4s)(9r + 4s)$	l) $(2a - 3b)(2a + 3b)$

3.

a) $3(x-3)(x+3)$	b) $5(n-2)(n+2)$	c) $4(2x-3)(2x+3)$
d) $5(25x^2+9y^2)$	e) $x^2(y+3)(y-3)$	f) $4(x+5y)(x-5y)$
g) $2(a - 3)(a + 3)$	h) $7(m - 2)(m + 2)$	i) $(3x - 5)(3x + 5)$
j) $4(9p^2+16q^2)$	k) $y^2(2z - 9)(2z + 9)$	l) $8(r - 5s)(r + 5s)$