EDGE Math 8

There are cases in which you need to apply two or more formulas for special products. This is shown in the next example.

Example

Multiply each of the following:

1. ${[(x + y) + z]}^{2}$

2. $(p + q + 3) (p + q - 3)$

Solution

1. Use ${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$ and let $a = x + y$ and $b = z .$

{[(x + y) + z]}^{2}

= {(x + y)}^{2} + 2 (x + y) (z) + z^{2}

Apply the same formula again to ${(x + y)}^{2},$ and let $a = x$ and $b = y .$ Thus,

{[(x + y) + z]}^{2}

= x^{2} + 2 x y + y^{2} + 2 x z + 2 y z + z^{2} .

2. Use

(a + b) (a - b) = a^{2} - a b + a b - b^{2} .

Let

a = p + q,

and

b = 3.

(p + q + 3) (p + q - 3)

= {(p + q)}^{2} - 3^{2}

Then apply the formula ${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$ to ${(p + q)}^{2},$ with $a = p$ and $b = q .$ Thus,

(p + q + 3) (p + q - 3)

= p^{2} + 2 p q + q^{2} - 9.