EDGE Math 8

Cubes of Binomials

To find the cube of a binomial, ${(a + b)}^{3},$ use the Product Rule and the formula for the square of a binomial, ${(a + b)}^{2} = a^{2} + 2 a b + b^{2} .$

\begin{array}{l} {(a + b)}^{3} = (a + b) (a + b) (a + b) \\ = (a^{2} + 2 a b + b^{2}) (a + b) \\ = a (a^{2} + 2 a b + b^{2}) + b (a^{2} + 2 a b + b^{2}) \\ = a^{3} + 2 a^{2} b + a b^{2} + a^{2} b + 2 a b^{2} + b^{3} \\ = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3} \end{array}

The same steps may be done to find the cube of a binomial, ${(a - b)}^{3} .$ Thus, you will have the following equations:

{(a - b)}^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}

{(a + b)}^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

Example

Multiply each of the following:

1. ${(x + 4)}^{3}$

2. ${(2 y - 3)}^{3}$

Solution

{(x + 4)}^{3}

= x^{3} + 3 (x^{2}) (4) + 3 (x) (4^{2}) + 4^{3}

= x^{3} + 12 x^{2} + 48 x + 64

{(2 y - 3)}^{3}

= {(2 y)}^{3} - 3 {(2 y)}^{2} (3)

+ 3 (2 y) (3^{2}) - 3^{3}

= 8 y^{3} - 36 y^{2} + 54 y - 27