EDGE Math 8

Using the FOIL method, the product of two binomials in the form $a + b$ and $a - b$ is obtained as follows:

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

If you will combine like terms, the equation can be simplified as

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

Note that the right-hand side of the equation above is called a difference of two squares.

Example

Multiply each of the following:

1. $(3 x + 8 y) (3 x - 8 y)$

2. $(5 a b + 4 t) (5 a b - 4 t)$

3. $(2 x - 7 z^{2}) (2 x + 7 z^{2})$

Solution

Each expression is a product of a sum and difference of two terms. Use the formula $(a + b) (a - b) = a^{2} - b^{2} .$

(3 x + 8 y) (3 x - 8 y)

= {(3 x)}^{2} - {(8 y)}^{2}

= 9 x^{2} - 64 y^{2}

(5 a b + 4 t) (5 a b - 4 t)

= {(5 a b)}^{2} - {(4 t)}^{2}

= 25 a^{2} b^{2} - 16 t^{2}

(2 x - 7 z^{2}) (2 x + 7 z^{2})

= {(2 x)}^{2} - {(7 z^{2})}^{2}

= 4 x^{2} - 49 z^{4}