Case III.

108. When a binomial is the difference of two squares.

1. Resolve into factors $x^2-y^2$.

lint Since, (x + y)(x - y) = x^2 - y^2,

the factors of $x^2-y^2$ are $x+y$ and $x-y$.

find the factors of a binomial when it is the difference of two squares, therefore,

Take the square root of the first term and the square root of the second term.

The sum of these roots will form the first factor;

The difference of these roots will form the second factor.

The square root of a monomial is one of the two equal factors of the monomial.

Thus $9x^8{y}^2=3x^{4}y\times 3x^{4}y$; and $3x^{4}y$ is the square root of $9x^8{y}^2$.

The rule for extracting the square root of a monomial, when a perfect square, is as follows:

Extract the square root of the coefficient, and divide the index of each letter by $2$.

Exercise 33.

Resolve into factors:

1. $4-x^2$.

2. $9-x^2$.

3. $9a^2-x^2$.

4. $25-x^2$.

5. $25x^2-a^2$.

6. $16a^4-121$.

7. $121a^4-16$.

8. $4a^2{b}^2-c^2{d}^2$.

9. $1-x^2{y}^2$.

10. $81x^2{y}^2-1$.