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Factoring the Difference of Two Squares

WHAT TO DO:

HOW TO DO IT:

1. Factor the binomial 49x^{4}
− 36y^{4}

The difference of two terms with â€œsquare
coefficientsâ€and â€œeven exponentsâ€ will always
be factored as the difference of squares.

49x^{4}
− 36y^{4}

49x^{4} − 36y^{4}

(7x^{2} + 6y^{2})(7x^{2} − 6y^{2})

2. Factor the binomial x^{8}
− y^{8}

Sometimes the factors themselves contain
a factorable binomial

→ difference of squares.

Another difference of squares. Continue factoring:

Continue factoring to prime factors:

x^{8}
− y^{8}

x^{8} − y^{8} = (x^{4} + y^{4})(x^{4}
− y^{4})

= (x^{4} + y^{4})(x^{2} + y^{2})(x^{2}
− y^{2})

= (x^{4} + y^{4})(x^{2} + y^{2})(x +
y)(x − y)

3. Factor the expression (t + 4)^{2} − 9

Some expressions can be factored as the
â€œdifference of squaresâ€ when one of the squared
terms is inside parentheses

Simplify:

(t + 4)^{2}
− 9

[(t + 4) + 3][(t + 4) − 3]

(t + 7) (t + 1)

4.
Factor

Watch for fractions in the expressions.

5. Factor completely: 16x^{3}y − 36xy^{3}

Factor out the common factors then note that
the term in parentheses can be factored as
the difference of squares. The remaining
factors are â€œprimeâ€ in rational numbers.