Factor differences of squares

A difference of two perfect squares, $a^2-b^2$ , factors as the product of the conjugate binomials $(a-b)(a+b)$ . This understanding includes recognizing squared monomials and numerical terms in polynomial expressions, applying the pattern with integer or rational coefficients, and checking the result by multiplication; it distinguishes differences of squares from sums of squares, without extending to complex-number factorization or more advanced generalizations.

Detailed Explanation: Factor differences of squares

A difference of squares has the form

a^2-b^2

It factors using the pattern

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

The expression must have:

subtraction between two terms, and
each term must be a perfect square.

Example: Factor $(25x^2-16)$

Step 1: Recognize the two squares.

Both terms are perfect squares:

25x^2=(5x)^2

and

16=4^2.

So rewrite the expression as

25x^2-16=(5x)^2-4^2.

Step 2: Apply the difference-of-squares pattern.

Here, $(a=5x)$ and $(b=4)$ . Therefore,

(5x)^2-4^2=(5x-4)(5x+4).

So the factored form is

\boxed{(5x-4)(5x+4)}.

Step 3: Check by multiplying.

Use the distributive property:

(5x-4)(5x+4) =25x^2+20x-20x-16 =25x^2-16.

The middle terms cancel, confirming the answer.

Remember: this pattern applies to a difference, or subtraction, of squares. An expression such as $(x^2+9)$ is a sum of squares and does not factor using this pattern over the real numbers.

Learn by doing: Factor differences of squares

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Polynomial Algebra Difference of Squares - Variables - Simplify

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