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08/14/2015

Algebra Help: Guide to Special Binomial Products

A binomial is a mathematical expression with two terms, as in . A binomial product is the result of multiplying two binomials together. There are three kinds of special binomial products you can remember to help factor and solve these kinds of equations. Let's look at them with some examples so you can keep them in mind.

Plus Square Binomial (+ Times +)

When you multiply a binomial by itself (a+b)(a+b), you get a^2+2ab+b^2. Use the FOIL method- first, outside, inside, last- to figure out why- a*a+ab+ba+b*b.

Minus Square Binomial (- Times -)

If both binomial terms are minus, as in (a-b)(a-b), then you get a^2-2ab+b^2.

Difference of Squares (+ Times -)

When the binomials have opposite terms, you get what is called a difference of squares. (a+b)(a-b)=a^2-b^2. This result is because the two ab terms cancel each other out.

Using the Special Products

Remember these three cases and they will help you solve many an equation. Let's look at how to use them. We gave you a and b as variables, because a and b could be anything. They could be simple, a=y and b=1. (y+1)^2=y^2+2y+1. Or they can be more complicated. Take (4y+2)(4y-2). Despite its complexity, we still see that we have a difference of squares binomial (a+b)(a-b), so we know our answer will be (4y)^2-2^2=16y^2-4.

Working Backwards (Factoring)

If you learn these cases well, you will be able to spot them in factoring. The product of what two binomials results in 4x^2-9. We know that 4x^2 is a square and 9 is a square, and in the form a^2-b^2, so we see that it is a difference of squares. Take the square root of each to determine a=2x and b=3. (2x+3)(2x-3) is our answer.

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