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Special Binomial Products

What happens when you multiply some binomials ... ?

Binomial

A binomial is polynomial with two terms

Binomial
example of a binomial

And "Product" means the result you get after multiplying!

We are going to look at three special cases of multiplying binomials ...

Multiplying a Binomial by Itself

What happens when you square a binomial (in other words, multiply it by itself) .. ?

(a+b)2 = (a+b)(a+b) = ... ?

The result:

(a+b)2 = a2 + 2ab + b2

 

And what happens if you square a binomial with a minus inside?

(a-b)2 = (a-b)(a-b) = ... ?

The result:

(a-b)2 = a2 - 2ab + b2

Add Times Subtract

And then there is one more special case... what if you multiply (a+b) by (a-b) ?

(a+b)(a-b) = ... ?

The result:

(a+b)(a-b) = a2 - b2

That was interesting! It ended up very simple. And it is called the "difference of two squares" (the two squares are a2 and b2).

The Three Cases

Here are the three results we just got:

(a+b)2  = a2 + 2ab + b2 } (the "perfect square trinomials")
(a-b)2  = a2 - 2ab + b2
(a+b)(a-b)  = a2 - b2 (the "difference of squares")

Remember those patterns, they will save you time and help you solve many algebra puzzles.

Using Them

So far we have just used "a" and "b", but they could be anything.

Example: (y+1)2

We can use the (a+b)2 case where "a" is y, and "b" is 1:

(y+1)2 = (y)2 + 2(y)(1) + (1)2 = y2 + 2y + 1


Example: (3x-4)2

We can use the (a-b)2 case where "a" is 3x, and "b" is 4:

(3x-4)2 = (3x)2 - 2(3x)(4) + (4)2 = 9x2 - 24x + 16


Example: (4y+2)(4y-2)

We know that the result will be the difference of two squares, because:

(a+b)(a-b) = a2 - b2

so:

(4y+2)(4y-2) = (4y)2 - (2)2 = 16y2 - 4

Sometimes you can recognize the pattern of the answer:

Example: can you work out which binomials to multiply to get 4x2 - 9

Hmmm... is that the difference of two squares?

Yes! 4x2 is (2x)2, and 9 is (3)2, so we have:

4x2 - 9 = (2x)2 - (3)2

And that can be produced by the difference of squares formula:

(a+b)(a-b) = a2 - b2

Like this ("a" is 2x, and "b" is 3):

(2x+3)(2x-3) = (2x)2 - (3)2 = 4x2 - 9

So the answer is that you can multiply (2x+3) and (2x-3) to get 4x2 - 9