Special Binomial Products

See what happens when we multiply some binomials ...

Binomial

A binomial is a polynomial with two terms

Algebraic expression 3x + 2 highlighting the two terms
example of a binomial

Product

Product means the result we get after multiplying.

In Algebra xy means x multiplied by y

And (a+b)(a−b) means (a+b) multiplied by (a−b). We use that a lot here!

Special Binomial Products

So when we multiply binomials we get ... Binomial Products!

And we'll look at three special cases of multiplying binomials ... so they are Special Binomial Products.

1. Multiplying a Binomial by Itself

What happens when we 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

Don't forget that middle term: a2 + 2ab + b2

This illustration shows why it works:

Square area model showing (a+b) squared as a squared plus 2ab plus b squared

2. Subtract Times Subtract

And what happens when we square a binomial with a minus inside?

(a−b)2 = (a−b)(a−b) = ... ?

The result:

(a−b)2 = a2 − 2ab + b2

The middle term has a minus: a2 − 2ab + b2

If you want to see why, then look at how the (a−b)2 square is equal to the big a2 square minus the other rectangles:

Area model for (a-b) squared showing subtraction from a larger square of area a squared

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

3. Add Times Subtract

And then there's one more special case ... what about (a+b) times (a−b) ?

(a+b)(a−b) = ... ?

The terms +ab and −ab cancel each other out, so the result is:

(a+b)(a−b) = a2 − b2

with no middle term

That was interesting! It ended up very simple.

And it is called the "difference of two squares" (the two squares are a2 and b2).

This illustration shows why it works:

Geometric rearrangement of a squared minus b squared into a rectangle of (a+b) by (a-b)
a2 − b2   can become   (a+b)(a−b)

We can swap (a−b) with (a+b) to also get:

(a−b)(a+b) = a2 − b2

The Three Cases

Here are the three results we just got:

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

The first two are the "perfect square trinomials" and the last is the "difference of squares"

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

But we can also do it the longer way using Multiplying Polynomials.

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 the result is the difference of two squares, because:

(a+b)(a−b) = a2 − b2

so:

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

Sometimes we can see the pattern of the answer:

Example: which binomials 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 we can multiply (2x+3) and (2x−3) to get 4x2 − 9

356, 357, 1110, 1111, 2284, 2285, 3200, 3201, 3202, 3203
Multiplying Polynomials Algebra Index