Special Cases of Binomial Multiplication Examples

What is (3x² + 4x)(3x² – 4x)?

Since the first terms of the binomials are the same and the second terms are additive inverses, the answer is:

(3x²)² – (4x)² =
9x⁴ – 16x²

Find two binomials that multiply together to give 25x² – 9.

This is a difference of two squares, so it looks like the answer to a multiplication problem of the form:

(▲ + ■)(▲ – ■)

We need to figure out what ▲ and ■ are. We'll need to work backwards for this one. Use your rear view mirror and proceed cautiously.

Since ▲² = 25x², we have▲ = ±5x.

Similarly, since ■² = 9, we must have ■ = ±3.

Two binomials that multiply together to give 25x² – 9 are (5x + 3) and (5x – 3).

To check our answer, we can perform the multiplication (5x + 3)(5x – 3).

We could have used -5x instead of 5x and/or -3 instead of 3. Either way will work, but for this sort of problem we pick the positive square roots because we're trying to make things look tidy and unnecessary negative signs are messy. Like they say, a messy expression is a sign of a disorganized mind.

Find (6x + 2)².

We know the form of the answer, so we can write it down:

(6x)² + 2(6x)(2) + (2)².

Then, we simplify to get

36x² + 24x + 4.

What is (3x³ – 5x)²?

We know the form of the answer because we were totally paying attention, so we write it down:

(3x³)² – 2(3x³)(5x) + (5x)²

Then we simplify:

9x⁶ – 30x⁴ + 25x²

Notice how much simpler things are now than when we started. Instead of having to write down the distributive property two or three times for every problem, we have quick and easy patterns to follow. Welcome to Shortcut City. Never been there? Follow us...we know a shortcut.