Multiplying Polynomials at a Glance

Our polynomial friends are so excited. They have a few more tricks up their tiny little sleeves. Before we can head on up Polynomial Mountain, we need to use these tricks to master the art of multiplying.

The first little trick is called MNCAE: Multiply the Numerical Coefficients and Add Exponents (of like variables).

Or maybe it was MNCAE: My New Cat Ate Eggs. What does this have to do with the planet of polynomials? Absolutely nothing. We haven't even seen a cat here, but this will still help us remember how to multiply our polynomials.

Time to try this out.

(2x³)(4x²)

Remember to MNCAE. Numbers are multiplied and exponents are added together. Multiply 2 × 4 = 8 and add 3 + 2 = 5.

8x⁵

One more time: MNCAE.

(3a²b⁴)(2ab⁵)

Multiply 3 × 2 = 6. Add 2 + 1 = 3 for the a terms, and add 4 + 5 = 9 for the b's.

6a³b⁹

We did it. Yes. It isn't enough, though. We want to pump *clap* you up.

Sample Problem

Multiply a² + a + 1 by a – 1.

We start by rewriting the problem as:

(a – 1)(a² + a + 1)

We'll use the distributive property and MNCAE to lick this problem. Not literally, though. Gross.

a(a² + a + 1) – 1(a² + a + 1) =

a³ + a² + a – a² – a – 1

Collect the like terms, and remember to only add and subtract the numerical coefficients.

a³ – 1

Another. We need another one. We're on a roll.

Sample Problem

Simplify: (x + y – z)(x – y + z).

Go, My New Cat. Get that polynomial.

x(x – y + z) + y(x – y + z) – z(x – y + z) =

x² – xy + xz + xy – y² + yz – xz + yz – z²

As before, we collect the like terms, and don't touch the exponents. Never again, we said.

(x²) + (-xy + xy) + (xz – xz) + (-y²) + (yz + yz) + (-z²) =

x² – y² + 2yz – z²

Sample Problem

Multiply 3x² + 2xy + y² by x² – 4xy + 2y².

These expressions are getting more complicated, but there's still no worries.

(3x² + 2xy + y²)(x² – 4xy + 2y²)

We know the word that makes the problem go away. No, not the bird, MNCAE.

3x²(x² – 4xy + 2y²) + 2xy(x² – 4xy + 2y²) + y²(x² – 4xy + 2y²) =

3x⁴ – 12x³y + 6x²y² + 2x³y – 8x²y² + 4xy³ + x²y² – 4xy³ + 2y⁴

Okay, the problem doesn't go away immediately. In fact, it looks even bigger. Maybe we should have asked the bird instead. We can fix this, though. Just collect the like terms.

(3x⁴) + (-12x³y + 2x³y) + (6x²y² – 8x²y² + x²y²) + (4xy³ – 4xy³) + (2y⁴) =

3x⁴ – 10x³y – x²y² + 2y⁴

Whew. How about a couple more, before we start up Polynomial Mountain?

Sample Problem

Multiply (5a – 2)(7a² + 3a – 4).

We're so buff now from all this exercise. Maybe after we climb this mountain, we'll return to Earth and do a thousand push-ups to celebrate. That sounds like a good use of our newfound muscle.

5a(7a² + 3a – 4) – 2(7a² + 3a – 4) =

35a³ + 15a² – 20a – 14a² – 6a + 8

Oh yeah, the problem. Multiply coefficients, add exponents, collect like terms.

(35a³) + (15a² – 14a²) + (-20a – 6a) + (8) =

35a³ + a² – 26a + 8

We've got this down. It's time to hike on up the mountain.

Sample Problem

Do you understand a concept better when there's a picture to help? We sure do. If so, multiplying polynomials will be the easiest procedure in all of math. Just use a table.

Think about this multiplication for a moment: (3x² – 2x + 1)(x + 5). You could find the product using the rainbows of multiplication, which look something like this:

Yeesh. It can be a little confusing. What a mess. After we draw our rainbows and multiply and double check that we didn't miss any terms, then we simplify.  Or we could not deal with any mess, and just make a picture and simplify. Let's try it.

First, make a table with as many columns as there are terms in your first polynomial. Write each term in its own column (in order, of course).

Then, in the same table, make as many rows as there are terms in your second polynomial. In this case we get:

And now, we just fill in the box like a multiplication table to find all our terms. Actually, it literally is a multiplication table.

If we had forgotten to multiply two terms together, there would be an empty box. Remember: when we use this method, empty boxes are evil. Anyway, now that we have our terms, we write those terms in an expression and simplify.

3x³ – 2x² + 1x + 15x² – 10x + 5 =
3x³ + 13x² – 9x + 5

Houston, we have our answer.