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Simple Polynomial Multiplication (page 1 of 3)

Sections: Simple multiplication, "FOIL" (and a warning), General multiplication

There were two formats for adding and subtracting polynomials: "horizontal" and "vertical". You can use those same two formats for multiplying polynomials. The very simplest case for polynomial multiplication is the product of two one-term polynomials. For instance:

• Simplify (5x²)(–2x³)

I've already done this type of multiplication when I was first learning about exponents, negative numbers, and variables. I'll just apply the rules I already know:

(5x²)(–2x³) = –10x⁵

The next step up in complexity is a one-term polynomial times a multi-term polynomial. For example:

• Simplify –3x(4x² – x + 10)

To do this, I have to distribute the –3x through the parentheses:

–3x(4x² – x + 10)
     =  –3x(4x²) – 3x(–x) – 3x(10)
     =  –12x³ + 3x² – 30x

The next step up is a two-term polynomial times a two-term polynomial. This is the simplest of the "multi-term times multi-term" cases. There are actually three ways to do this. Since this is one of the most common polynomial multiplications that you will be doing, I'll spend a fair amount of time on this.

• Simplify (x + 3)(x + 2)

The first way I can do this is "horizontally"; in this case, however, I'll have to distribute twice, taking each of the terms in the first parentheses "through" each of the terms in the second parentheses:   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

(x + 3)(x + 2)
     =  (x + 3)(x) + (x + 3)(2)
     =  x(x) + 3(x) + x(2) + 3(2)
     =  x² + 3x + 2x + 6
     =  x² + 5x + 6

This is probably the most difficult and error-prone way to do this multiplication. The "vertical" method is much simpler. First, think back to when you were first learning about multiplication. When you did small numbers, it was simplest to work horizontally, as I did in the first two polynomial examples above:

3 × 4 = 12

But when you got to larger numbers, you stacked the numbers vertically and, working from right to left, took one digit at a time from the lower number and multiplied it, right to left, across the top number. For each digit in the lower number, you formed a row underneath, stepping the rows off to the left as you worked from digit to digit in the lower number. Then you added down.

For instance, you would probably not want to try to multiply 121 by 32 horizontally, but it's easy when you do it vertically:

You can multiply polynomials in this same manner, so here's the same exercise as above, but done "vertically" this time:

• Simplify (x + 3)(x + 2)

I need to be sure to do my work very neatly.

I'll set up the multiplication:

...and then I'll multiply:

I get the same answer as before:  x² + 5x + 6

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