Multiplying Simple Polynomials

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• Simplify the following: 2 (3x + 1)

You may already have seen this sort of computation when you learned about simplifying with parentheses.

(You likely won't need to use so many steps as I did above, at least not once you're comfortable with the process, and your instructor almost certainly won't be expecting this much. I'm being overly complete in this lesson in hopes that, by the time you're done, you're sure of what's going on and are comfortable with the process.)

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Moving up in complexity, we can multiply two single-term polynomials (called "monomials").

• Simplify (5x²)(–2x³)

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Usually (and in contrast to the exercise just completed), a monomial that's going to be taken through a parenthetical doesn't have parentheses around it. Instead, the multiplication is indicated simply by the "juxtaposition" of the monomial with (that is, by putting the monomial right next to) the parenthetical expression. This is called "multiplication by juxtaposition", and looks like this:

• Simplify –3x (1 – x)

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The next step up in complexity is multiplying a monomial (rather than a plain number) through a multi-term polynomial.

• Simplify –3x(4x² – x + 10)

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The next step up in complexity is the multiplication of one two-term polynomial by another two-term polynomial (that is, one binomial by another binomial). 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 likely will be doing, I'll spend a fair amount of time on this.

• Simplify (x + 3)(x + 2)

This horizontal multiplication, while mathematically valid, is probably the most difficult and error-prone way to do this multiplication. The "vertical" method is much simpler. Think back to when you were first learning about multiplication. When you did small numbers, it was simplest to work horizontally:

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 (first the ones digit, then the tens digits, then the hundreds digits, and so forth), you formed a new row underneath, stepping the rows off to the left as you worked from right-most digit to left-most 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)