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General Polynomial Multiplication (page 3 of 3)

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

Sometimes you will have to multiply one multi-term polynomial by another multi-term polynomial. You can do this horizontally if you want, but there is so much room for error that I usually switch over to vertical multiplication once the polynomials get big. For bigger multiplications, vertical is usually faster, and is much more likely to give you a correct answer. Here are some examples:

• Simplify (4x² – 4x – 7)(x + 3)

Here's what it looks like when done horizontally:

(4x² – 4x – 7)(x + 3)
  =  (4x² – 4x – 7)(x) + (4x² – 4x – 7)(3)
  =  4x²(x) – 4x(x) – 7(x) + 4x²(3) – 4x(3) – 7(3)
  =  4x³ – 4x² – 7x + 12x² – 12x – 21
  =  4x³ – 4x² + 12x² – 7x – 12x – 21
  =  4x³ + 8x² – 19x – 21

Painful, huh? Now I'll do it the other way. Vertical:

Much nicer! So the answer is: 4x³ + 8x² – 19x – 21

• Simplify (x + 2)(x³ + 3x² + 4x – 17)

I'm just going to do this one vertically. Note that, since order doesn't matter for multiplication, you can still put the "x + 2" polynomial on the bottom for vertical multiplication, just as you always put the smaller number on the bottom when you were doing regular vertical multiplication with just plain numbers.

So the answer is: x⁴ + 5x³ + 10x² – 9x – 34

• Simplify (3x² – 9x + 5)(2x² + 4x – 7)

Take your time, and do your work neatly:   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

So the answer is:  6x⁴ – 6x³ – 47x² + 83x – 35

• Simplify (x³ + 2x² + 4)(2x³ + x + 1)

Notice that these polynomials have gaps in their terms. The first polynomial has no x term, and the second polynomial has no x² term. LEAVE GAPS in your set-up, because you will almost certainly need the space. Here's what I mean:

See how I needed the gaps? See how it helped that I had everything lined up according to the degree? If I hadn't left gaps, my terms could easily have become misaligned. Take the care to write things neatly, and you'll save yourself so much trouble!

The answer is:  2x⁶ + 4x⁵ + x⁴ + 11x³ + 2x² + 4x + 4

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