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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.

• 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, no? Now I'll do it vertically:

Much nicer! But, either way, 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)

I'll take my time, and do my work neatly:   Copyright © Elizabeth Stapel 2000-2011 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 an x³ term, an x² term, and a constant term, but no x term; and the second polynomial has an x³ term, an x term, and a constant term, but no x² term. When I do the vertical multiplication, I will need to leave spaces in my set-up, corresponding to the "gaps" in the degrees of the polynomials' terms, because I will almost certainly need the space.

(This is similar to using zeroes as "place holders" in regular numbers. You might have a thousands digit of 3, a hundreds digit of 2, and a units digit of 5, so you'd put a 0 in for the tens digits, creating the number 3,205.)

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. Warning: Take the care to write things neatly, and you'll save yourself from many needless difficulties.

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

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