What is an example polynomial division problem?

The GCF of two positive integers can be found using this method:

• Divide the larger number by the smaller to give a quotient and remainder.

• If the remainder is `0` then the smaller number is the GCF.

• Otherwise repeat with the smaller number and remainder.

For example:

`342 / 24 = 13` with remainder `12`

`24 / 12 = 2` with remainder `0`

So the GCF of `342` and `24` is `12`.

We can do the same with polynomials.

For example:

What is the GCF of `2x^4+7x^3+17x^2+16x-6` and `x^4+4x^2-8x+12` ?

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Solution

We can divide polynomials by dividing their coefficients, not forgetting to include `0` for any missing powers of `x`.

In the following long divisions I have premultiplied the dividend in the second division by `7^2=49` to avoid having to deal with fractions. This does not compromise the goal of finding the GCF polynomial, as scalar factors are not important to us.

So:

`(2x^4+7x+17x^2+16x-6)/(x^4+4x^2-8x+12) = 2" "` with remainder `7x^3+9x^2+32x-30`

`(49(x^4+4x^2-8x+12))/(7x^3+9x^2+32x-30) = 7x-9" "` with remainder `53x^2+106x+318 = 53(x^2+2x+6)`

`(7x^3+9x^2+32x-30)/(x^2+2x+6) = 7x-5" "` with remainder `0`

So the GCF is `x^2+2x+6`