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#