# What is an example polynomial division problem?

Oct 17, 2016

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

#### Explanation:

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:

$\frac{342}{24} = 13$ with remainder $12$

$\frac{24}{12} = 2$ 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 $2 {x}^{4} + 7 {x}^{3} + 17 {x}^{2} + 16 x - 6$ and ${x}^{4} + 4 {x}^{2} - 8 x + 12$ ?

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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:

$\frac{2 {x}^{4} + 7 x + 17 {x}^{2} + 16 x - 6}{{x}^{4} + 4 {x}^{2} - 8 x + 12} = 2 \text{ }$ with remainder $7 {x}^{3} + 9 {x}^{2} + 32 x - 30$

$\frac{49 \left({x}^{4} + 4 {x}^{2} - 8 x + 12\right)}{7 {x}^{3} + 9 {x}^{2} + 32 x - 30} = 7 x - 9 \text{ }$ with remainder $53 {x}^{2} + 106 x + 318 = 53 \left({x}^{2} + 2 x + 6\right)$

$\frac{7 {x}^{3} + 9 {x}^{2} + 32 x - 30}{{x}^{2} + 2 x + 6} = 7 x - 5 \text{ }$ with remainder $0$

So the GCF is ${x}^{2} + 2 x + 6$