6. (2*x*^{2} - 3*x* + 5)/(*x*^{2} + 4*x* - 6)

Use long division of polynomials.

The polynomial on the left, the one that would be on top of the fraction, is called the *dividend*, and the one to the right, which would be in the bottom, is called the *divisor*. We divide the divisor into the dividend. The long division format looks like

Divide the first term in the divisor into the fisrt term in the dividend. To divide teerms, simply reverse the process for multiplying them. divide the coefficients and subtract the exponents. Then we will see that 2*x*^{2}/*x*^{2} is 2. This is the first term in the answer, which is called the quotient. Put a 2 above the 2*x*^{2}.

Dividing is the oppposite of multiplying, so we simply reverse the steps in multiplying. When you multply thisterm in the quotient times all the terms in the divisor, we then subtract these products from the dividend.

Remember that subtract means "change the sign and add." I will generally keep the original sign and then put the changed sign above it so that I can tell which one is the original sign and which one is the changed sign.

In general, one would then repeat the process with the remainder and keep repeating untiil one got a remainder whose degree was less than the degree of the divisor. Howeve, in this problem, that has happened after the first term inthe quotient, so we are done. There are several ways of expressing the answer. One would be,

2 R -11*x* + 17

or

One can check the answer to a division problem by multiplying the answer by the divisor and seeing if we get the dividend.

We have to multiply the divosor by both of the terms inside the parenthesses.

= 2*x*^{2} + 8*x* - 12 - 11*x* + 17

= 2*x*^{2} - 3*x* + 5

and it checks.