Division of a polynomial by binomial is done by long-division method.

In this method to find the first quotient term, divide the first dividend term by the first divisor term.

Multiply the divisor with the term obtained by dividing the first dividend term with the first quotient term.

The product obtained is subtracted from the dividend.

Repeat the same process to divide the other terms of the dividend.

Calculation:

Consider the polynomial \(\frac{x^{2}-7x-78}{x+6}\).

The following steps are used to solve the problem.

First multiply the divisor by x and write the product \(x^{2} + 6x\) under the dividend and subtract.

The value obtained is equals to —13x.

Now multiply the divisor by -13 and write the product —13x — 78 under the dividend and simplify.

The polynomial \(x^{2} — 7x — 78\) is divided as follows,

Then,

\(\frac{x-13}{x}+6)(x^{2}-7x-78)\)

\(\frac{x^{2}+6x}{-13x-78}\)

\(\frac{-13x-78}{0}\rightarrow\ Remainder\)

The simplified value of the polynomial \(\frac{x^{2}-7x-78}{x+6}\) is (x - 13).

Final statement:

The simplified value of the polynomial after division is equals to (x — 13).

Final statement:

The simplified value of the polynomial after division is equals to (x-13).