Question 4.3.2: Using Long Division Divide x^4-13 x²+x+35 by x²-x-6.

Because the dividend does not contain an x³ term, we use a zero coefficient for the missing term.

$\begin{aligned}x ^ { 2 } – x – 6 ) \overset{x^{2}+x-6 \quad \leftarrow \text {Quotient}} {\overline{x ^ { 4 } + 0 x ^ { 3 } – 13 x ^ { 2 } + x + 35}}\\ x^{4}-x^{3}-6 x^{2}\\\overline{x^{3}-7 x^{2}+x+35}\\ x^3-x^2-6x\\\overline{-6 x^{2}+7 x+35 } \\\underline{-6 x^{2}+6 x+36} \\         x – 1 ←\text{Remainder}\end{aligned}$

The quotient is x²+x-6, and the remainder is x – 1.

We can write this result in the form

$\frac{x^{4}-13 x^{2}+x+35}{x^{2}-x-6}=x^{2}+x-6+\frac{x-1}{x^{2}-x-6} .$