Polynomial Long Division in Algebra

Question

How do i even begin to fathom these questions? How do i begin to answer them? Help would be much appreciated!

1. Divide $X^5 - X^4 - 6X^3 - 8X^2 + 8X +48$ , by $X^2 - X - 6$. Hence fully factorise $X^5 - X^4 - 6X^3 - 8X^2 + 8X +48$.

2. Find all four factors of $36X^4 - 289X^2 + 400$ , if $4X^2 - 25$ divides evenly into the quartic expression.

Answer 1

1. If has a factor $ax+b$ where $a,b$ are integers, then $b$ must divide the constant term and $a$ must divide the coefficient of the highest power. So $a$ must be 1 and $b$ must divide 48. Since $a=1$, it is probably quicker to check if $-b$ is a root. So try $\pm1,\pm2,\pm3,\dots$. You should quickly find that $2,-2,3$ are roots, so $(x-2)(x+2)(x-3)=x^3-3x^2-4x+12$ must be a factor.

The other term must be a quadratic. The constant term must be 48/12=4, and the coefficient of $x^2$ must be 1. It is easy to find that it must be $x^2+2x+4$. If you try to use the formula for the roots of that you find they are complex. So we cannot factorise further.

1. This is much easier. It must be a quadratic. The original has only even powers, so we can start by factorising it into the product of two factors of $ax^2+b$ and $cx^2+d$. We are given one of them, so it is easy to find the other $9x^2-16$. Now check if these two quadratic factors factorise. They both do: $2x\pm5$ and $3x\pm4$.