Factorizing an Algebraic Function

[lifeiseasy]

Homework Statement

Find the divisor in each of the following.
Dividend = x^3 - 4x^2 + x - 1, quotient = x - 6, remainder = 10x + 17

Homework Equations

Dividend = divisor x quotient + remainder

The Attempt at a Solution

By division algorithm, we have
f(x) = [(x^3 - 4x^2 + x - 1) - (10x + 17)] / (x - 6)
= (x^3 - 4x^2 - 9x - 18) / (x - 6)

At this point I can't factorize the numerator. Any help would be appreciated. Thanks!

 

[Mentallic]

So it is obvious from the question that the cubic should have a factor of x-6 and you can even check it by showing f(6)=0.

Anyway, to factorize it all you have to do is solve

$$x^3 - 4x^2 - 9x - 18=(x-6)(ax^2+bx+c)$$

You just need to find the values of the constants a,b,c and you can quickly find two by noting that the cubic power on the right will be (x)(ax²) and this is equivalent to x³ so of course a=1, and the constant on the right is (-6)(c) and this is equal to -18 from the left, so c=3. Now just find the value of b by expanding the right side.

This is called equating coefficients by the way.

 

[lifeiseasy]

Thanks! I just found out that the solution is absolutely obvious if we use factor theorem.