# Problem involving the factor and remainder theorem

So am struggling to solve this problem as it has been a while since I have been this enticed with mathematics and hence is a bit weary of the subject.

It involves using the factor and remainder theorem to find two constants (a and b) when: $$\ f(x) = x^3 + ax + b$$

and you are given the fact that $(x + 1)$ is a factor of $f(x)$ and you also know that when: $$\frac {f(x)} {(x - 3)}$$ The remainder is 16.

Initially i decided to use the remainder theorem to see if i could work out $a$ and $b$: $$\ f(3) = 3^3 + a(3) + b = 27 + 3a + b$$

but I ended up getting some unfamiliar equation which I didn't think was right.

I'd appreciate it if someone could inform me of how I should ideally try and approach this question as right now am a bit stuck in the mud.

## 1 Answer

Your way is fine, and is the standard way to approach the problem. After getting $$f(3)=27+3a+b=16 \iff 3a+b=-11$$ Use the factor theorem to get that $$f(-1)=-1-a+b=0 \iff b=a+1$$ Putting this back into our original equation, we get $$4a+1=-11 \iff a=-3$$ And thus $a=-3, b=a+1=-2$. Thus $$f(x)=x^3-3x-2$$ Done.

• Oh yes i see now, thank you very much Sheen! – Zochonis Mar 1 '17 at 9:05