# How to solve this zero polynomial problem

Let $a$, $b$ and $c$ be real numbers such that:

$$P(x)=4x^3-b^2\cdot x^a+c-b$$

If this polynomial is a zero polynomial, what is the least value of $a\cdot b+c$?

I'm really stumped on how to go about solving this problem. I haven't solved a zero polynomial problem before so this is a tad bit too confusing for me to solve it.

I'd post my attempts at solving this but they were worthless and this has taken a good chunk of my time already.

Thanks in advance!

## 2 Answers

If $a\ne 3$ then we can't have zero polynomial. So $a=3$. Now $b^2=4$ and $c=b$, so $b=2=c$.

• "If a≠3 then we can't have zero polynomial." Can you explain to me what you mean by this? – dimwitt04 Dec 8 '17 at 18:28
• For example if $a$ is at least 4 then we have polynomial of the degree at least 4. If $a=2$ we have $deg(P(X)) = 2$. – Maria Mazur Dec 8 '17 at 18:29
• What about $b=c=-2$? – dromastyx Dec 8 '17 at 18:40
• Yes, that is actually correct is in this case we get smaller value. – Maria Mazur Dec 8 '17 at 18:41

Hint: All coefficients in the zero polynomial are zero.