Finding value of expression with roots of a given polynomial.

Question

If $p(x) = x^3-3x^2+2x+5$ and $p(a)=p(b)=p(c)=0$, what is the value of $(2-a)(2-b)(2-c)$?

At first glance, it seemed to me that I needed to find the roots of the given polynomial. But this polynomial cannot be factorized using factors of the constant term. Is there any other method to find the values of a, b and c ?

Answer 1

Here is an approach assuming there are no other roots of $p(x)$ except $a,b,c$. By Vieta's formulae, the product of the roots is the constant terms of the polynomial, so you can compute the constant term of $p(2-x)$ and that will be your answer.

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UPDATE

@ajotatxe's answer is much better.

Answer 2

Hint:

$$p(x)=(x-a)(x-b)(x-c)$$

(Assuming that $p$ has no more roots, real or complex).

Answer 3

If $a,b,c$ are distinct, you know that $p(x)=(x-a)(x-b)(x-c)$ and so the value we want to compute is simply $(2-a)(2-b)(2-c) = p(2)$, which we know to be equal to 5. In this case, two of these numbers will be complex conjugates.