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 ?

I don't know Vieta's formula. What is it? – SPat04 Jun 06 '19 at 14:35 — SPat04, Jun 06 '19 at 14:35
Vieta's formulas relate polynomial coefficients to roots – J. W. Tanner Jun 06 '19 at 14:37 — J. W. Tanner, Jun 06 '19 at 14:37

gt6989b · Answer 1 · 2019-06-06T16:08:20.620

4

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.

UPDATE

@ajotatxe's answer is much better.

edited Jun 06 '19 at 16:08

answered Jun 06 '19 at 14:35

gt6989b

54,422

@gt6968b I'll delete the comments in order to reduce the "noise" – PierreCarre Jun 06 '19 at 16:11

ajotatxe · Answer 2 · 2019-06-06T15:01:18.393

4

Hint:

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

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

edited Jun 06 '19 at 15:01

answered Jun 06 '19 at 14:35

ajotatxe

65,084

+1, much better than mine, very clever, should be the accepted answer. – gt6989b Jun 06 '19 at 16:06

PierreCarre · Answer 3 · 2019-06-06T14:48:15.313

2

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.

edited Jun 06 '19 at 14:48

answered Jun 06 '19 at 14:35

PierreCarre

20,974
1
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Thank you. I understand the problem now. – SPat04 Jun 06 '19 at 15:03

Finding value of expression with roots of a given polynomial.

3 Answers3