How do you find the polynomial function with roots -5, 1, 2?

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How do you find the polynomial function with roots -5, 1, 2?

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Answer 1 · 2015-12-19T21:33:24

Begin with the factored form to find that one such polynomial is

$P (x) = x^{3} + 2 x^{2} - 13 x + 10$ $P(x) = x^3 +2x^2 - 13x + 10$

Explanation:

A simple way of generating a polynomial with given roots is to begin with it in factored form. If $(x - x_{0})$ $(x-x_0)$ is a factor of a polynomial $P (x)$ $P(x)$ , then $P (x_{0}) = 0$ $P(x_0) = 0$ and so $x_{0}$ $x_0$ is a root of $P (x)$ $P(x)$ .

Applying this here, we get

$P (x) = (x - (- 5)) (x - 1) (x - 2)$ $P(x) = (x-(-5))(x-1)(x-2)$

Again, plugging in any of the desired roots immediately yields $0$ $0$ , meaning the polynomial has the desired properties. Then, all that remains is to multiply it out.

$P (x) = (x + 5) (x - 1) (x - 2)$ $P(x) = (x+5)(x-1)(x-2)$

$= x^{3} + 2 x^{2} - 13 x + 10$ $= x^3 + 2x^2 -13x + 10$

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How do you find the polynomial function with roots -5, 1, 2?

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