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How do you write a polynomial function of least degree and leading coefficient 1 when the zeros are 1, -1, 2, -2, 3?

1 Answer

Jul 5, 2016

$f (x) = x^{5} - 3 x^{4} - 5 x^{3} + 15 x^{2} + 4 x - 12$ $f(x) = x^5-3x^4-5x^3+15x^2+4x-12$

Explanation:

Each zero $a$ $a$ corresponds to a linear factor $(x - a)$ $(x-a)$ , so we can write:

$f (x) = (x - 1) (x + 1) (x - 2) (x + 2) (x - 3)$ $f(x) = (x-1)(x+1)(x-2)(x+2)(x-3)$

$= (x^{2} - 1) (x^{2} - 4) (x - 3)$ $=(x^2-1)(x^2-4)(x-3)$

$= (x^{4} - 5 x^{2} + 4) (x - 3)$ $=(x^4-5x^2+4)(x-3)$

$= x^{5} - 3 x^{4} - 5 x^{3} + 15 x^{2} + 4 x - 12$ $=x^5-3x^4-5x^3+15x^2+4x-12$

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