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How do you write a polynomial equation of least degree given the roots -1, 1, 5?

1 Answer

Aug 11, 2018

$x^{3} - 5 x^{2} - x + 5 = 0$ $x^3-5x^2-x+5 = 0$

Explanation:

If $a$ $a$ is a zero of a polynomial in $x$ $x$ , then $(x - a)$ $(x-a)$ is a factor and vice versa.

So a polynomial of minimum degree with zeros $- 1$ $-1$ , $1$ $1$ and $5$ $5$ is:

$(x + 1) (x - 1) (x - 5) = (x^{2} - 1) (x - 5) = x^{3} - 5 x^{2} - x + 5$ $(x+1)(x-1)(x-5) = (x^2-1)(x-5) = x^3-5x^2-x+5$

and a polynomial equation of minimum degree with roots $- 1$ $-1$ , $1$ $1$ and $5$ $5$ is:

$x^{3} - 5 x^{2} - x + 5 = 0$ $x^3-5x^2-x+5 = 0$

Any non-zero constant multiple of this cubic equation is also a soltuion.

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