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How do you write a polynomial equation of least degree given the roots -1, 1, 3, -3?

1 Answer

Aug 11, 2018

$x^{4} - 10 x^{2} + 9 = 0$ $x^4-10x^2+9 = 0$

Explanation:

A polynomial in $x$ $x$ has a zero $a$ $a$ if and only if it has a factor $(x - a)$ $(x-a)$ .

So a polynomial in $x$ $x$ with zeros $- 1$ $-1$ , $1$ $1$ , $3$ $3$ and $- 3$ $-3$ must be a multiple of:

$(x + 1) (x - 1) (x - 3) (x + 3) = (x^{2} - 1) (x^{2} - 9) = x^{4} - 10 x^{2} + 9$ $(x+1)(x-1)(x-3)(x+3) = (x^2-1)(x^2-9) = x^4-10x^2+9$

So a polynomial equation of minimum degree with roots $- 1$ $-1$ , $1$ $1$ , $3$ $3$ and $- 3$ $-3$ is:

$x^{4} - 10 x^{2} + 9 = 0$ $x^4-10x^2+9 = 0$

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