How do you find a polynomial function that has zeros 0, 10?

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How do you find a polynomial function that has zeros 0, 10?

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Answer 1 · 2017-01-10T03:43:48

$x^{2} - 10 x$ $x^2-10x$

Explanation:

A polynomial $P (x)$ $P(x)$ has a zero $x_{0}$ $x_0$ if and only if $(x - x_{0})$ $(x-x_0)$ is a factor of $P (x)$ $P(x)$ . Using that, we can work backwards to make a polynomial with given zeros by multiplying each necessary factor of $(x - x_{0})$ $(x-x_0)$ .

As our desired polynomial has $0$ $0$ and $10$ $10$ as zeros, it must have $(x - 0)$ $(x-0)$ and $(x - 10)$ $(x-10)$ as factors. Multiplying these, we get

$(x - 0) (x - 10) = x (x - 10) = x^{2} - 10 x$ $(x-0)(x-10) = x(x-10) = x^2-10x$

This is a polynomial of least degree which has $0$ $0$ and $10$ $10$ as zeros. Note that multiplying this by any other polynomial or constant will also result in a polynomial with $0$ $0$ and $10$ $10$ as zeros.

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How do you find a polynomial function that has zeros 0, 10?

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