What are the rational zeros of a polynomial function?

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What are the rational zeros of a polynomial function?

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Answer 1 · 2016-05-30T19:39:16

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Explanation:

A polynomial in a variable $x$ $x$ is a sum of finitely many terms, each of which takes the form $a_{k} x^{k}$ $a_kx^k$ for some constant $a_{k}$ $a_k$ and non-negative integer $k$ $k$ .

So some examples of typical polynomials might be:

$x^{2} + 3 x - 4$ $x^2+3x-4$

$3 x^{3} - \frac{5}{2} x^{2} + 7$ $3x^3-5/2x^2+7$

A polynomial function is a function wholse values are defined by a polynomial. For example:

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

$g (x) = 3 x^{3} - \frac{5}{2} x^{2} + 7$ $g(x) = 3x^3-5/2x^2+7$

A zero of a polynomial $f (x)$ $f(x)$ is a value of $x$ $x$ such that $f (x) = 0$ $f(x) = 0$ .

For example, $x = - 4$ $x=-4$ is a zero of $f (x) = x^{2} + 3 x - 4$ $f(x) = x^2+3x-4$ .

A rational zero is a zero that is also a rational number, that is, it is expressible in the form $\frac{p}{q}$ $p/q$ for some integers $p, q$ $p, q$ with $q \neq 0$ $q != 0$ .

For example:

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

has two rational zeros, $x = \frac{1}{2}$ $x=1/2$ and $x = - 1$ $x=-1$

Note that any integer is a rational number since it can be expressed as a fraction with denominator $1$ $1$ .

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What are the rational zeros of a polynomial function?

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