# How do you write a polynomial function in standard form with Zero: 5, multiplicity 3?

Jul 15, 2016

${x}^{3} - 15 {x}^{2} + 75 x - 125$

#### Explanation:

A zero, say $a$ of a polynomial $f \left(x\right)$ is one for which $f \left(a\right) = 0$.

Let us assume that zeros are $\left\{a , b , c , d\right\}$, then it is apparent that such a polynomial could be $\left(x - a\right) \left(x - b\right) \left(x - c\right) \left(x - d\right)$.

As we need to write a polynomial with zero $5$ with multiplicity $3$, the polynomial is

${\left(x - 5\right)}^{3}$ and using identity ${\left(x - a\right)}^{3} = {x}^{3} - 3 a {x}^{2} + 3 {a}^{2} x - {a}^{3}$, this can be expanded
as

x^3-3×x^2×5+3×x×5^2-5^3 or

${x}^{3} - 15 {x}^{2} + 75 x - 125$.