# How do you find a polynomial function that has zeros x=-2, 4, 7 and degree n=3?

Feb 21, 2017

${x}^{3} - 9 {x}^{2} + 6 x + 56$

#### Explanation:

If a polynomial has zeros: $\textcolor{red}{x = - 2} , \textcolor{b l u e}{x = 4} , \mathmr{and} \textcolor{g r e e n}{x = 7}$
then it has factors
$\textcolor{w h i t e}{\text{XXX}} \left(x - \left(\textcolor{red}{- 2}\right)\right) , \left(x - \textcolor{b l u e}{4}\right) , \mathmr{and} \left(x - \textcolor{g r e e n}{7}\right)$

Furthermore, if the polynomial has degree $3$, then these $3$ factors are the only factors and the polynomial is
$\textcolor{w h i t e}{\text{XXX}} \left(x - \left(\textcolor{red}{- 2}\right)\right) \times \left(x - \textcolor{b l u e}{4}\right) \times \left(x - \textcolor{g r e e n}{7}\right)$

{: (,underline(xx),underline(" | "),underline(x),underline(color(red)(+2))), (,x," | ",x^2,+2x), (,underline(color(blue)(-4)),underline(" | "),underline(-4x),underline(-8)), (," | ",x^2,-2x,-8), (,,,,), (,,,,), (underline(xx),underline(" | "),underline(x^2),underline(-2x),underline(-8)), (x," | ",x^3,-2x^2,-8x), (underline(color(green)(-7)),underline(" | "),underline(-7x^2),underline(+14x),underline(+56)), (,x^3,-9x^2,+6x,56) :}