# How do you write a polynomial with Zeros: -2, multiplicity 2; 4, multiplicity 1; degree 3?

##### 1 Answer

#### Explanation:

For a polynomial, if *zero* of the function, then *factor* of the function.

We have two unique zeros: *twice*.

Follow the colors to see how the polynomial is constructed:

#"zero at "color(red)(-2)", multiplicity "color(blue)2#

#"zero at "color(green)4", multiplicity "color(purple)1#

#p(x)=(x-(color(red)(-2)))^color(blue)2(x-color(green)4)^color(purple)1#

Thus,

#p(x)=(x+2)^2(x-4)#

Expand:

#p(x)=(x^2+4x+4)(x-4)#

#p(x)=x^3-12x-16#

We can graph the function to understand multiplicities and zeros visually:

graph{x^3-12x-16 [-6, 6, -43.83, 14.7]}

The zero at

The zero at

Note that the function *does* have three zeros, which it is guaranteed by the Fundamental Theorem of Algebra, but one of such zeros is represented twice.