Question

For the polynomial function ƒ(x) = x4 − 16x2, find the zeros. Then determine the multiplicity at each zero and state whether the graph displays the behavior of a touch or a cross at each intercept.

Answer 1

Our function f(x) can be rewritten if we factor out a common x^2 from each term:
$f(x) = x^2(x^2-16)$
Now inside the parentheses we have a polynomial of the form a^2 - b^2, or the difference of two perfect squares, which can be factored as (a+b)(a-b) so we have:
$f(x)=x^2(x-4)(x+4)$
Setting our function equal to zero gives us the roots x = 0, x = 4, and x = -4.
The multiplicity of the root zero is two since it occurs twice, and the others are one since they occur only once. If you graph the function you can see that it will only touch the x-axis at x = 0, but will cross the x-axis at x = 4 and x = -4.