Question

For the polynomial function ƒ(x) = x4 −25x2, 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

ANSWER

The zeros are $x=-5,x=0,x=5$

EXPLANATION

Given;

$f(x)=x^4-25x^2$.

We can rewrite the function as

$f(x)=x^2(x^2-25)$

$\Rightarrow f(x)=x^2(x^2-5^2)$

$\Rightarrow f(x)=x^2(x-5)(x+5)$

The zeros are found by equating the function to zero.

$\Rightarrow x^2(x-5)(x+5)=0$

$\Rightarrow (x-5)=0$

The multiplicity is 1, since it is odd the graph crosses at this intercept. which is $x=5$

Or

$\Rightarrow (x+5)=0$

The multiplicity is 1, since it is odd the graph crosses at this intercept. which is $x=-5$

Or

$\Rightarrow x^2=0$

This last root has a multiplicity of 2.

That is

$x=0$ repeats two times.

Since the multiplicity is even, the graph touches the x-axis at the point $x=0$.

See graph.