Question

For the following polynomial function, find A) the degree of the polynomial, B) all x-intercepts, and C) the y-intercepts? f(x)=(x^2-25)(x^2-9)

Answer 1

This is a fourth-degree polynomial

X-intercepts are - 5, -5, 3, and -3

The Y-intercept is 225

Explanation:

Since it's already in the factored form you can find the zeros by separating each binomial and solving for x

`x^2-25=0`
`x^2=25`
`x=+-5`

`x^2-9=0`
`x^2=9`
`x=+-3`
so `x= +-5, +-3`

To find the y-intercept and the degree of the polynomial we need to convert the factored form into standard form

`f(x)=(x^2-25)(x^2-9)`
`f(x)=x^4-9x^2-25x^2+225`
`f(x)=x^4-34x^2+225`

The degree of a polynomial is just the leading coefficients power which is 4 in this equation

In order to find the y-intercept we just need to allow `x=0` because that is when any equation will cross the y-axis

`f(x)=0^4-34(0)^2+225`
`f(x)=225`