Question

ALGEBRA 2 help please;Question in the attached

Answer 1

`f(x) = x^(5) -3x^(2) -2` has 5 real and 0 non-real zeroes.

Explanation:

Here, the given polynomial is
`f(x) = x^(5) -3x^(2) -2`

Now, by FUNDAMENTAL THEOREM OF ALGEBRA:

A polynomial of degree n can have at most n roots.

So, here the number of roots f(x) can have = 5

Now, examine the change in the sign of f(x) and f(-x)

`f(x) = x^(5) -3x^(2) -2`

Signs of f(x) is + - -.

So for f(x) the sign changes only once.

Now,
`f(-x) = (-x)^(5) -3(-x)^(2) -2`

or,
`f(-x) = -x^(5) -3x^(2) -2`

Here, signs are - - -

So,for f(-x) the SIGNS DO NOT CHANGE.

So it has no negative real zero.

Hence,
`f(x) = x^(5) -3x^(2) -2` has 5 real and 0 non-real zeroes.