Question

I NEED HELP ASAP....

The graph of a polynomial function of degree 10 has 5 x-intercepts, 4 of which have multiplicity 1 and one which has multiplicity 2. Describe the nature and number of all the function's zeros.
A) The function has 6 real zeros.
B) The function has 4 imaginary zeros.
C) The function has 6 real and 4 imaginary zeros.
D) The function has 5 real and 5 imaginary zeros.

Answer 1

You're told there is a 10th degree polynomial. There are four roots of multiplicity 1 and one of multiplicity 2 (a double root).

Anytime there is a root of multiplicity 1 of a polynomial, its graph crosses the x-axis at that root. Anytime there is one of multiplicity 2, it means that we count the root twice and the graph has a tangency point.

The degree of a polynomial tells you how how many roots it has. Ours is degree 10, so it has ten roots. We have the four roots of multiplicity 1 and the one of multiplicity two, for a total of 6. (four and two).

So there are six real roots.

The rest of the roots are imaginary and non-real, and 10 - 6 = 4. So there are four imaginary roots.

Thus choice C is best.