Question

1. the function h has a zero at x = -3 with a multiplicity 1, a zero at x = 2 with multiplicity 2, and a zero at x = 4 with multiplicity 3.

a. is it possible for h to have a degree of 4? give a reason for your answer.
b. is it possible for h to have a degree of 7? give a reason for your answer.

Explanation:

Step1: Recall degree - multiplicity relation

The degree of a polynomial is the sum of the multiplicities of its zeros.

Step2: Calculate sum of multiplicities

The multiplicities are 1 (for $x = - 3$), 2 (for $x = 2$) and 3 (for $x = 4$). So the sum is $1+2 + 3=6$.

a.

Since the sum of the multiplicities of the known zeros is 6, it is not possible for $h$ to have a degree of 4 because the degree of a polynomial is at least the sum of the multiplicities of its known real - valued zeros.

Answer:

No, because the sum of the multiplicities of the given zeros is 6 which is greater than 4.

b.

It is possible for $h$ to have a degree of 7. The sum of the multiplicities of the given zeros is 6. There could be one more zero of multiplicity 1 (or other combinations of additional zeros that would make the total sum of multiplicities equal to 7).