Question

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The graph of a polynomial function of degree 5 has three x-intercepts, all with multiplicity 1. Describe the nature and number of all its zeros.

Answer 1

The answer is C.

Explanation:

Recall that as consequence of the Fundamental Theorem of Algebra, a polynomial of n-th degree has n complex roots (some roots, or all of them, can be real).

Now, as we have a polynomial of degree 5, it has 5 roots. From the statement we know that its graph has three intercepts with the X-axis, which is equivalent to the existence, at least, of three real roots. Moreover, we know that the multiplicity of each root is one. Then, we have exactly three real roots.

The above deduction means that there are other two roots, and those must be complex.

Answer 2

Answer: C

Explanation:

Degree of 5 means there are 5 roots.

If 3 roots are real, then there are 2 remaining - which must be imaginary.