Question

which statement is true about the graphed cubic function?

• the function has two distinct real zeros and one complex zero.
• the function has two distinct real zeros and no complex zeros.
• the function has one distinct real zero and two complex zeros.
• the function has three distinct real zeros and no complex zeros.

Explanation:

Identify the real zeros from the graph

Using the Zeros of Polynomials knowledge point

$$
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$$

Determine the number of complex zeros

A cubic function is a polynomial of degree 3. By the Fundamental Theorem of Algebra, a degree 3 polynomial has exactly 3 zeros in the complex number system (counting multiplicity). Since there is exactly 1 real zero, the remaining 2 zeros must be non-real complex conjugate zeros. Thus, the function has one distinct real zero and two complex zeros.

Answer:

• The function has two distinct real zeros and one complex zero.
• The function has two distinct real zeros and no complex zeros.
• The function has one distinct real zero and two complex zeros. (Correct answer)
• The function has three distinct real zeros and no complex zeros.