Question

which could represent a polynomial function with 3 real distinct zeros and a real zero with multiplicity 2? select all that apply.
a. graph a
b. graph b
c. graph c
d. graph d

Explanation:

Step1: Recall zero multiplicity rules

A zero with odd multiplicity crosses the x-axis; a zero with multiplicity 2 (even) touches the x-axis and bounces back. We need 3 distinct zeros that cross the axis, plus 1 zero that touches (multiplicity 2), so total unique real zeros = 4, with one having multiplicity 2.

Step2: Analyze Graph A

Graph crosses x-axis at 3 distinct points, no x-axis touches. Does not match.

Step3: Analyze Graph B

Graph touches x-axis at 1 point (multiplicity 2) and crosses at 2 distinct points. Only 3 unique zeros total. Does not match.

Step4: Analyze Graph C

Graph crosses x-axis at 3 distinct points, and touches x-axis at 1 point (multiplicity 2). Matches the requirement.

Step5: Analyze Graph D

Graph crosses x-axis at 2 distinct points, and touches x-axis at 2 points (each multiplicity 2). Does not match.

Answer:

C. <The graph labeled C that crosses the x-axis at 3 points and touches the x-axis at 1 point>