4. Polynomial Functions

Understanding Polynomial Functions

4. Polynomial Functions

Understanding Polynomial Functions

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Multiple Choice

Find the zeros of the given polynomial function and give the multiplicity of each. State whether the graph crosses or touches the x-axis at each zero. $f\left(x\right)=x^2\left(x-1\right)^3\left(2x+6\right)$

Cross at $x=0,$ Cross at $x=1,$ Cross at $x=3$

Touch at $x=0,$ Cross at $x=-1,$ Cross at $x=3$

Cross at $x=0,$ Touch at $x=1,$ Touch at $x=-3$

Touch at $x=0,$ Cross at $x=1,$ Cross at $x=-3$

Verified step by step guidance

Identify the factors of the polynomial function f(x) = x^2(x-1)^3(2x+6). The factors are x^2, (x-1)^3, and (2x+6).

Find the zeros of each factor. For x^2, the zero is x = 0. For (x-1)^3, the zero is x = 1. For (2x+6), set 2x+6 = 0, which gives x = -3.

Determine the multiplicity of each zero. The zero x = 0 has a multiplicity of 2, x = 1 has a multiplicity of 3, and x = -3 has a multiplicity of 1.

Analyze the behavior of the graph at each zero. If the multiplicity is odd, the graph crosses the x-axis at that zero. If the multiplicity is even, the graph touches the x-axis at that zero.

Conclude the behavior: The graph touches the x-axis at x = 0 (even multiplicity), crosses at x = 1 (odd multiplicity), and crosses at x = -3 (odd multiplicity).