Question

Find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=−3(x+1/2)(x−4)3

Accepted Answer

Identify the zeros of the polynomial function by setting each factor equal to zero. For the function $$f(x) = -3\left(x + \frac{1}{2}\right)\left(x - 4\$$right)^3$$, the zeros come from solving x$$ + \frac{1}{2} = 0$$ and x - 4 = 0$. Solve each equation to find the zeros: x$$ + \frac{1}{2} = 0$$ gives x$$ = -\frac{1}{2}$$, and x - 4 = 0$ gives x = 4$. Determine the multiplicity of each zero by looking at the exponent on each factor. The factor $$\left(x + \frac{1}{2}\right)$$ has an implied exponent of 1, so its multiplicity is 1. The factor $$\left(x - 4\$$right)^3$$ has an exponent of 3, so its multiplicity is 3. Use the multiplicity to describe the behavior of the graph at each zero: if the multiplicity is odd (like 1 or 3), the graph crosses the x-axis at that zero. If the multiplicity were even, the graph would touch the x-axis and turn around at that zero. Summarize the results: at $$x = -\frac{1}{2} ($$multiplicity 1), the graph crosses the x-axis; at $$x = 4 ($$multiplicity 3), the graph also crosses the x-axis but with a flatter, cubic behavior near the zero.

Find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=−3(x+1/2)(x−4)³

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Key Concepts

Zeros of a Polynomial Function

Multiplicity of Zeros

Behavior of the Graph at Zeros