Find the zeros for the 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) = 4(x + 6)(x + 5)3

Question

Find the zeros for the 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) = 4(x + 6)(x + 5)3

BlackTom AI · Accepted Answer

The question asks us to find all zeros of the polynomial function, state their multiplicities, and describe the x-axis behavior (crossing vs touching and turning around) at each zero.

First, identify the zeros from the factored form: f(x) = 4(x + 6)(x + 5)^3.
- The factor (x + 6) equals zero when x = -6. Since this factor appears to the first power, the multiplicity of the zero x = -6 is 1.
- The factor (x + 5) appears with exponent 3, so the zero x = -5 has multiplicity 3.

Next, determine the crossing vs touching behavior at each zero. A simple rule: if the multiplicity is odd, the graph crosses the x-axis at that zero; if even, it touches and turns around there.
- At x = -6, the multiplicity is 1 (odd), so the graph crosses the x-axis at this zero.
- At x = -5, the multiplicity is 3 (also odd), so the graph also crosses the x-axis at this zero.

With these observations, you can describe the zeros and their behavior:
- Zero x = -6, multiplicity 1, crosses x-axis.
- Zero x = -5, multiplicity 3, crosses x-axis.

Note on the provided data: the answer options field is empty, yet the listed answer in the nearby data suggests exactly these conclusions. This discrepancy means no multiple-choice choices are given to select from here, so the reasoning below stands independently of a provided option list.

If you were to compare to a completed option set, you would expect an option similar to: "-6, multiplicity 1, crosses x-axis; -5, multiplicity 3, crosses x-axis" to be correct, based on the factorization and parity of multiplicities.

Find the zeros for the 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) = 4(x + 6)(x + 5)3单项选择题

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