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) = 6(x - 9)(x + 2)^3)

determine the zero(s).

the zero(s) is/are
(type integers or decimals. use a comma to separate answers as needed.)

Explanation:

Find the zeros of the function

Using the Polynomial Functions knowledge point

$$
LATEXBLOCK0
$$

Determine the multiplicity of each zero

The exponent of each linear factor determines its multiplicity.

• For the factor \((x - 9)^1\), the exponent is \(1\), so the zero \(x = 9\) has multiplicity \(1\).
• For the factor \((x + 2)^3\), the exponent is \(3\), so the zero \(x = -2\) has multiplicity \(3\).

Analyze the graph behavior at each zero

The behavior of the graph at an \(x\)-intercept depends on whether the multiplicity of the zero is odd or even.

• Since the multiplicity of \(x = 9\) is \(1\) (odd), the graph crosses the \(x\)-axis at \(x = 9\).
• Since the multiplicity of \(x = -2\) is \(3\) (odd), the graph crosses the \(x\)-axis at \(x = -2\).

Answer:

Determine the zero(s).

The zero(s) is/are <blank>\(9, -2\)</blank>
(Type integers or decimals. Use a comma to separate answers as needed.)