Questions: 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-1)(x-9)^2 Determine the zero(s). The zero(s) is/are (Type integers or decimals. Use a comma to separate answers as needed.) Determine the multiplicities of the zero(s). Select the correct choice below and, if necessary, fill in the answer box(es) within your choice. A. There are two zeros. The multiplicity of the smallest zero is The multiplicity of the largest zero is (Simplify your answers.) B. There are three zeros. The multiplicity of the smallest zero is The multiplicity of the largest zero is The multiplicity of the other zero is . (Simplify your answers.) C. There is one zero. The multiplicity of the zero is (Simplify your answer.) Determine the behavior of the function at each zero. Select the correct choice below and, if necessary, fill in the answer boxes within your choice. A. The graph touches the x-axis and turns around at all zeros. B. The graph crosses the x-axis at all zeros.

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-1)(x-9)^2 Determine the zero(s). The zero(s) is/are (Type integers or decimals. Use a comma to separate answers as needed.) Determine the multiplicities of the zero(s). Select the correct choice below and, if necessary, fill in the answer box(es) within your choice. A. There are two zeros. The multiplicity of the smallest zero is The multiplicity of the largest zero is (Simplify your answers.) B. There are three zeros. The multiplicity of the smallest zero is The multiplicity of the largest zero is The multiplicity of the other zero is . (Simplify your answers.) C. There is one zero. The multiplicity of the zero is (Simplify your answer.) Determine the behavior of the function at each zero. Select the correct choice below and, if necessary, fill in the answer boxes within your choice. A. The graph touches the x-axis and turns around at all zeros. B. The graph crosses the x-axis at all zeros.
Transcript text: Submit quiz 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. \[ f(x)=4(x-1)(x-9)^{2} \] Determine the zero(s). The zero(s) is/are $\square$ (Type integers or decimals. Use a comma to separate answers as needed.) Determine the multiplicities of the zero(s). Select the correct choice below and, if necessary, fill in the answer box(es) within your choice. A. There are two zeros. The multiplicity of the smallest zero is $\square$ The multiplicity of the largest zero is $\square$ (Simplify your answers.) B. There are three zeros. The multiplicity of the smallest zero is $\square$ The multiplicity of the largest zero is $\square$ The multiplicity of the other zero is $\square$ . (Simplify your answers.) C. There is one zero. The multiplicity of the zero is $\square$ (Simplify your answer.) Determine the behavior of the function at each zero. Select the correct choice below and, if necessary, fill in the answer boxes within your choice. A. The graph touches the $x$-axis and turns around at all zeros. B. The graph crosses the $x$-axis at all zeros.
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Solution

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Solution Steps

Step 1: Identify the zeros of the polynomial

The polynomial function is given by: \[ f(x) = 4(x-1)(x-9)^2 \] To find the zeros, set \( f(x) = 0 \): \[ 4(x-1)(x-9)^2 = 0 \] Since \( 4 \neq 0 \), the zeros occur when: \[ (x-1) = 0 \quad \text{or} \quad (x-9)^2 = 0 \] Solving these equations: \[ x = 1 \quad \text{and} \quad x = 9 \] Thus, the zeros are \( 1 \) and \( 9 \).

Step 2: Determine the multiplicities of the zeros

The multiplicity of a zero is determined by the exponent of its corresponding factor in the polynomial:

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

Thus, the multiplicities are:

  • The multiplicity of the smallest zero (\( 1 \)) is \( 1 \).
  • The multiplicity of the largest zero (\( 9 \)) is \( 2 \).
Step 3: Determine the behavior of the graph at each zero

The behavior of the graph at each zero depends on the multiplicity:

  • 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 and turns around at that zero.

For this polynomial:

  • At \( x = 1 \) (multiplicity \( 1 \)), the graph crosses the \( x \)-axis.
  • At \( x = 9 \) (multiplicity \( 2 \)), the graph touches the \( x \)-axis and turns around.

Thus, the graph crosses the \( x \)-axis at \( x = 1 \) and touches the \( x \)-axis and turns around at \( x = 9 \).

Final Answer

The zero(s) is/are \(1, 9\).
A. There are two zeros. The multiplicity of the smallest zero is \(1\). The multiplicity of the largest zero is \(2\).
A. The graph touches the \(x\)-axis and turns around at \(x = 9\) and crosses the \(x\)-axis at \(x = 1\).

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