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Consider the graph of p(x). Given that (x−3) is a factor of p(x), is the multiplicity of (x−3) even or odd? Why?

(1 point)
Responses

The multiplicity of (x−3) is even because the graph of p(x) touches the x-axis at (3,0) and then turns around.
The multiplicity of left parenthesis x minus 3 right parenthesis is even because the graph of p ( x ) touches the x -axis at left parenthesis 3 comma 0 right parenthesis and then turns around.

The multiplicity of (x−3) is odd because the graph of p(x) touches the x-axis at (3,0) and then turns around.
The multiplicity of left parenthesis x minus 3 right parenthesis is odd because the graph of p ( x ) touches the x -axis at left parenthesis 3 comma 0 right parenthesis and then turns around.

The multiplicity of (x−3) is even because the associated zero, x=3, is odd.
The multiplicity of left parenthesis x minus 3 right parenthesis is even because the associated zero, x equals 3 , is odd.

The multiplicity of (x−3) is odd because the associated zero, x=3, is odd.

1 answer

To determine whether the multiplicity of \( (x - 3) \) is even or odd, we can analyze the behavior of the graph of \( p(x) \) at the x-intercept \( (3, 0) \).

If \( (x - 3) \) is a factor of \( p(x) \) and it behaves in such a way that the graph of \( p(x) \) touches the x-axis and turns around at \( (3, 0) \), this indicates that the multiplicity is even. When the multiplicity is even, the graph does not cross the x-axis at that point but merely touches it.

Conversely, if the graph of \( p(x) \) were to cross the x-axis at \( (3, 0) \), this would imply that the multiplicity is odd.

Given the provided response options, the following statements capture the correct reasoning:

  1. The multiplicity of \( (x - 3) \) is even because the graph of \( p(x) \) touches the x-axis at \( (3,0) \) and then turns around.

This statement accurately reflects the situation if \( (x - 3) \) is an even multiplicity. Hence, if you observe that the graph behaves this way, the multiplicity of \( (x - 3) \) is indeed even.

Therefore, the correct answer is: The multiplicity of (x−3) is even because the graph of p(x) touches the x-axis at (3,0) and then turns around.