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)ResponsesThe 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.The multiplicity of left parenthesis x minus 3 right parenthesis is odd because the associated zero, x equals 3 , is odd.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.

Question

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)ResponsesThe 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.The multiplicity of  left parenthesis x minus 3 right parenthesis is odd because the associated zero, x equals 3 , is odd.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.

GPT-4o mini · Answer

To determine the multiplicity of the factor \((x−3)\) in the polynomial \(p(x)\), we consider how the graph of \(p(x)\) behaves at the root \(x = 3\):

If the graph touches the x-axis at the point \((3, 0)\) and turns around, this indicates that the multiplicity of the root \(x = 3\) is even.
Conversely, if the graph crosses the x-axis at the point \((3, 0)\), this indicates that the multiplicity of the root is odd.

Based on the information provided in the responses, the correct choice would be:

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 indicates that the factor \((x-3)\) has an even multiplicity.