Given k(x)=−0.5x5(3x+1)2(1−4x)

Question

, find the multiplicity of (3x+1)
and describe the behavior of the graph of k(x) at the associated x-intercept.(1 point)
Responses

The multiplicity of (3x+1)
is 2. At the associated x-intercept, the graph of k(x) crosses the x-axis.
The multiplicity of left parenthesis 3 x plus 1 right parenthesis is 2. At the associated x -intercept, the graph of k ( x ) crosses the x -axis.

The multiplicity of (3x+1)
is −13
. At the associated x-intercept, the graph of k(x) crosses the x-axis.
The multiplicity of left parenthesis 3 x plus 1 right parenthesis is negative Start Fraction 1 over 3 End Fraction . At the associated x -intercept, the graph of k ( x ) crosses the x -axis.

The multiplicity of (3x+1)
is −13
. At the associated x-intercept, the graph of k(x) touches the x-axis and turns around.
The multiplicity of left parenthesis 3 x plus 1 right parenthesis is negative Start Fraction 1 over 3 End Fraction . At the associated x -intercept, the graph of k ( x ) touches the x -axis and turns around.

The multiplicity of (3x+1)
is 2. At the associated x-intercept, the graph of k(x) touches the x-axis and turns around.

Answer 1

To find the multiplicity of \((3x + 1)\) in the function \(k(x) = -0.5x^5(3x + 1)^2(1 - 4x)\), we look at the factor \((3x + 1)\) in the expression.

The factor appears as \((3x + 1)^2\), which indicates that the multiplicity of the root associated with \((3x + 1)\) is 2.

When determining the behavior of the graph at the associated x-intercept, we consider the following rules:

• If the multiplicity is odd, the graph crosses the x-axis at that point.
• If the multiplicity is even, the graph touches the x-axis and turns around.

Since the multiplicity of \((3x + 1)\) is 2 (which is even), the graph of \(k(x)\) will touch the x-axis at the x-intercept associated with this factor and turn around.

Thus, the correct answer is:

The multiplicity of \((3x + 1)\) is 2. At the associated x-intercept, the graph of \(k(x)\) touches the x-axis and turns around.