Given k(x)=−0.5x5(3x+1)2(1−4x)
, 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 −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 −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 2. 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 2. 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) crosses the x-axis.
The multiplicity of (3x+1) is 2. At the associated x-intercept, the graph of k(x) crosses the x-axis.
The multiplicity of (3x+1) is -1/3. At the associated x-intercept, the graph of k(x) touches the x-axis and turns around.
To find the multiplicity of a factor in a polynomial, we need to determine the power to which the factor is raised in the polynomial expression. In this case, we are looking for the multiplicity of (3x+1) in the polynomial k(x) = -0.5x^5(3x+1)^2(1-4x).
The multiplicity of (3x+1) is determined by the exponent of the factor. In this case, the factor (3x+1) is raised to the power of 2. Therefore, the multiplicity of (3x+1) is 2.
Now, let's describe the behavior of the graph of k(x) at the associated x-intercept, which occurs when (3x+1) equals 0.
To find the x-intercept, we set (3x+1) equal to 0 and solve for x:
3x + 1 = 0
3x = -1
x = -1/3
At the associated x-intercept of x = -1/3, the graph of k(x) either touches the x-axis and turns around (if the multiplicity is odd) or crosses the x-axis (if the multiplicity is even).
Since the multiplicity of (3x+1) is 2, which is an even number, the graph of k(x) crosses the x-axis at x = -1/3.
Therefore, the correct response is: The multiplicity of (3x+1) is 2. At the associated x-intercept, the graph of k(x) crosses the x-axis.