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 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 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 −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.

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GPT-4o mini · Answer

To analyze the polynomial $ k(x) = -0.5x^5(3x+1)^2(1-4x) $, we will focus on the factor $ (3x+1) $. 1. **Finding the multiplicity of $ (3x+1) $**: The factor $ (3x+1) $ appears in $ k(x) $ as $ (3x+1)^2 $. Therefore, the multiplicity of $ (3x+1) $ is 2. 2. **Describing the behavior of the graph at the associated x-intercept**: When the multiplicity of a factor is even, the graph will touch the x-axis at the x-intercept corresponding to that factor and turn around (bounce off). The x-intercept associated with $ (3x+1) $ is found by setting it to zero: $$ 3x + 1 = 0 \quad \Rightarrow \quad 3x = -1 \quad \Rightarrow \quad x = -\frac{1}{3} $$ At this x-intercept $ x = -\frac{1}{3} $, since the multiplicity is even (specifically, 2), the graph of $ k(x) $ will touch the x-axis and turn around. **Final conclusion**: - The multiplicity of $ (3x+1) $ is 2. - At the associated x-intercept, the graph of $ k(x) $ touches the x-axis and turns around.