Question

what could be the equation of p? choose 1 answer: a $p(x) = (x - 1)(2x + 7)(x + 1)^2$ b $p(x) = (x - 1)(2x + 7)^2(x + 1)^2$ c $p(x) = (x - 1)(2x + 7)^2(x + 1)$ d $p(x) = (x - 1)^2(2x + 7)^2(x + 1)$

Explanation:

Step1: Recall Polynomial Roots and Multiplicities

For a polynomial \( p(x) \) in factored form, each factor \((x - a)\) corresponds to a root \( x = a \), and the exponent of the factor is the multiplicity of the root. If the graph touches the x - axis at a root, the multiplicity is even; if it crosses the x - axis, the multiplicity is odd. (Assuming the original problem has a graph where we can infer multiplicities, but since we need to choose from the options, we analyze the exponents.)

Step2: Analyze Each Option's Multiplicities

• Option A: \( p(x)=(x - 1)(2x + 7)(x + 1)^2 \)
• Root \( x = 1 \): multiplicity 1 (odd, crosses x - axis)
• Root \( x=-\frac{7}{2} \): multiplicity 1 (odd, crosses x - axis)
• Root \( x=-1 \): multiplicity 2 (even, touches x - axis)
• Option B: \( p(x)=(x - 1)(2x + 7)^2(x + 1)^2 \)
• Root \( x = 1 \): multiplicity 1 (odd, crosses)
• Root \( x=-\frac{7}{2} \): multiplicity 2 (even, touches)
• Root \( x=-1 \): multiplicity 2 (even, touches)
• Option C: \( p(x)=(x - 1)(2x + 7)^2(x + 1) \)
• Root \( x = 1 \): multiplicity 1 (odd, crosses)
• Root \( x=-\frac{7}{2} \): multiplicity 2 (even, touches)
• Root \( x=-1 \): multiplicity 1 (odd, crosses)
• Option D: \( p(x)=(x - 1)^2(2x + 7)^2(x + 1) \)
• Root \( x = 1 \): multiplicity 2 (even, touches)
• Root \( x=-\frac{7}{2} \): multiplicity 2 (even, touches)
• Root \( x=-1 \): multiplicity 1 (odd, crosses)

(Assuming the original problem's graph: If the graph touches the x - axis at \( x =-\frac{7}{2} \) and \( x=-1 \), and crosses at \( x = 1 \), then the multiplicities of \( x=-\frac{7}{2} \) and \( x=-1 \) should be even, and multiplicity of \( x = 1 \) should be odd. Option B has \( x=-\frac{7}{2} \) (multiplicity 2, even), \( x=-1 \) (multiplicity 2, even), and \( x = 1 \) (multiplicity 1, odd), which fits the typical behavior of a polynomial graph with those touch/cross properties.)

Answer:

B. \( p(x)=(x - 1)(2x + 7)^2(x + 1)^2 \)