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Difficulty: HardPolynomial Factors and Graphs

The polynomial function pp is defined by p(x)=x3+bx2+cx+dp(x) = x^3 + bx^2 + cx + d, where bb, cc, and dd are constants. In the xyxy-plane, the graph of y=p(x)y = p(x) has xx-intercepts at (2,0)(2, 0) and (−3,0)(-3, 0). If the remainder when p(x)p(x) is divided by x−1x - 1 is −8-8, what is the value of dd?

  1. −6-6Answer
  2. B
    66
  3. C
    −2-2
  4. D
    1818

Answer

−6-6
The correct value is −6-6. Since the graph has xx-intercepts at (2,0)(2, 0) and (−3,0)(-3, 0), the Factor Theorem dictates that (x−2)(x - 2) and (x+3)(x + 3) are factors of p(x)p(x). Since the leading coefficient of the cubic polynomial p(x)p(x) is 11, it can be written in factored form as p(x)=(x−2)(x+3)(x−r)p(x) = (x - 2)(x + 3)(x - r), where rr is the third root. According to the Remainder Theorem, the remainder when p(x)p(x) is divided by x−1x - 1 is equal to p(1)=−8p(1) = -8. Substituting x=1x = 1 yields −4(1−r)=−8-4(1 - r) = -8, which simplifies to 1−r=21 - r = 2, so r=−1r = -1. The polynomial is thus p(x)=(x−2)(x+3)(x+1)p(x) = (x - 2)(x + 3)(x + 1). The constant term dd is equal to the value of the function when x=0x = 0, which is p(0)=(−2)(3)(1)=−6p(0) = (-2)(3)(1) = -6.

Step-by-Step Solution

1
Determine the factors corresponding to the given xx-intercepts of the polynomial.
The factors are (x−2)(x - 2) and (x+3)(x + 3).
Since the graph of y=p(x)y = p(x) has xx-intercepts at (2,0)(2, 0) and (−3,0)(-3, 0), we know that p(2)=0p(2) = 0 and p(−3)=0p(-3) = 0. According to the Factor Theorem, (x−2)(x - 2) and (x+3)(x + 3) must be factors of p(x)p(x).
2
Set up the factored form of the cubic polynomial function p(x)p(x).
p(x)=(x−2)(x+3)(x−r)p(x) = (x - 2)(x + 3)(x - r)
Because p(x)p(x) is a cubic polynomial (degree 3) with a leading coefficient of 11, it can be written as the product of three linear factors: (x−2)(x - 2), (x+3)(x + 3), and (x−r)(x - r), where rr is the unknown third root.
3
Apply the Remainder Theorem to solve for the third root rr.
r=−1r = -1
By the Remainder Theorem, the remainder when p(x)p(x) is divided by x−1x - 1 is equal to p(1)p(1). We are given that this remainder is −8-8, so p(1)=−8p(1) = -8. Substituting x=1x = 1 into our factored expression gives: p(1)=(1−2)(1+3)(1−r)=−8⇒−1⋅4⋅(1−r)=−8⇒−4(1−r)=−8⇒1−r=2⇒r=−1p(1) = (1 - 2)(1 + 3)(1 - r) = -8 \Rightarrow -1 \cdot 4 \cdot (1 - r) = -8 \Rightarrow -4(1 - r) = -8 \Rightarrow 1 - r = 2 \Rightarrow r = -1.
4
Determine the value of the constant coefficient dd.
d=−6d = -6
Using the root r=−1r = -1, the complete factored expression is p(x)=(x−2)(x+3)(x+1)p(x) = (x - 2)(x + 3)(x + 1). The constant term dd is equivalent to p(0)p(0): d=p(0)=(0−2)(0+3)(0+1)=(−2)(3)(1)=−6d = p(0) = (0 - 2)(0 + 3)(0 + 1) = (-2)(3)(1) = -6.

Key Concept

Using the Factor Theorem and Remainder Theorem to find unknown coefficients in a polynomial function.

Alternative Method

Alternatively, you can expand the general form p(x)=(x−2)(x+3)(x−r)=(x2+x−6)(x−r)=x3+(1−r)x2−(r+6)x+6rp(x) = (x - 2)(x + 3)(x - r) = (x^2 + x - 6)(x - r) = x^3 + (1 - r)x^2 - (r + 6)x + 6r. Comparing this to p(x)=x3+bx2+cx+dp(x) = x^3 + bx^2 + cx + d, we see that d=6rd = 6r. Since the remainder when p(x)p(x) is divided by x−1x - 1 is −8-8, we have p(1)=−8p(1) = -8. Substituting x=1x = 1 into our expanded form gives 13+(1−r)(1)2−(r+6)(1)+6r=−8⇒1+1−r−r−6+6r=−8⇒4r−4=−8⇒4r=−4⇒r=−11^3 + (1 - r)(1)^2 - (r + 6)(1) + 6r = -8 \Rightarrow 1 + 1 - r - r - 6 + 6r = -8 \Rightarrow 4r - 4 = -8 \Rightarrow 4r = -4 \Rightarrow r = -1. Thus, d=6(−1)=−6d = 6(-1) = -6.
Estimated Time:2m 0s
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Topics

Polynomial Factors and Graphs

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