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Difficulty: MediumPolynomial Factors and Graphs

The polynomial function ff is defined by f(x)=x3−2x2−13x+kf(x) = x^3 - 2x^2 - 13x + k, where kk is a constant. If x−4x - 4 is a factor of f(x)f(x), what is the value of f(−2)f(-2)?

  1. A
    −22-22
  2. 3030Answer
  3. C
    4646
  4. D
    5454

Answer

The correct value of f(−2)f(-2) is 3030.
The correct answer is 3030. According to the Factor Theorem, if x−4x - 4 is a factor of f(x)f(x), then f(4)=0f(4) = 0. Substituting x=4x = 4 into f(x)=x3−2x2−13x+kf(x) = x^3 - 2x^2 - 13x + k gives 43−2(4)2−13(4)+k=04^3 - 2(4)^2 - 13(4) + k = 0. Simplifying this expression results in 64−32−52+k=064 - 32 - 52 + k = 0, which gives −20+k=0-20 + k = 0, so k=20k = 20. The polynomial is therefore f(x)=x3−2x2−13x+20f(x) = x^3 - 2x^2 - 13x + 20. Evaluating this function at x=−2x = -2 gives f(−2)=(−2)3−2(−2)2−13(−2)+20=−8−8+26+20=30f(-2) = (-2)^3 - 2(-2)^2 - 13(-2) + 20 = -8 - 8 + 26 + 20 = 30.

Step-by-Step Solution

1
Apply the Factor Theorem to set up an equation for kk.
f(4)=0f(4) = 0
According to the Factor Theorem, if x−cx - c is a factor of a polynomial f(x)f(x), then f(c)=0f(c) = 0. Since x−4x - 4 is a factor, substituting x=4x = 4 into f(x)f(x) must yield 00.
2
Substitute x=4x = 4 into the polynomial and solve for kk.
k=20k = 20
Evaluating f(4)f(4) gives 43−2(4)2−13(4)+k=04^3 - 2(4)^2 - 13(4) + k = 0. This simplifies to 64−32−52+k=064 - 32 - 52 + k = 0, which simplifies further to −20+k=0-20 + k = 0. Solving this equation gives k=20k = 20.
3
Write the complete polynomial function using the solved value of kk.
f(x)=x3−2x2−13x+20f(x) = x^3 - 2x^2 - 13x + 20
Replacing the constant kk with 2020 in the original function definition gives the complete polynomial expression.
4
Evaluate the polynomial function at x=−2x = -2.
f(−2)=30f(-2) = 30
Substituting x=−2x = -2 into the polynomial gives f(−2)=(−2)3−2(−2)2−13(−2)+20f(-2) = (-2)^3 - 2(-2)^2 - 13(-2) + 20. Evaluating the terms gives −8−8+26+20=30-8 - 8 + 26 + 20 = 30.

Key Concept

The Factor Theorem states that a polynomial f(x)f(x) has a factor x−cx - c if and only if f(c)=0f(c) = 0. This can be used to determine unknown coefficients in a polynomial before evaluating the function at a different value.
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Topics

Polynomial Factors and Graphs

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