SAT Math Multiple Choice Question 941: Answer and Explanation

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Question: 941

9. If x – 4 is a factor of x2kx + 2k, where k is a constant, what is the value of k?

  • A. –4
  • B. 4
  • C. 8
  • D. 12

Correct Answer: C



Difficulty: Hard

Category: Passport to Advanced Math / Quadratics

Strategic Advice: You could substitute each of the answer choices for k and factor the resulting expression, but this will use up valuable time on Test Day. Instead, think about what it means for x - 4 to be a factor of the expression.

Getting to the Answer: If x - 4 is a factor of x2 - kx + 2k, then x2 - kx + 2k can be written as the product (x - 4)(x - a) for some real number a. Expanding the product (x - 4)(x - a) yields x2 - 4x - ax + 4a, which can be rewritten as x2 - (4 + a)x + 4a. Substituting this for the factored form of the original expression results in the equation x2 - (4 + a)x + 4a = x2 - kx + 2k. Two quadratic equations are equal if and only if the coefficients of their like terms are equal, so 4 + a = k and 4a = 2k. You now have a system of equations to solve. The first equation is already solved for k, so substitute 4 + a into the second equation for k and solve for a:

The questions asks for the value of k, and k = 4 + a, so k = 4 + 4 = 8, which is (C).

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