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Nonlinear Equations in One Variable and Systems of Equations

SAT Math· difficulty 4/5

The polynomial p(x)=x3−2x2+kx+7p(x) = x^3 - 2x^2 + kx + 7 has the same remainder when divided by x−1x - 1 and x+1x + 1. What is the value of kk?

  • A

    kk can be any value

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  • B

    k=1k = 1

  • C

    k=0k = 0

  • D

    k=−1k = -1

Explanation

p(1)=1−2+k+7=6+kp(1) = 1 - 2 + k + 7 = 6 + k. p(−1)=−1−2−k+7=4−kp(-1) = -1 - 2 - k + 7 = 4 - k. Setting equal: 6+k=4−k⇒2k=−2⇒k=−16 + k = 4 - k \Rightarrow 2k = -2 \Rightarrow k = -1. So choice should be k=−1k = -1.

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