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Nonlinear Equations in One Variable and Systems of Equations

SAT Math· difficulty 5/5

When polynomial p(x)p(x) is divided by x2−1x^2 - 1, the quotient is x+2x + 2 and the remainder is 3x−13x - 1. What is p(x)p(x)?

  • A

    x3+2x2+3x−1x^3 + 2x^2 + 3x - 1

  • B

    x3−2x2+2x+3x^3 - 2x^2 + 2x + 3

  • C

    x3+2x2−3x−1x^3 + 2x^2 - 3x - 1

  • D

    x3+2x2+2x−3x^3 + 2x^2 + 2x - 3

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Explanation

p(x)=(x2−1)(x+2)+(3x−1)=x3+2x2−x−2+3x−1=x3+2x2+2x−3p(x) = (x^2 - 1)(x + 2) + (3x - 1) = x^3 + 2x^2 - x - 2 + 3x - 1 = x^3 + 2x^2 + 2x - 3.

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