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Polynomials and Functions Flashcards

6 cards from real AIME practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.

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  1. Factor completely: 2x^3 - 8x.

    Answer: 2x(x-2)(x+2)

    Factor out 2x: 2x(x^2-4)=2x(x-2)(x+2).

  2. If P(x) = x^3 + 2x^2 - 5x - 6 and P(-1) = 0, fully factor P(x).

    Answer: (x+1)(x-2)(x+3)

    Since x=-1 is a root, (x+1) is a factor; dividing gives x^2+x-6=(x-2)(x+3).

  3. What is the degree of the product of x^3 + 1 and x^2 - x + 1?

    Answer: 5

    The degree of a product equals the sum of degrees: 3+2=5.

  4. Find all rational roots of f(x) = x^3 - 6x^2 + 11x - 6.

    Answer: {1, 2, 3}

    By the Rational Root Theorem, test divisors of 6; x=1,2,3 all satisfy f(x)=0.

  5. If f(x) = 2x + 3, find f^(-1)(11).

    Answer: 4

    Solve 2x+3=11: x=4, so f^(-1)(11)=4.

  6. What is the sum of the roots of 4x^3 - 8x^2 + 5x - 1 = 0?

    Answer: 2

    By Vieta's formulas, sum of roots = -(-8)/4 = 2.