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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. Find all values of x where f(x) = x^3 - 3x is increasing.

    Answer: x 1

    f'(x)=3x^2-3=3(x-1)(x+1)>0 when |x|>1, i.e., x1.

  2. A polynomial P(x) of degree 4 with leading coefficient 1 has roots 1, -1, 2, and -2. Write P(x).

    Answer: x^4 - 5x^2 + 4

    P(x)=(x-1)(x+1)(x-2)(x+2)=(x^2-1)(x^2-4)=x^4-5x^2+4.

  3. The polynomial x^3 + px + q has a double root at x = 2. Find p.

    Answer: -12

    A double root at 2 means (x-2)^2(x-r) with sum of roots giving r+4=0, r=-4; expanding gives x^3-8x+... wait: (x-2)^2(x+4)=x^3+0x^2-12x+... check: p=-12.

  4. For what value(s) of k does kx^2 - 4x + 1 = 0 have exactly one real solution?

    Answer: 4

    Set discriminant to zero: 16-4k=0, k=4.

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

    Answer: 65/8

    f(3)=8 and f(-3)=1/8; sum=8+1/8=65/8.

  6. Find the domain of f(x) = √(4 - x^2).

    Answer: -2 ≤ x ≤ 2

    Require 4-x^2≥0, so x^2≤4, giving -2≤x≤2.