Algebra & Functions Flashcards
7 cards from real AMC 12 practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.
Read the first 7 Algebra & Functions flashcards as text
What is the remainder when x^3 - 2x^2 + 4x - 5 is divided by x - 2?
Answer: 3
By the Remainder Theorem, P(2) = 8 - 8 + 8 - 5 = 3.
A polynomial P satisfies P(1) = 3 and P(-1) = 5. What is the remainder when P(x) is divided by x^2 - 1?
Answer: -x + 4
Writing the remainder as ax + b gives a + b = 3 and -a + b = 5, so a = -1 and b = 4.
What is the value of ⌊√1⌋ + ⌊√2⌋ + ... + ⌊√10⌋?
Answer: 19
Three 1s, five 2s, and two 3s give 3 + 10 + 6 = 19.
In an arithmetic sequence, a_3 = 7 and a_10 = 28. What is a_20?
Answer: 58
The common difference is 21/7 = 3, so a_20 = 7 + 17(3) = 58.
If f(x) = 1/(1 - x), what is f(f(x)) for x ≠ 0, 1?
Answer: (x - 1)/x
f(f(x)) = 1/(1 - 1/(1 - x)) = (1 - x)/(-x) = (x - 1)/x.
What is the sum of all real x satisfying |2x - 3| = 7?
Answer: 3
The solutions are x = 5 and x = -2, which sum to 3.
What is the vertex of the parabola y = 2x^2 - 8x + 3?
Answer: (2, -5)
The vertex x-coordinate is -b/(2a) = 2, and y(2) = 8 - 16 + 3 = -5.
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