AIME Algebra 2 — Questions and Answers
Question 1: Find the remainder when x^3 - 4x + 2 is divided by x - 2.
- 2 (Correct answer)
- 0
- 4
- -2
Correct answer: 2
By the Remainder Theorem, substitute x=2: 8-8+2=2.
Question 2: Solve |2x - 3| = 7. What is the sum of all solutions?
- 3 (Correct answer)
- 7
- 5
- 10
Correct answer: 3
2x-3=7 gives x=5 and 2x-3=-7 gives x=-2; sum is 5+(-2)=3.
Question 3: The roots of 3x^2 - 5x + 1 = 0 are r and s. Find r^2 + s^2.
- 19/9 (Correct answer)
- 25/9
- 7/3
- 13/9
Correct answer: 19/9
r+s=5/3, rs=1/3; r^2+s^2=(r+s)^2-2rs=25/9-2/3=25/9-6/9=19/9.
Question 4: For what value of c does cx^2 + 8x + 4 = 0 have a double root?
- 4 (Correct answer)
- 2
- 8
- 1
Correct answer: 4
A double root requires discriminant=0: 64-16c=0 gives c=4.
Question 5: If x + 1/x = 3, what is x^2 + 1/x^2?
- 7 (Correct answer)
- 9
- 6
- 11
Correct answer: 7
Square x+1/x=3: x^2+2+1/x^2=9, so x^2+1/x^2=7.
Question 6: How many integer solutions does x^2 < 16 have?
- 7 (Correct answer)
- 6
- 8
- 4
Correct answer: 7
x can be -3,-2,-1,0,1,2,3, giving 7 integer solutions.
AIME Algebra 2
Find the remainder when x^3 - 4x + 2 is divided by x - 2.