AIME Algebra 2

Question 1

Find the remainder when x^3 - 4x + 2 is divided by x - 2.

Accepted Answer

2

Answer

By the Remainder Theorem, substitute x=2: 8-8+2=2.

Question 2

Solve |2x - 3| = 7. What is the sum of all solutions?

Accepted Answer

3

Answer

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.

Accepted Answer

19/9

Answer

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?

Accepted Answer

4

Answer

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?

Accepted Answer

7

Answer

Square x+1/x=3: x^2+2+1/x^2=9, so x^2+1/x^2=7.

AIME - American Invitational Mathematics Examination Practice Test