IMO Practice Test — Polynomials
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
hard
If P(x) = x³ - 6x² + 11x - 6, find the sum of its zeros
6
-6
11
-11
Explanation: Sum of zeros = -coefficient of x²/coefficient of x³ = 6
Question 2 of 6
hard
Find the value of a such that (x - 2) is a factor of x³ - 5x² + ax + 2
5
-5
7
-7
Explanation: P(2)=8-20+2a+2=0 → -10+2a=0 → 2a=10 → a=5? Wait recalc: 8-20+2a+2 = -10+2a=0 → a=5, but not in options. Let me check: 8-20=-12, -12+2=-10, so -10+2a=0 → a=5. Options: A=5 → A
Question 3 of 6
hard
The polynomial ax³ + bx² + cx + d has zeros 1, 2, and 3. If a = 1, find b + c + d
-6
6
-36
36
Explanation: P(x)=(x-1)(x-2)(x-3)=x³-6x²+11x-6. So b=-6, c=11, d=-6. Sum = -6+11-6=-1
Question 4 of 6
hard
Simplify: (x² - y²) ÷ (x - y)
x + y
x - y
x² + y²
x² - y²
Explanation: x²-y² = (x-y)(x+y), so ÷ (x-y) gives x+y
Question 5 of 6
hard
If x + 1/x = 5, find x² + 1/x²
23
25
27
29
Explanation: (x+1/x)² = x² + 2 + 1/x² = 25 → x²+1/x²=23
Question 6 of 6
hard
Find the remainder when x⁹⁹ + 99 is divided by x + 1
98
99
100
101
Explanation: Remainder = P(-1)=(-1)⁹⁹+99 = -1+99=98