Stanford University EPGY Math Olympiad

Math Olympiad Problem SolvingStanford University EPGY Summer Institutes 2008

Problem Set: Working with Polynomials

1. (1988 AHSME #5) Suppose that b and c are constants and

(x + 2)(x + b) = x2 + cx + 6.

What is c?

(A) -5 (B) -3 (C) -1 (D) 3 (E) 5

2. (1988 AHSME #15) Suppose that a and b are integers such that x2 − x− 1 isa factor of ax3 + bx2 + 1. What is b?

(A) -2 (B) -1 (C) 0 (D) 1 (E) 2

3. (2003 AMC 12B #9) Suppose that P (x) is a linear polynomial with P (6) −P (2) = 12. What is P (12)− P (2)?

(A) 12 (B) 18 (C) 24 (D) 30 (E) 36

4. (1974 AHSME #4) What is the remainder when x51 + 51 is divided by x + 1?

(A) 0 (B) 1 (C) 49 (D) 50 (E) 51

5. (1999 AHSME #17) The polynomial P (x) has remainder 99 when divided byx− 19 and remainder 19 when divided by x− 99. What is the remainder whenP (x) is divided by (x− 19)(x− 99)?

(A) −x + 80 (B) x + 80 (C) −x + 118 (D) x + 118 (E) 0

6. (2000 AMC 10 #24) Suppose that P (x/3) = x2 + x + 1. What is the sum of allvalues of x for which P (3x) = 7?

(A) -1/3 (B) -1/9 (C) -0 (D) 5/9 (E) 5/3

7. (1977 AHSME #21) For how many values of the coefficient a do the equations

0 = x2 + ax + 1 and 0 = x2 − x− a

have a common real solution?

Summer 2008 1 Problem Set: Polynomials

Stanford University EPGY Math Olympiad

(A) 0

(B) 1

(C) 2

(D) 3

(E) infinitely many

8. Find the remainder when you divide x81 + x49 + x25 + x9 + x by x3 − x.

9. (1977 USAMO) If a and b are two roots of x4 + x3 − 1 = 0, show that ab is aroot of x6 + x4 + x3 − x2 − 1 = 0.

10. If p(x) is a polynomial of degree n such that p(k) = 1/k, k = 1, 2, . . . , n+1, findp(n + 2).

11. The polynomial p(x) satisfies p(−x) = −p(x). When p(x) is divided by x − 3the remainder is 6. Find the remainder when p(x) is divided by x2 − 9.

Summer 2008 2 Problem Set: Polynomials