The Art and Craft of Problem Solving

5.4.9 Find the remainder when you divide x^81 +x^49 +

x^25 +x^9 +x by x^3 -x.

5.4.10 Let p(x) = x^6 + x^5 + ... + 1. Find the remain­

der when p(x^7 ) is divided by p(x).

5.4.11 (Gerald Heuer) Find an easier solution to Ex­

ample 5.4.3 on page 169 by first showing that the quar­

tic polynomial factors into the two quadratics (x^2 +

ax - 32)(x^2 + bx - 62).

5.4. 12 Find a polynomial with integral coefficients

whose zeros include v'2 + v'5.

5.4. 13 Prove that if a monic polynomial has a rational

zero, then this zero must in fact be an integer.

5.4.14 Let p(x) be a polynomial with integer coeffi­

cients satisfying p(O) = p( I) = 1999. Show that p has

no integer zeros.

5.4. 15 Let p(x) be a I 999-degree polynomial with in­

teger coefficients that is equal to ± 1 for 1999 different

integer values of x. Show that p(x) cannot be factored

into the product of two polynomials with integer coef­

ficients.

5.4. 16 (Hungary 1899) Let r and s be the roots of

x^2 - (a+d)x+ (ad -be) = O.

Prove that r^3 and s^3 are the roots of

i -(a^3 +d^3 + 3abc+ 3bcd)y+ (ad - bc)^3 = O.

5.4. 17 Let a,b,c be distinct integers. Can the poly­

nomial (x - a)(x - b)(x - c) - I be factored into the

product of two polynomials with integer coefficients?

5.5 Inequalities

5.5 INEQUALITIES 173

5.4. 18 Let p(x) be a polynomial of degree n, not nec­

essarily with integer coefficients. For how many con­

secutive integer values of x must a p(x) be an integer

in order to guarantee that p(x) is an integer for all in­

tegers x?

5.4. 19 (lMO 1993) Let f(x) = x" + 5x"-^1 + 3 where

n> I is an integer. Prove that f(x) cannot be expressed

as the product of two polynomials, each of which has

all its coefficients integers and degree at least I.

5.4.20 (USAMO 1977) If a and b are two roots of

x^4 +x^3 - I = 0, prove that ab is a root of x^6 +x^4 +

x^3 -x^2 -1=0.

5.4.21 (Canada 1970) Let P(x) = Xl + al1_ 1 xn-^1 +

... + alx + ao be a polynomial with integral coeffi­

cients. Suppose that there exist four distinct integers

a, b, c, d with Pta) = P(b) = P(c) = P(d) = 5. Prove

that there is no integer k with P(k) = 8.

5.4.22 (USAMO 84) P(x) is a polynomial of degree

3n such that

P(O) = P(3)

P(I) = P(4)

P(2) = P(5)

and

Determine n.

P(3n)

= P(3n-2)

= P(3n-l )

2,

I,

0,

P(3n+ I) = 730.

5.4.23 (American Mathematical Monthly, October

1962) Let P (x) be a polynomial with real coefficients.

Show that there exists a nonzero polynomial Q(x) with

real coefficients such that P(x)Q(x) has terms that are

all of a degree divisible by 10^9.

Inequalities are important because many mathematical investigations involve estima­

tions, optimizations, best-case and worst-case scenarios, limits, etc. Equalities are

nice, but are really quite rare in the "real world" of mathematics. A typical example

was the use of rather crude inequalities to establish the divergence of the harmonic

series (Example 5 .3.4 on page 161). Another example was Example 2.3. 1 on page 41,

where we proved that the equation b^2 + b + 1 = a^2 had no positive integer solutions by

showing that the alleged equality was, in fact, an inequality.

Here is another solution to that problem, one that uses the tactic of looking for

perfect squares: the equation b^2 + b + 1 = a^2 asserts that b^2 + b + 1 is a perfect square.

But