# Worksheet 6 Polynomials

Move everything to one side. Factor . Expand. Complete the square!

x^{n} − y^{n} =?, x^{2m+1} + y^{2m+1} =?. A degree n polynomial is determined

by its values at n + 1 points. Coefficients of a polynomial in terms of

its roots. How to get sums of powers of roots ? Rational roots theorem

(if a polynomial with integer coefficients has a rational root then ???).

Long division of polynomials .

**1.** 1/(x + 1)(x + 2)(x + 3) =?/(x + 1)+?/(x + 2)+?/(x + 3).

**2.** Consider two polynomials, P(x) and Q(x). Suppose that each of

them has the property that the sum of its coefficients at odd powers

of x is equal to the sum of its coefficients at even powers of x . For

example, we can take P(x) = 1+3x+2x^{2} and Q(x) = −1+4x^{2} +3x^{3}.

Is it true that P(x)Q(x) has the same property?

**3.** Show that each number in the sequence 49, 4489, 444889, 44448889,

... is a perfect square .

**4.** Find the remainder when you divide x^{81} + x^{49} + x^{25} + x^{9} + x by

x^{3} − x.

**5.** If P(x) is a polynomial of degree n such that P(k) = k/(k + 1)

for k = 0, . . . , n, determine P(n + 1).

**6.** Prove that is irrational but

is rational .

**7.** It is known that a quadratic equation has either 0, 1, or 2 unique

real solutions . But consider the equation

where a, b, and c are distinct. Notice that x = a, x = b, and x = c are

all solutions — how can this equation have three solutions?

**8.** (The interpolation formula) Suppose a_{1}, ..., a_{n} are distinct num-

bers, and b_{1}, ..., b_{n} are given numbers, and P(x) is a degree at most

n − 1 polynomial such that P(a_{i}) = b_{i} for all i. Show that

**9.** A repunit is a positive integer whose digits in base 10 are all ones.

Find all polynomials f with real coefficients such that if n is a repunit,

then so is f(n).

**10.** Solve

(x^{2} − 3x − 4)(x^{2} − 5x + 6)(x^{2} + 2x) + 30 = 0.

**11.** Let k be a positive integer. Prove that there
exist polynomials

P_{0}(n), P_{1}(n), . . . , P_{k-1}(n) (which may depend
on k) such that for any

integer n,

( means the largest integer ≤ a.)

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