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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_{k1}(n) (which may depend
on k) such that for any
integer n,
( means the largest integer ≤ a.)
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