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Prove a fundamental result about polynomials
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We will now prove a fundamental result about polynomials: every
non-zero polynomial of degreen (over a field F) has at most n roots. If
you don‘t know what a field is, you can assume in thefollowing that F
= R (the real numbers).
(a) Show that for any α ∈ F, there exists some polynomial Q(x) of
degree n−1 and some b ∈ Fsuch that P(x) = (x−α)Q(x) +b.
(b) Show that if α is a root of P(x), then P(x) = (x−α)Q(x).
(c) Prove that any polynomial of degree 1 has at most one root. This is
your base case.(d) Now prove the inductive step: if every polynomial
of degree n−1 has at most n−1 roots, thenany polynomial of degree n
has at most n roots
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QNT 295 The H2 Hummer limousine has eight tires on it
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The H2 Hummer limousine has eight tires on it. A fleet of 1230 H2
limos was fit with a batch of tires that mistakenly passed quality
testing. The following table lists the frequency distribution of the
number of defective tires on the 1230 H2 limos.
Number of defective tires
0
1
2
3
4
5
6
7
8