4.1 Polynomials
1.
a. f + g = 4 + 2x + 2x2 + 5x3,
f g = 3 + 2x + 4x2 + 4x3 + 3x4 + 4x6.
2.
a. (1 + x)5 = 1 + 5x + 10x2 + 10x3 + 5x4 + x5 = 1 + x5 in 
c. From the hint: 
. Hence p divides 
for 1 ≤ k ≤ p − 1. Since p is a prime and p does not divide k, this shows that p divides 
. Hence 
in 
for 1 ≤ k ≤ p − 1. The binomial theorem gives
. Hence p divides for 1 ≤ k ≤ p − 1. Since p is a prime and p does not divide k, this shows that p divides . Hence in for 1 ≤ k ≤ p − 1. The binomial theorem gives
3.
a. The polynomials are a0 + a1x + a2x2 + a3x3 where 
, for all i and a3 ≠ 0. Hence there are 5 choices for each of a0, a1 and a2, and 4 choices for a3, for 53 · 4 = 500 in all.
, for all i and a3 ≠ 0. Hence there are 5 choices for each of a0, a1 and a2, and 4 choices for a3, for 53 · 4 = 500 in all.
4.
a. If f = (x − 4)(x − 5) then f(4) = 0 = f(5) is clear. In 
Thus the roots ...
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