Page 87 - Linear Algebra Done Right

Exercises
Exercises
1. Suppose m and n are positive integers with m ≤ n. Prove that 73
there exists a polynomial p ∈P n (F) with exactly m distinct
roots.
2. Suppose that z 1 ,...,z m+1 are distinct elements of F and that
w 1 ,...,w m+1 ∈ F. Prove that there exists a unique polynomial
p ∈P m (F) such that
p(z j ) = w j
for j = 1,...,m + 1.

3. Prove that if p, q ∈P(F), with p = 0, then there exist unique
polynomials s, r ∈P(F) such that

q = sp + r

and deg r< deg p. In other words, add a uniqueness statement
to the division algorithm (4.5).
4. Suppose p ∈P(C) has degree m. Prove that p has m distinct
roots if and only if p and its derivative p have no roots in com-

mon.
5. Prove that every polynomial with odd degree and real coefficients
has a real root.