Question

Let f(x) and g(x) be polynomials and suppose that we have f(a) = g(a) for all real numbers a. In this case prove that f(x) and g(x) have exactly the same coefficients. [Hint: Consider the polynomial h(x) = f(x) − g(x). If h(x) has at least one nonzero coefficient then the equation h(x) = 0 has finitely many solutions.]

Answer #1

Let
F be a field, and let f (x), g(x) ∈ F [x] be nonzero polynomials.
Then it must be the case that deg(f (x)g(x)) = deg(f (x)) +
deg(g(x)).

Let
F be a field and let a(x), b(x) be polynomials in F[x]. Let S be
the set of all linear combinations of a(x) and b(x). Let d(x) be
the monic polynomial of smallest degree in S. Prove that d(x)
divides a(x).

Let f(x) g(x) and h(x) be polynomials in R[x].
Show if gcd(f(x), g(x)) = 1 and gcd(f(x), h(x)) = 1, then
gcd(f(x), g(x)h(x)) = 1.

Let P be the vector space of all polynomials in x with real
coefficients. Does P have a basis? Prove your answer.

1 Approximation of functions by polynomials
Let the function f(x) be given by the following:
f(x) = 1/ 1 + x^2
Use polyfit to approximate f(x) by polynomials of degree k = 2,
4, and 6. Plot the approximating polynomials and f(x) on the same
plot over an appropriate domain. Also, plot the approximation error
for each case. Note that you also will need polyval to evaluate the
approximating polynomial.
Submit your code and both plots. Make sure each of...

Let f(x) be a cubic polynomial of the form x^3 +ax^2 +bx+c with
real coefficients.
1. Deduce that either f(x) factors in R[x] as the product of
three degree-one
polynomials, or f(x) factors in R[x] as the product of a
degree-one
polynomial and an irreducible degree-two polynomial.
2.Deduce that either f(x) has three real roots (counting
multiplicities) or
f(x) has one real root and two non-real (complex) roots that are
complex
conjugates of each other.

Suppose V is a vector space over F, dim V = n, let T be a linear
transformation on V.
1. If T has an irreducible characterisctic polynomial over F,
prove that {0} and V are the only T-invariant subspaces of V.
2. If the characteristic polynomial of T = g(t) h(t) for some
polynomials g(t) and h(t) of degree < n , prove that V has a
T-invariant subspace W such that 0 < dim W < n

Let P(R) denote the family of all polynomials (in a single
variable x) with real coefficients. We have seen that with the
operations of pointwise addition and multiplication by scalars,
P(R) is a vector space over R. Consider the 2 linear maps D, I :
P(R) to P(R), where D is differentiation and I is
anti-differentiation. In detail, for a polynomial p =
a0+a1x1+...+anxn,
we have D(p) =
a1+2a2x+....+nanxn-1
and I(p) =
a0x+(a1/2)x2+...+(an/(n+1))xn+1.
a. Show that D composed with I...

Let
a1, a2, ..., an be distinct n (≥ 2) integers. Consider the
polynomial
f(x) = (x−a1)(x−a2)···(x−an)−1 in Q[x]
(1) Prove that if then f(x) = g(x)h(x)
for some g(x), h(x) ∈ Z[x],
g(ai) + h(ai) = 0 for all i = 1, 2, ..., n
(2) Prove that f(x) is irreducible over Q

Let f ∈ Z[x] be a nonconstant polynomial with the property that
all the roots (in comlex plane) for the equation f(x) = 0 are
distinct. Prove that there exist infinitely many positive integers
n such that f(n) is not a perfect square.

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