Question

The polynomial of degree 4, P(x) has a root of multiplicity 2 at
x=2 and roots of multiplicity 1 at x=0 and x=-4 It goes through the
point (5,324).

Find a formula for P(x)

Answer #1

The polynomial of degree 5, P(x), has leading coefficient 1, has
roots of multiplicity 2 at x=5 and x=0, and a root of multiplicity
1 at x=−1.
Find a possible formula for P(x).

Find the exact solution(s) to the equation:
17x2=1720−x17x2=1720-x
x=x=
The polynomial P(x)P(x) of degree 4 has
a root of multiplicity 2 at x = 4
a root of multiplicity 1 at x = 0 and at x = -3
It goes through the point (5, 12)
Find a formula for P(x)P(x).
Leave your answer in factored form.
P(x)=P(x)=

the linear transformation, L(p(x))=d/dx p(x)+p(0). maps a
polynomial p(x) of degree<= 2 into a polynomial of degree
<=1, namely, L:p2 ~p1. find the marix representation of L with
respect to the order bases{x^2,x,1}and {x,1}

Find an equation for f(x), the polynomial of smallest degree
with real coefficients such that f(x) bounces off of the x-axis at
5, breaks through the x-axis at −1, has complex roots of −5−3i and
−4+2i and passes through the point (0,89).

Find a polynomial in R[x] of minimum degree which has roots 3,
i+1 and -3i.

Find a polynomial f (x) of degree 4 with real coefficients and
the following zeros.
4 (multiplicity 2) , i
f(x)=

Let p be an odd prime. Let f(x) ∈ Q(x) be an irreducible
polynomial of degree p whose Galois group is the dihedral group
D_2p of a regular p-gon. Prove that f (x) has either all real roots
or precisely one real root.

Find an equation for f(x), the polynomial of smallest degree
with real coefficients such that f(x) breaks through the x-axis at
−5, breaks through the x-axis at −4, has complex roots of 5−i and
−3−5i and passes through the point (0,68).

5a: f(x) is a 4th degree polynomial with 3 distinct roots: -1,
2, 2^(.5)i ; and f(1) = 12.
f(x) = ? Provide the answer
in factored form.
5b: Suppose you don’t know that f(1) = 12.
What is the most general
formula for f(x)?
Leave the answer in
un-factored form.

Consider the polynomial
P(x)=−4x5+5x4−4x+5P(x)=−4x5+5x4−4x+5.
(a) Verify that 12√(1+i)12(1+i) is a root of P(x)?(?).
b) Given that 12√(1+i)12(1+i) is also a root of P(−x)P(−x),
without calculation list 44 distinct roots of P(x)P(x). Explain
your answer.
(c) Prove that P(x)P(x) has no real roots in (−∞,0](−∞,0] or in
[3,∞)[3,∞).

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