For the following exercises, use the given information about the polynomial graph to write the equation.Degree 4. Roots of multiplicity 2 at x = 1/2 and roots of multiplicity 1 at x = 6 and x = −2. y-intercept at (0,18).
The Polynomial f(x) = X^3 - X^2 - X -1 has one real root a,
which happens to be positive. This real number a satisfies the
following properties:
- for i = 1,2,3,4,5,6,7,8,9,10, one has {a^i} not equal to
zero
- one has
[a] = 1, [a^2] = 3, [a^3] = 6, [a^4] = 11, [a^5] = 21, [a^6] =
7, [a^7] = 71, [a^8] = 130
(for a real number x, [x] denotes the floor of x and {x}...
Degree 5. Roots of multiplicity 2 at x = −3 and x = 2 and a root of multiplicity 1 at x=−2. y-intercept at (0, 4). For the above exercises, use the given information about the polynomial graph to write the equation.
Prove the following:
Let f(x) be a polynomial in R[x] of positive degree n.
1. The polynomial f(x) factors in R[x] as the product of
polynomials of degree
1 or 2.
2. The polynomial f(x) has n roots in C (counting multiplicity).
In particular,
there are non-negative integers r and s satisfying r+2s = n such
that
f(x) has r real roots and s pairs of non-real conjugate complex
numbers as
roots.
3. The polynomial f(x) factors in C[x] as...