1. Let f be a polynomial ao+ajz+...+ anz" of degree n > 1 with complex coefficients...
Please help with this proof.
Thank you
Problem 3 Prove that a polynomial anz" + an-121-1 +...+ajz+ao is a continuous function on the entire complex plane.
let fx be a polynomial of degree <= to n
whats the value of f(Xo, X1....Xn). explain
Let f(x) = ao tai xt...... + Anxh be a polynomial of | degree less than or equal to n, and let {xo.xi... n} be distinct points What is the value of f[xo, X.. Xn] Justify / Explain.
3. Any polynomial with real coefficients of degree k can be factored com- pletely into first-degree binomials which may include complex numbers. That is, for any real ao, Q1, ..., āk ao + a1x + a22² + ... + axxk = C(x – 21)(x – z2....(x – zk) for some real C and 21, 22, ... Zk possibly real or complex. Therefore, up to multiplicity, every polynomial of degree k has exactly k-many roots, includ- ing complex roots. Find all...
Problem 1. Given a polynorial p(x) anz" + an-lz"-ı + + aix + ao, where the coefficients are a,'s, Horner's method is an efficient algorithm for evaluating the polynomial at a number c that works as follows: Multiply an by c, then add an-1. Then multiply the result by c and add an-2. Then multiply the result by c and add an-3 and so on until you reach a0. This over all gives an O(n) algorithm for evaluation of p(c)...
Please answer problem 4, thank you.
2. The polynomial p of degree n that interpolates a given function f at n+1 prescribed nodes is uniquely defined. Hence, there is a mapping f -> p. Denote this mapping by L and show that rl Show that L is linear; that is, 3. Prove that the algorithm for computing the coefficients ci in the Newton form of the interpolating polynomial involves n long operations (multiplications and divisions 4. Refer to Problem 2,...
Let f (x) be a monic polynomial of degree n with distinct zeros ai,..., an. Prove -1
Let f (x) be a monic polynomial of degree n with distinct zeros ai,..., an. Prove -1
Theorem. Let p(x) = anr" + … + ao be a polynomial with integer coefficients, i, e. each ai E Z. If r/s is a rational root of p (expressed in lowest terms so that r, s are relatively prime), then s divides an and r divides ao Use the rational root test to solve the following: + ao is a monic (i.e. has leading coefficient 1) polynomial with integer coefficients, then every rational root is in fact an integer....
Let A be an n × n matrix with characteristic polynomial
f(t)=(−1)nt n + an−1t n−1 + ··· + a1t + a0. (a) Prove that A is
invertible if and only if a0 = 0. (b) Prove that if A is
invertible, then A−1 = (−1/a0)[(−1)nAn−1 + an−1An−2 + ··· + a1In].
324 Chap. 5 Diagonalization (c) Use (b) to compute A−1 for A = ⎛ ⎝
12 1 02 3 0 0 −1 ⎞ ⎠ .
#18 a, b...
3. Let Tn(x) be the degree n Chebyshev polynomial. Evaluate Tn (0.5) for 2 <n < 10, by applying the three-term recurrence directly with x = 0.5, starting with T.(0.5) = 1 and Ti(0.5) = 0.5.
Please prove the theorems,
thank you
6.1 Theorem. Let anx+an-1- +ag he a polynomial of degree n0 with integer coefficients and assume an0. Then an integer r is a Poot of (x) if and only if there exists a polynomlal g(x) of degree n - with integer coeficients such that f(x) (x)g(x). This next theorem is very similar to the one above, but in this case (xr)g(x) is not quite equal to f(x), but is the same except for the...