Let f (x) be a monic polynomial of degree n with distinct zeros ai,..., an. Prove -1
ZEROS OF POLYNOMIAL FUNCTIONS 1. Find a polynomial function f(x) of degree 3 that has the indicated zeros and satisfies the given condition Zeros: -5, 2, 4 Condition: f(3) = -24 f(x) = 2. Find a polynomial function f(x) of degree 3 that has the indicated zeros and satisfies the given condition. Zeros: -1, 2, 3 Condition: f(-2) = 80 f(x) = 3. Find a polynomial function f(x) of degree 3 that has the indicated zeros and satisfies the given...
Form a polynomial f(x) with real coefficients having the given degree and zeros. Degree 4; zeros: 1, multiplicity 2; 2i Enter the polynomial. f(x) = all (Type an expression using x as the variable. Use integers or fractions for any numbers in Its
(i) Find a non-zero polynomial in Z3 x| which induces a zero function on Z3. f(x), g(x) R have degree n and let co, c1,... , cn be distinct elements in R. Furthermore, let (ii) Let f(c)g(c) for all i = 0,1,2,...n. g(x) Prove that f(x - where r, s E Z, 8 ± 0 and gcd(r, s) =1. Prove that if x is a root of (iii) Let f(x) . an^" E Z[x], then s divides an. aoa1
(i)...
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.
with distinct nodes, prove there is at most one polynomial of
degree ≤ 2n + 1 that interpolates the data. Remember the
Fundamental Theorem of Algebra says a nonzero polynomial has number
of roots ≤ its degree. Also, Generalized Rolle’s Theorem says if r0
≤ r1 ≤ . . . ≤ rm are roots of g ∈ C m[r0, rm], then there exists ξ
∈ (r0, rm) such that g (m) (ξ) = 0.
1. (25 pts) Given the table...
Form a polynomial f(x) with real coefficients having the given degree and zeros. Degree 5; zeros: - 7; - i; 6+ i Enter the polynomial. F(x) =a (Type an expression using x as the variable. Use integers or fractions for any numbers in the expression. Simplify your answer.)
Let f(x) = do +212 + ... + and" be a polynomial of degree less than or equal to n, and let {x0, 21, ..., Un} be distinct points. What is the value of f (x0, X1, ... , Xn]? Explain.
Write a polynomial f(x) that satisfies the given conditions. Polynomial of lowest degree with zeros of (multiplicity 2) and 1 (multiplicity 1) and with f(0) = -2. 4 $(x) = a
Suppose that P is a polynomial of degree n and that P has n distinct real roots. Prove that P(k) has n-k distinct real roots for 1≤ k ≤ n-1.
Form a polynomial whose zeros and degree are given. Zeros: -4,4,6; degree: 3 Type a polynomial with integer coefficients and a leading coefficient of 1 in the box below. f(x) = (Simplify your answer.)