Question

Let z0 be a zero of the polynomial P(z)=a0 +a1z+a2z2 +···+anzn of degree n (n ≥...

Let z0 be a zero of the polynomial

P(z)=a0 +a1z+a2z2 +···+anzn of degree n (n ≥ 1). Show in the following way that

P(z) = (z − z0)Q(z) where Q(z) is a polynomial of degree n − 1.

(an ̸=0)

(k=2,3,...).

(a) Verify that
zk−zk=(z−z)(zk−1+zk−2z +···+zzk−2+zk−1)

00000 (b) Use the factorization in part (a) to show that

P(z) − P(z0) = (z − z0)Q(z)
where Q(z) is a polynomial of degree n − 1, and deduce the desired result from this.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let p(x) be an irreducible polynomial of degree n over a finite field K. Show that...
Let p(x) be an irreducible polynomial of degree n over a finite field K. Show that its Galois group over K is cyclic of order n and then show how the Galois group of x3 − 1 over Q is cyclic of order 2.
problem 2 In the polynomial ring Z[x], let I = {a0 + a1x + ... +...
problem 2 In the polynomial ring Z[x], let I = {a0 + a1x + ... + anx^n: ai in Z[x],a0 = 5n}, that is, the set of all polynomials where the constant coefficient is a multiple of 5. You can assume that I is an ideal of Z[x]. a. What is the simplest form of an element in the quotient ring z[x] / I? b. Explicitly give the elements in Z[x] / I. c. Prove that I is not a...
Prove that the set V of all polynomials of degree ≤ n including the zero polynomial...
Prove that the set V of all polynomials of degree ≤ n including the zero polynomial is vector space over the field R under usual polynomial addition and scalar multiplication. Further, find the basis for the space of polynomial p(x) of degree ≤ 3. Find a basis for the subspace with p(1) = 0.
Recall that (by the Fundamental Theorem of Algebra) the only polynomial P(t) of degree n−1 that...
Recall that (by the Fundamental Theorem of Algebra) the only polynomial P(t) of degree n−1 that vanishes at n distinct points t1,...,tn ∈ R is P(t) ≡ 0. Using this, show that given any values b1,...,bn ∈ R, there is a polynomial Q(t) = ξ1 + ξ2t + ... + ξntn-1 such that Q(ti) = bi (and thus Q(t) takes precisely the given values at the given points). Hint: Show that the coefficient matrix of the corresponding system is nonsingular.
1. Let D ⊂ C be an open set and let γ be a circle contained...
1. Let D ⊂ C be an open set and let γ be a circle contained in D. Suppose f is holomorphic on D except possibly at a point z0 inside γ. Prove that if f is bounded near z0, then f(z)dz = 0. γ 2. The function f(z) = e1/z has an essential singularity at z = 0. Verify the truth of Picard’s great theorem for f. In other words, show that for any w ∈ C (with possibly...
1. a) Draw a sketch of: {z∈C|Im((3−2i)z)>6}. b) If 2i is a zero of p(z)=az^2+z^3+bz+16, find...
1. a) Draw a sketch of: {z∈C|Im((3−2i)z)>6}. b) If 2i is a zero of p(z)=az^2+z^3+bz+16, find the real numbers a,b. c) Let p(z)=z^4−z^3−2z^2+a+6z, where a is real. Given that 1+i is a zero of p(z): Find value of a, and a real quadratic factor of p(z). Express p(z)as a product of two real quadratic factors to find all four zeros of p(z).
1. A zero of a polynomial p(x) ∈ R[x] is an element α ∈ R such...
1. A zero of a polynomial p(x) ∈ R[x] is an element α ∈ R such that p(α) = 0. Prove or disprove: There exists a polynomial p(x) ∈ Z6[x] of degree n with more than n distinct zeros. 2. Consider the subgroup H = {1, 11} of U(20) = {1, 3, 7, 9, 11, 13, 17, 19}. (a) List the (left) cosets of H in U(20) (b) Why is H normal? (c) Write the Cayley table for U(20)/H. (d)...
Let a1, a2, ..., an be distinct n (≥ 2) integers. Consider the polynomial f(x) =...
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
(§2.1) Let a,b,p,n ∈Z with n > 1. (a) Prove or disprove: If ab ≡ 0...
(§2.1) Let a,b,p,n ∈Z with n > 1. (a) Prove or disprove: If ab ≡ 0 (mod n), then a ≡ 0 (mod n) or b ≡ 0 (mod n). (b) Prove or disprove: Suppose p is a positive prime. If ab ≡ 0 (mod p), then a ≡ 0 (mod p) or b ≡ 0 (mod p).
Let P(R) denote the family of all polynomials (in a single variable x) with real coefficients....
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...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT