Question

Prove that if f(x) = akx^k +ak−1x^k+1 +ak−2x^k+2+...+a1x+a0 is a polynomial in Q[x] and ak ̸=...

Prove that if f(x) = akx^k +ak−1x^k+1 +ak−2x^k+2+...+a1x+a0 is a polynomial in Q[x] and ak ̸= 0, and f (x) factors as f (x) = g(x)h(x), where g(x) and h(x) are polynomials in Q[x], then deg f = deg g+ deg h.

Math   Algebra   
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
Prove: If f(x) = anx^n + an−1x^n−1 + ··· + a1x + a0 has integer coefficients...
Prove: If f(x) = anx^n + an−1x^n−1 + ··· + a1x + a0 has integer coefficients with an ? 0 ? a0 and there are relatively prime integers p, q ∈ Z with f ? p ? = 0, then p | a0 and q | an . [Hint: Clear denominators.]
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...
True or False, explain: 1. Any polynomial f in Q[x] with deg(f)=3 and no roots in...
True or False, explain: 1. Any polynomial f in Q[x] with deg(f)=3 and no roots in Q is irreducible. 2. Any polynomial f in Q[x] with deg(f)-4 and no roots in Q is irreducible. 3. Zx40 is isomorphic to Zx5 x Zx8 4. If G is a finite group and H<G, then [G:H] = |G||H| 5. If [G:H]=2, then H is normal in G. 6. If G is a finite group and G<S28, then there is a subgroup of G...
7. Suppose p(x) = a0 + a1x + a 2 x 2 + · · ·...
7. Suppose p(x) = a0 + a1x + a 2 x 2 + · · · + akx k is a polynomial of degree k. Find the Taylor series of p(x), and find its radius and interval of convergence.
Let p(x) = a0 + a1x + a2x2 and q(x) = b0 + b1x + b2x2...
Let p(x) = a0 + a1x + a2x2 and q(x) = b0 + b1x + b2x2 be vectors in P2 with p, q = a0b0 + a1b1 + a2b2. Determine whether the polynomials form an orthonormal set, and if not, apply the Gram-Schmidt orthonormalization process to form an orthonormal set. (If the set is orthonormal, enter ORTHONORMAL in both answer blanks.) {−2 + x2, −2 + x} u1= u2=
Let p(x) = a0 + a1x + a2x2 and q(x) = b0 + b1x + b2x2...
Let p(x) = a0 + a1x + a2x2 and q(x) = b0 + b1x + b2x2 be vectors in P2 with p, q = a0b0 + a1b1 + a2b2. Determine whether the given second-degree polynomials form an orthonormal set, and if not, then apply the Gram-Schmidt orthonormalization process to form an orthonormal set. (If the set is orthonormal, enter ORTHONORMAL in both answer blanks.) { 3 (x2−1), 3 (x2 + x + 2)} u1 = u2 =
Let p(x) = a0 + a1x + a2x2 and q(x) = b0 + b1x + b2x2...
Let p(x) = a0 + a1x + a2x2 and q(x) = b0 + b1x + b2x2 be vectors in P2 with p, q = a0b0 + a1b1 + a2b2. Determine whether the given second-degree polynomials form an orthonormal set, and if not, then apply the Gram-Schmidt orthonormalization process to form an orthonormal set. (If the set is orthonormal, enter ORTHONORMAL in both answer blanks.) { square root 3 (x2−1), square root 3 (x2 + x + 2)} u1 = u2...
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
Q7) Factorise the polynomial f(x) = x3 − 2x2 + 2x − 1 into irreducible polynomials...
Q7) Factorise the polynomial f(x) = x3 − 2x2 + 2x − 1 into irreducible polynomials in Z5[x], i.e. represent f(x) as a product of irreducible polynomials in Z5[x]. Demonstrate that the polynomials you obtained are irreducible. I think i manged to factorise this polynomial. I found a factor to be 1 so i divided the polynomial by (x-1) as its a linear factor. So i get the form (x3 − 2x2 + 2x − 1) = (x2-x+1)*(x-1) which is...
. Consider the sequence defined recursively as a0 = 5, a1 = 16 and ak =...
. Consider the sequence defined recursively as a0 = 5, a1 = 16 and ak = 7ak−1 − 10ak−2 for all integers k ≥ 2. Prove that an = 3 · 2 n + 2 · 5 n for each integer n ≥ 0
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT