Factor into irreducibles:X^2 − 7 in R[X];X^2 − 7 in Q[X];2X^3 + X^2 + 2X + 2 in Z5[X];X^4 + X^2 + 1 in Z3[X].

Q: Need only handwritten solution only (not typed one).

A: The given function is .We have to find the inverse function.

Q: Show that x2 + 2x8 + 1 does not have multiple zeros in any extension of Z3.

A: As per the question we are given a polynomial x2 + 2x8 + 1 and we have to show that there is no…

Q: 11. Find all the zeros of f(x)=x²-x²-4x² +8x²-32r+48 using synthetic division. Write fix) in…

A:

Q: 3. Let p(x) = 6xª – 13x³ + 13x² – 2. - (a) Find q(x) such that p(x) = (2x – 1)q(x) using synthetic…

A:

Q: 4. Find the product of (x +8)(x- 3) (X)

A:

Q: For the polynomial below, 2 is a zero. h(x)=x'+4x+ 6x- 36 Express h (x) as a product of linear…

A:

Q: What is the value of k so that the polynomial P \ left (x \ right) = x ^ 5−23x ^ 4 + \ left (2k − 3…

A: Given : P(x)=x5−23x4+(2k−3)x3−17x2−54x+40 Q(x)=x-2 Remainder theorem states that x -a is the divisor…

Q: Test the polynomial x – 2x3 + 6x – 3 for irreducibility over Q.

A: We have to check whether the polynomial x4-2x3+6x-3 is irreducible over the set of rational numbers…

Q: Which of the following quadratic polynomials are irreducible? To check, complete the square if…

A:

Q: Which of the following is a partially factored form of the function f(x) = x² + 5x² - 2x-24?…

A:

Q: Write g(x) = x² + x³ + 2x²+x+2 as a product of irreducible polynomials over 2.₂. за

A: Write as a product of irreducible polynomials over .

Q: Use synthetic division to show that f(x) = x³ + 3x² – 13x – 15 is divisible by the linear factor x +…

A: Given query is to use synthetic division on f(x) , and find its factors.

Q: The value of r for which x2 - rx - 2 and x3 + x2 - x - 1 have a common factor is?

A: Given two equations x2-rx-2=0 and x3+x2-x-1=0

Q: Completly factorize x5 + 4x4 + 7x³ + 8x² + 4x + 2 in Q[x]. Justify that the found factors are…

A:

Q: Express x³ + 4x4 + 4x³ −x² − 4x + 1 € Z₁[x] as a product of irreducible polynomials over Z5. Be sure…

A: We have to express polynomial in Z5 given as  x5+4x4+4x3-x2-4x+1  As a product of irreducible…

Q: Find the coefficient of x11 in (x2 - 3x) / (1 - x )4

A: Solution:

Q: Determine Q(x) when P(x)=x³-5x2-3x+2 is divided by x-1 and then expressed in the form Select one: O…

A:

Q: 5. а) Factor the polynomial f (x) = 3xª+2x³+2x² +2x-1 into irreducible polynomials in C[r], R[7),…

A: In complex number, .

Q: Show that Q[x]/⟨x 2 − 2⟩ ≅ Q[x]/⟨x 2 + 4x + 2⟩.

A:

Q: Show that x2 − 4  is a factor of  x4− 5x3+ 2x2 + 20x − 24.

A:

Q: 16. Show that (x + 1) is a factor of p(x) = x³ + 2x² + 3x +2

A: We have to show that x+1 is a factor of p(x)=x2+2x2+3x+2 .

Q: 3. Divide f(x)= x³ + 2x² - 2x+4 By (x-2) using synthetic division. What is the remainder?

A:

Q: 0 2 -1 et A 1 1 1 and suppose B is a 3 x 3 matrix with |B| = 6. 22 5 a) 5) Find the determinant of A…

A: Given,            A=02-1111225,  and  B is a 3×3 matrix with B=6.

Q: Construct the addition table for Z5. Using this table, or otherwise, determine the following x in…

A: First we find addition table for Z5. Then find values of x.

Q: - Express P(x) = xª +.2x³ + 2x² - x - 3 in terms of Legendre's polynomials

A: In this problem, we use Rodrigue's formula to express Legendre's polynomials.