The polynomial f = x³ + x² + 1 is irreducible over Z2. Let u be a root of f in some extension Z₂(u) of Z₂. How many elements does this extension have? Be able to describe this set via set builder notation on your solutions.

Q: Consider Z[x] = {a,x" + a₁₁x¹+...a₁x+ao: a; € Z,i = 0,1,2,...,n), the set of all polynomials in…

A: Inverse elements are 1 and -1

Q: Let p > 2 be a prime number. If f(x) = 1' + x* + x² - x +1 € Z,[F] is divisible by the polynomial z…

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Q: Exercise 4.14. Suppose ƒ is any polynomial. (a) If d is a nonzero constant polynomial, what are the…

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Q: t if the complex number z is a root of the polynomial f(

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Q: 2. Let P(x) = ax² + bx + c be an irreducible polynomial in Q[x). Prove that (a) P(x) has two…

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Q: If f(x) is a polynomial of degree <3, f(a) a x-a

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Q: The first three Legendre polynomials are 1, x, and x2 - ½. Choose c so that the fourth polynomial x3…

A: The first three Legendre polynomials are 1,x,x2-12. The fourth polynomial is x3-cx which is…

Q: Let p be a prime and let q = p². Show that Z,[x] has a unit that is not a constant polynomial.

A: Given that, Let p be a prime and let q = p ^ 2 . Show that Zq[x] has a unit that is not a constant…

Q: Let 0 be a root of the polynomial x³ + 3x + 7. Prove that 0 is prime in Q(0).

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Q: 6. Use the fact that every odd degree polynomial with real coefficients has a real root to show that…

A: Show that the function f:ℝ→ℝ defined as fx=x5-2x2+x is onto for all x∈ℝ. Definition:  A function…

Q: According to the integral zero theorem, the possible integral zeros of f(x) = 4x³+5x²+7x+6 are…

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Q: 2. Let R be a U.F.D. and f (x), g (x) e R [x]. If f(x) g (x) is a primitive polynomial, then f (x)…

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Q: List all the polynomials of degree 2 in Z2[x]. Which of these areequal as functions from Z2 to Z2?

A: We know that a polynomial of degree 2 has the representation fx=ax2+bx+c, where a, b and c are…

Q: For all z € C, let P(z) = ao+a₁2+...+anz" be a polynomial, where a, &C for all j= 0,..., n. Suppose…

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Q: for which value of L are the legendre polynomial orthogonal on [ -L, L ]  interval? also find || P2…

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Q: Consider the following theorem: Let f be a quadratic polynomial with a positive leading coefficient.…

A: Given theorem: Let f be a quadratic polynomial with a positive leading coefficient. Then f has a…

Q: Let p: R → R be a polynomial of odd degrees. Prove that the equation p(x) = 0 has a solution in real…

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Q: 2. Let char F = 0 and f(x) E F[x] be a monic polynomial having positive degree. If d(x) = gcd(f(x),…

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Q: Please help me answer 2i) I get the incorrect answer

A: Topic:- polynomial  Only 2(i) is asked

Q: Compute the first 5 Legendre Polynomials, Po(x), P₁(x), -P₁(x). (These These are > Solutions, P,…

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Q: 3. Let P(x) be a polynomial of degree n. If n is even, prove P(x) cannot have an inverse function.

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Q: Let T„(x) be the Maclaurin polynomial of a function f(x) satisfying f(5) (x) < 1 for all x. Which of…

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Q: نقطة واحدة If f(z) = evsin 2z. Then 4 d Vsin 2z Vcot 2z. evain 2z %3D dz TO FO

A: ANSWER 2 :-

Q: 8. Let f be an entire function such that f(z)| < A + B|z|" for all z E C, where A and B are positive…

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Q: Junction f (x) given that f (2) = 12 f(0) = 2, f(1)= 3, and f(3) = 35. Hence find f(5).

A: f(5)=52 ....

Q: V5 + v6 Find a monic integral polynomial for which x is a root and prove that x is algebraically…

A: We have x = 5+6. The objective here is to show that x is an algebraically irrational number.  An…

Q: Let P(z) = z" +am-12"-1 + ... + a1z + ao be a polynomial of degree n 2 1 with a; # 0 for at %3D…

A: We will solve Q9 as mentioned. Given Pz=zn+an-1zn-1+…+a1z+a0 be a polynomial of degree n≥1 with ai≠0…

Q: Let f be an entire function such that |f(2)| < A + B|z|" for all z E C, where A and B are positive…

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Q: Under what conditions (if any) are polynomials (over R) injective? Surjective? Bijective? Explain…

A: Let us consider the n degree polynomial px=anxn+an−1xn−1+......+a1x+a0. Here, an≠0 and x∈R…

Q: Show that the following polynomials are irreducible over Q. (a) f(r) = 13r + 15a + 20a? + 21a + 19…

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