A. Show that if p(n) is a polynomial of the form n3 + c2 n² + C1 n + co, then p(n) = ©(n³).

Q: Let a be a root of the polynomial x3+x+3, which is irreducible over Q. The polynomial x2-2 is also…

A: Given: a is a root of the polynomial x3+x+3, which is irreducible over ℚ. The polynomial x2-2 is…

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

A: Let P(x)=ax2+bx+c be an irreducible polynomial in Q[x] a) To show P(x) has two distinct roots, let…

Q: Consider the Legendre polynomials of degree n Ph(x) = Σ (-1)m 0<m< where n is a nonnegative integer…

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

A: We will use the basic knowledge of polynomials and rings to answer this question properly, in…

Q: If p(x) is a polynomial in Zp[x] with no multiple zeros, show thatp(x) divides xpn - x for some n.

A: As per the question we are given a polynomial p(x) in the field ℤp[x] with no multiple zeros.  And…

Q: a. Let V be the the set of polynomials of degree less than or equal to 2, with operations defined as…

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Q: Using Newton's Divided Difference, the polynomial that interpolates x + sinx at the points x, = 1,x2…

A: The objective is to find the value of 2c3-c2

Q: If p(x) is a polynomial in Zp[x] with no multiple zeros, show thatp(x) divides xpn - x for some n.

A: Given that px is a polynomial in Zpx with no multiple zeros. We have to show that px divides xpn−x…

Q: Suppose P(z) is a polynomial satisfying • P( – 4) = - 32 • P'( – 4) = 44 • P''( – 4) = – 90 • P(3) (…

A: The polynomial function P(x) where polynomial is a function of x , input any value of x the value of…

Q: Let Br(X) denotes k-th Bernoulli polynomial and B be the k-th Bernoulli

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Q: A quadratic polynomial y = 4x² + 7x – 3 has roots p and q. Find a quadratic g(x) = x² + ax + ß with…

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Q: Show that P(x) = (2n + 1) P(x) + P_1(x) n-1 where P(x) is a Legendre polynomial of degree n.

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Q: Find all the possible rational zeros of the following polynomial: f(x) = x³ – 5x² − 3x+14 O ±1, ±2,…

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Q: Expand the f(x) = 10 + 8x +9x² in a seried of Legendre polynomials. O a 13P,x)– 8P,(x) – 6 P, (x) O…

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Q: A quadratic polynomial y = 4x2 + 7x – 3 has roots p and q. Find a quadratic g(x) x2 + ax + Bwith…

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Q: Find a 4th degree polynomial f(x) with integer coefficients, such that 2i, -1, and 2 are roots of…

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Q: Suppose the denominator of a rational function can be factored into distinct linear factors D(x) =…

A: Induction is used to to proove the statement. As at n=1 the value of N(x) is a for distinct real…

Q: Which polynomial function has the following end behavior? Assume a and k are positive constants. If…

A: Given: If x→-∞ then p(x)→+∞,   If x→+∞ then p(x)→-∞

Q: A quadratic polynomial y = 4x2 + 7x – 3 has roots p and q. Find a quadratic g(x) x2 + ax B with…

A: topic- quadratic equations for equation ax2+bx+c=0sum of roots=-baproduct of roots=ca

Q: Show that if p(n) is a polynomial of the form n³ + c₂ n² + c ₁ n + Co, then p(n) = O(n³).

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Q: 4. Find a polynomial p(x) of degree at most 2, which is not in the span of P₁(x) = x² - 2x +1,…

A: p1x=x2-2x+1p2x=x-1p3x=x2-1

Q: Prove the following for Legendre polynomial P(x): 0 for m # n P,(x)P(x)dx 2 for m = n 2n+1

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Q: Find a polynomial, f (x), such that f(x) has three roots and two of these roots are x = 1 and x =…

A: It is given that polynomial has three roots and two of these roots are  x= 1 &amp; x=-3, and leading…

Q: Consider the polynomial P(x) = 2x² + 5x³ + x² + 10x - 6. (b) Show that if z is a complex number for…

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Q: 4) For Bernstein polynomial, B₁ (f; x) = Σk=o (k) x* (1 − x)"-kƒ (), we have B₁ (2t² + 1;x) = 2x² +…

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Q: A quadratic polynomial y = 4x² + 7x – 3 has roots p and q. Find a quadratic g(x) = x2 + ax + B with…

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Q: Find all the possible rational zeros of the following polynomial: f(x) = 2x²¹ - 5x³+2x+2 O +1, +2…

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Q: Exercise 3. Find all polynomial pE P(R) such that p(2?) = (2² + 1)p(2). You should first find the…

A: We have to find the degree of polynomial first.

Q: A zero of the polynomial 8x3-10x2-x+3 in Z1s(x) is O a.x=-1 O O b.x-1 OCx-2 Od. None

A: From the option, we put the value of x and then find out value of equation , If zero will come its…

Q: [23-R] [CLO 6] Find all of the solutions of the equation 3(x + 5)3 (6 − x)9 (4x − 2) = 0. (Note:…

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Q: Assume that the polynomial 7? + bx +c = 0 in which b, cE Z, has two integer solutions. Which one of…

A: We have to equation as - x2 + bx + c = 0.......(1) The above equation (1) is a quadratic equation…