Prove that the polynomial f(x) 2an +a2", n € N, a E R \ {0} is not divisible by g(z) = r+a

Q: Let T(x) denote the Chebyshev polynomial of degree k. Express the composite polynomial Tn(Tm(x)) of…

A: The Chebyshev polynomial of first kind defined as  Tmx=2xTm-1x-Tm-2x, with initial conditions given…

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: Letf(x), g(x) and h(x) be quadratic polynomials having positive leading coefficients and real and…

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Q: Show that the polynomial 2x + 1 in Z4[x] has a multiplicative inversein Z4[x].

A:

Q: Prove that if a > 0 and n is any positive integer, then the polynomial function p(x) = x2n+1+ ax…

A: Roots of the polynomial are values are when put for variable gives 0 value for the polynomial. At…

Q: Find a monic polynomial f(x) of least degree over C that has the given numbers as zeros, and a monic…

A: Given: The zeros of the polynomial 3-i ,  i   and 2

Q: Find the monic quadratic polynomial, q(x), such that •1

A: Given:

Q: The coordinates of the P(x)=x³ – 2x2 + 3x polynomial with respect to the base B-[1, x, x?,x'] are as…

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Q: Find a polynomial with integer coefficients that satisfies the given conditions. S has degree 4 and…

A: Given, 4 degree zeroes = 2i and 3i

Q: Use the linear factorization theorem to construct a polynomial function with the given condition.…

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Q: Determine the third Taylor polynomial of the function at x = 0. f(x) = x + 36 %3D 6 + x2+ x3 1728…

A: Topic = Series  Correct option = 1st option

Q: Use the intermediate value theorem to show that the polynomial has a real zero between the given…

A: f(x)=4x3-2x-5 We need to use the intermediate value theorem to show that the polynomial has a real…

Q: The minimal polynomial of v2 + v-5 over Q is:

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Q: Find a monic polynomial f(x) of least degree over C that has the given numbers as zeros, and a monic…

A: Given, Zeros, -3i,4 it has three zeros ±3i, 4 these zeros are factor of the equation…

Q: Given the polynomial f(x) = -xr² + x + 2, what is the smallest positive integer a such that the…

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Q: .Find a monic polynomial f(x) of least degree over C that has the given numbers as zeros, and a…

A: Find a monic polynomial f(x) of least degree over C that has the given numbers as zeros, and a monic…

Q: Show that there are no polynomials f(x) and g(x) in Z[x] such that 1 = 2f(x) + xg(x)

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Q: Let f(x) = x' + 2x + 4 and g(x) - 3x + 2 in Z,[x]. Then, upon dividing f(x) by glx), we get 7- The…

A: I am solving part7 & part8 as you mentioned.

Q: Find the recursion formula and a fourth degree polynomial approximadion for the power serîies…

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Q: If f(z) is a polynomial In z such that when it divides by z-i and z+i,the remainder when f(z) is…

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Q: 13) Use the Intermediate Value Theorem to show that the polynomial function has a zero in the given…

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Q: .5) a) Express the polynomial f(x) in terms of Legendre polynomials where f(x) = 1-3x + 3x2

A: Given ,            f(x) = 1-3x+3x²

Q: Let ro = -a; xi = 0; and x2 = a. Compute the interpolation polynomial for f(x) = -r* using Lagrange…

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Q: Let xo < x1. Find a second degree polynomial P(x) such that P(xo) = fo, P(x1) = f1, and P'(xo) = fó-

A: Consider a second degree polynomial in x as: P(x)=ax2+bx+c                                     ..(1)…

Q: Q3: Factorize f (x) = 2x* + x³ + 2x + 4 € Z5[x] as a polynomials product to irreducible over Z5?

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Q: .Find a monic polynomial f(x) of least degree over C that has the given numbers as zeros, and a…

A: Monic Polynomial: "Mono" stands for "Single" Therefore it is a type of polynomial having single…

Q: Consider the polynomials a(x) = r' + x³ + x + 4 and b(x) = 2r³ + 4.x² + x + 2 from Z5[r]. a) Find…

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Q: Express the polynomial x3+2x+3 in Z5[x] as a product of irreducible polynomials of Z5[x] .

A: Explanation of the answer is as follows

Q: 5. а) Factor the polynomial f (x) = 3x*+2a³ +2x² +2x– 1 into irreducible polynomials %3D in C[r],…

A: "Since you have asked multiple questions, we will solve the first question  for you. If you want any…

Q: Use the Intermediate Value Theorem to show that the given polynomial f(x) = x3 - x - 1, has a real…

A: We have to show that the given polynomial fx=x3-x-1 has a real zero between 1 and 2 Finding the…

Q: Let f (x) = x4 –- 6x3 + 6x2 4x + 1. The coefficient of x7 in the 7th Taylor polynomial p7 is

A: Solve the following

Q: Prove that for any polynomial P of degree <n, | P' (x) | < n(1 || P || - x²)-1/2 || P ||, where max…

A: We will construct another function in terms of P(x) and prove the given statement.

Q: Let f (x) = x4 − 7x3 + 6x2 − 4x + 1. The coefficient of x8 in the 8th Taylor polynomial p8 is

A: Application of derivative

Q: Find all n EN for which is the polynomial x2n + x" +1 divisible by (x)f g(x) = x² + x + 1?

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Q: Prove that if the polynomial f(x) = x²n + a₁x²n-² + ... + an-1x² + an is divisible by g1(x) = x - 1,…

A: To Prove- Prove that if the polynomial fx = x2n + a1x2n-2 + ⋯⋯ + an-1x2 + an is divisible by g1x = x…

Q: Prove that if a &gt; 0 and n is any positive integer, then the polynomial function p(x) = x2n+1 + ax…

A: The given polynomial function is px=x2n+1+ax+b where a &gt; 0 and n is any positive integer. Claim:…

Q: Prove using the generating function of the Legendre Polynomials: t2 [xP(x) – nP,(x)]t": (1- 2xt +…

A: We have to prove ∑n=0∞xPn'(x)-nPn(x)tn=t21-2xt+t232 We know the generating function of the Legendre…

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: Find a monic polynomial f(x) of least degree over C that has the given numbers as zeros, and a monic…

A: Given :-   The zeros of the polynomial 3i , 1+2i. The complex roots occur in pair, so conjugate of…

Q: Use the intermediate value theorem to show that the polynomial has a real zero between the given…

A: The given function is

Q: a) Factor a +x³ + x into irreducible polynomials in Z2[r];

A: i) Linear factors are irreducible. ii) Just substitute values in Zp and see the expression becomes…

Q: Let f(x) = 4x4+x3+2x+1 and g(x) = 3x2+2x+3 be two polynomials in Z5[x].  Determine the quotient and…

A: We have to solve given problem:

Q: Find a monic polynomial f(x) of least degree over C that has the given numbers as zeros, and a monic…

A: It is known that complex zeros of a polynomial with real coefficients occurs in pairs. Given that i…

Q: Consider the polynomial f(x) = 25x - 9x* + 3x? - 12 %3D show that it is irreducible in Q[x]

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Q: Find the value ofn EN such that the polynomial f(x) = (x+1)"+ (x- 1)" is divisible by g(x) = x.

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Q: a) Factor a + r³ + x into irreducible polynomials in Z2[x];

A: We shall solve first Que only, if you want others to be solved kindly post it again by mentioning…

Q: prove that if the polynomial f(x) = x2n + a1x(2n-2)  +....+an-1x2 + an  is divisible by g1(x)= x-1…

A: prove that if the polynomial f(x) = x2n + a1x(2n-2)  +....+an-1x2 + an  is divisible by g1(x)= x-1…

Q: Find the Maclaurin polynomial of order 3 generated by f. f(x) = e3z² O P (x)= 1+ 3x² + §æª + §æ® O…

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Q: Let a E Z2 be a zero of the irreducible polynomial x3 + x +1 over Z2. Show that the elements of B =…

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.