The following polynomial 2 points below is: p (x) = a + 3x + 2 not irreducible polynomial over the polynomial of integers modulo 7 both of them irreducible polynomial over the polynomial of integers modulo 7

Q: Find a polynomial with integer coefficients that satisfies the given conditions. P has degree 2 and…

A: 3+i is a zero, so [x-(3+i)] or (x-3-i) is a factor of P. 3-i is a zero, so [x-(3-i)] or (x-3+i) is a…

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: 3. Every polynomial function of odd degree with real coefficients has at least real zero(s).

A: Given: 3.every polynomial function of odd degree with real coefficient has atleast_________real…

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: For any positive integer n, how many polynomials are there of degree n over Z2? How many distinct…

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

A: A polynomial is basically a function, which can either be quadratic, cubic and so on in which the…

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: 6. Find and fully expand a polynomial with rational coefficients and lowest possible degree that…

A: Given  Remainder = -7 when divisor (x-2)  Roots are x= 3/2 and -5  To find  polynomial with expanded…

Q: 1. If a polynomial function p(x) has real roots of -7, 2, and 3, which of the following must be a…

A: Given that: The given polynomial p(x) has real roots of -7, 2 and 3.

Q: A real number x ∈ R is called algebraic if there exist integersa0, a1, a2, . . . , an ∈ Z, not all…

A: An= the set of zereos of polynomials with integer coefficents of degree atmost n.

Q: Find a polynomial with integer coefficients that satisfies the given conditions. Q has degree 3 and…

A: Given:Q is a polynomial of degree3roots :-9,1+i

Q: Find the minimal polynomial for  ∛2 + ∛4  over Q.

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Q: Suppose a polynomial function of degree 4 with rational coefficients has the following given numbers…

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Q: Find a polynomial with integer coefficients that satisfies the given conditions. U has degree 5,…

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

A: We know that If a complex number a+ib is a zero of a polynomial, then a-ib is also zero of that…

Q: Find a polynomial with integer coefficients that satisfies the given conditions. Q has degree 3 and…

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Q: Create a polynomial that has the following characteristics: Has a zero of c = 0, with a multiplicity…

A: Leading term of a polynomial is the term with the highest power of x. For example: f(x)=2x3 +2x+4…

Q: Which of the following polynomials is guaranteed by the Intermediate Value Theorem to have a Select…

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

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Q: Consider the following polynomials over Z8, where a is written for [a] in Z8:   f(x) = 2x3+ 7x + 4,…

A: The given polynomials over Z8 are fx=2x3+7x+4,   gx=4x2+4x+6  and  hx=6x2+3. Compute fx gx+hx over…

Q: For any positive integer n, how many polynomials are there ofdegree n over Z2? How many distinct…

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

A: The objective is to find the polynomial R(x) With integer coefficients that satisfies the condition,…

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

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Q: Find a degree 3 polynomial with real coefficients having zeros 1 and 3i and a lead coefficient of 1.…

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Q: Find a polynomial P(x) of degree 3 that has integer coefficients and zeros 1/2 and 3 - i.

A: 3-i+32+1=-i2+1=-1+1=0 So 3-i  is a zero of " x+32+1 ". . . . . . (1). 2·12-1=1-1=0 So 12 is a zero…

Q: Suppose that f is a degree 4 polynomial with integer coefficients such that for some distinct…

A: Suppose that f is a degree 4 polynomial with integer coefficients such that for some distinct…

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: Show that the polynomial p(x) = -x^3+x^2+1 has a root in [1,2]

A: We can check for the values of p(x) :

Q: Consider the following polynomials over Z8 where a is written for [a] in Z8:-   f(x) = 2x3 + 7x + 4,…

A: When adding two polynomials coefficients of like powers are added. With coefficients of polynomials…

Q: Find a polynomial with integer coefficients that satisfies the given conditions. Q has degree 3 and…

A: Since, we know that if a zero of the polynomial is a complex number,the its complex conjugae will be…

Q: Find a polynomial with integer coefficients that satisfies the given conditions. Q has degree 3 and…

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Q: Find a polynomial with integer coefficients that satisfies the given conditions. P has degree 2 and…

A: Given,         P has degree 2 and zeros 2+i and 2-i.

Q: Find a polynomial p(x) = ao + a1x + aza² + azx³ of degree < 3 (here ao, a1, a2, a3 are real numbers)…

A: In this question , use the given values and solve the equations and find the polynomial.

Q: Find a polynomial with integer coefficients that satisfies the given conditions. P has degree 2 and…

A: Given that the zeros of polynomial are 1 + i and 1 - i. To find the polynomial of P(x) has degree.

Q: Find a polynomial with integer coefficients that satisfies the given conditions P has degree 3 and…

A: Use factor form for polynomial

Q: 4. If the polynomial function P has real coefficients and if a + bi is a zero of P, then is also a…

A: We know, complex roots occurs in conjugate pairs. So if,x+iy, is a root of given polynomial…

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

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Q: Find a polynomial with integer coefficients that satisfies the given conditions. Q has degree 3 and…

A: Here we have to find the polynomial with integer coefficients that satisfies the given conditions. Q…

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

A: If a complex is a root of polynomial then it complex conjugate is also a root.

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: Find a polynomial with integer coefficients that satisfies the given conditions. T has degree 4,…

A: The given conditions of a polynomial T with integer coefficients are,

Q: For any non-negative integer m show that In [x2m] is a polynomial in n of -degree m with integer…

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Q: the legendre polynomial pn (x) has .a none of these .b n zeros of which only one is between – 1 and…

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Q: In a ring, the characteristic is the smallest integer n such that nx=0 for all x in the ring.  Is it…

A: Definition of Polynomial ring-

Q: Find a polynomial of degree 3 that has zeros – 1, 1 and 2 such that the coefficents are integers= 2'

A: Let the polynomial equation be(x-α)(x-β)(x-γ)=0where α,β,γ are zeros

Q: In the set of polynomials in the form: P(x) = ax + bx' + cx² + dx + e, x E R there is a subset M,…

A: Given the polynomial M=px=ax4+bx3+cx2+dx+e, x∈ℝ|x,p(x)=-2,4,-1,0,1,-2,3,6. (a) Consider a cubic…

Q: the following polynomial irreducible over Q? (Prove your statement)! 14x6 – 18x3 + 9x² + 6x – 60 - -

A: The polynomial is irreducible over Q by using Eisenstein's Criterion for irreducibility test.

Q: 1 The cubic Legendre polynomial is P2(x) = (5x³ – 3). Prove that it is orthog onal (over [-1, 1]))…

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Q: A subset of the real numbers SCR is defined by the statement that x E S whenever x is a real root of…

A: As per the policy we are solving first three subparts. Please specify which subparts are to be…

Q: 7. Let a be a nonzero real number. Evaluate limr→ ∞ f(x) and limr+ -∞ f(x), where f (x) = ax – 3x* +…

A: See the attachment

Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.