Let p(x) be a polynomial that is divisible by (x-a)n where n∈ N and n >= 2. Prove that p'(x) is divisible by (x-a)n-1.

Q: Verify that the factorization for f (x) = x3 + x2 + 1 over Z2

A: Given that, fx=x3+x2+1. Also ℤ2=0, 1 To verify the factorization for fx over ℤ2.

Q: There is the positive real roots of a family of polynomials of the form: p(n) = x^n - x^(n-1) - ...…

A: We will use basic arithmetic to solve this equation and besides this the knowledge of plotting the…

Q: Form the interpolating polynomial of degree k that satisfies pG) (xo) = f(G) (xo) for 0<j< k.

A: To form: The interpolating polynomial of degree k that satisfies pjx0=fjx0 for 0≤j≤k

Q: . Factor x12 -1 as a product of irreducible polynomials over Z.

A: Given: x12-1. We have to factor x12-1 as a product of irreducible polynomials over Z.

Q: Suppose that x and y are real numbers such that y is 6 greater than x. What is the smallest possible…

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Q: ind a polynomial p(x) with zeros as -2,-3,1,3 ?

A: To find a polynomial p(x) with zeros as -2,-3,1,3

Q: Consider the polynomial P (x)=kx' - 5x +3x + 2. Find the value of k such that the remainder is 4…

A: The polynomial is given by Px = kx3-5x2+3x+2 The remainder obtained after dividing the polynomial Px…

Q: ve that if x5 – 3x4 + 2x

A: Introduction: In an equation, the two expressions are assumed to be equal, as indicated by the…

Q: product of irreducible polynomials.

A:

Q: The president of a major housing construction firm reports that the number of construction jobs (in…

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Q: If p(x) is a polynomial in Zp[x] with no multiplezeros, show that p(x) divides xp^n-x for some n.

A: Px is a polynomial  Px=x-ax-bx-c...x-ux-vx-w where   a,b,c...∈Zp but u,v,w∉Zp Let Rx=x-ux-vx-w has…

Q: Let f(x) = x* - 3x2 + 2 be a polynomial function. Which one the following intervals, the function…

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Q: Define the upper bound for the error in a Chebyshev polynomial?

A: To Define: An upper bound for the Chebyshev polyomial.

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: Find the integer a in Z5 so that (x-a) become a divider for f(x)=x^3+2x^2+3x+3 over Z5[x]

A: We have to find an a∈Z5 such that x-a is a divider for the polynomial fx=x3+2x2+3x+3 over Z5x. If…

Q: Suppose X₁ ~ N(6, 1) and X₂ ~ N(7,2), and X₁ is independent of X₂. Find P(X₁ < X₂) and P(X₁ < X₂ +…

A: We know linear combination of normal distribution is also a normal distribution. Therefore any…

Q: Find the polynomial f(x) for which x· f(x – 1) = (x – 3) · f(x), Væ |

A: find the polynomial f(x) for which  x fx-1=x-3 fx ∀x for x=0 x fx-1=x-3 fx ∀x

Q: Find a value for c so that the polynomial y^2+10y+c factors to (y+5)2.

A:

Q: 2. Find a polynomial that interpolates (-1,4) and (3,6), then find the value of f(x) at x=0 and x=…

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Q: A polynomial which goes through the point (1,4) and has zeros at x=0,x=3 and x=-2

A: Given:A polynomial pass through the point (1,4) and has zeroes at x=0, x=3 and x=-2.

Q: Factor x12 - 1 as a product of irreducible polynomials over Z.

A: Concept: A branch of mathematics which deals with symbols and the rules for manipulating those…

Q: Find p(1) and p (-3) of each polynomial. p(x) = x4 - x3 + x - 1

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Q: The general polynomial of degreen has the form = a,x" + a,-1x"- + .. + azx? + ajx + ao, where a, +…

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Q: E Show that x'+ 3x+ 2 has 4 zerosin Ze

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Q: Given (xi, x2024) Vi = 0,...,n, where x0 = -1, x₁ = 0, x2 = 1, x3 = 2, ..., xn = n = 1. Write down…

A: (a). When n=1, we have to points: (x0​,x02024​)=(−1,1) and (x1​,x12024​)=(0,0).  The interpolating…

Q: Determine the number a in Z5 so that (x - a) becomes a divisor of f (x) = x ^ 3 + 2x ^ 2 + 3x + 3 in…

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Q: Express f(v) 4 + 6 x² + Z x + 2 in terms of Legendre Polynomials.

A: