Let S = – 5,1 – 2x2, p(x) }. For which polynomial p(x) the set S spans P2? Select one: O 2x + x O 2x 1 none of the others O x²

Q: Find k such that f(x) = x3 - kx2 + kx + 2 has the factor x - 2.

A: The general polynomial equation of degree n  is an equation of the form…

Q: The polynomial p(x) = 2-(x+1)+x(x+1)- 2x(x+1)(x–1) interpolates the first four points in the table:…

A: Given, the polynomial p(x)=2-(x+1)+x(x+1)-2x(x+1)(x-1)   interpolates the first four points in the…

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: 4. In each part, determine whether the polynomial is a linear combination of P, = 2+x+x², P, = 1-x²,…

A: We have to determine whether the polynomial is a linear combination of given polynomials.

Q: Q1-Using the nodes (x, =2, x-=2.5 and x; =4), to find the second interpolation polynomial for flx) =

A: Given nodes are x0=2, x1=2.5, x2=4 and function is f(x)=1x Now table is x 2 2.5 4 f(x) 0.5 0.4…

Q: y = exp( 2-x ) at x = [-1,2].

A: Given   y=e(2-x) , in the interval x=-1,2 So the value of y at x=-1,0,1,2 are…

Q: Find the zeros of the polynomial f (x) = x³ – x2 + 4x – 4 О 1,2, -2 O -1, 2, -2 О 1,21, -2i O -1,…

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Q: 19. Which polynomial function has zeros 1, 1+i, and 1-1? A P(x)=x³ -x² +2x + 1 B P(x) =x -4x² + 3x –…

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Q: Find a polynomial P(x)=x3+ax2+bx+c satisfying ALL of the following properties:

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Q: Let p, = 2t – 1 and p2 = t3 +2. Write three polynomials that are in span{p1, P2}-

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Q: Write a general polynomial q(z) of degree n with coefficients that are bo, b,, b2, ..b,. What are…

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Q: Find all values of k such that f (x) is divisible by the given linear polynomial .

A: Given polynomial is fx=k2x3-4kx+3. To find all values of k such fx is divisible by x-1. Solution:…

Q: Let p be a polynomial function defined by p(x) = 2x³ + 5x²–6x-9. Use the fact that p(-4) = –33 ,…

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Q: Find k such that f(x) = x4 - kx3 + kx2 + 1 has the factor x + 2.

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Q: Determine the linear interpolating polynomial that passes through the points (2,4) and (5,1). p(x) =…

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Q: Find the critical points, A and B, of the following polynomial (with A < B). f(x) = 4x° – 588x – 6 -…

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Q: Find k such that f1x2 = x4 - kx3 + kx2 + 1 has the factorx + 2.

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Q: Show that for each positive integer n the polynomial (x-1)2 is a divisor of q(x)=nxn + 1 -(n + 1) xn…

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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: Determine the zero(s) of k(x) = x° – 7x2 + 10x. - The zeros are x =

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Q: If c is a zero of the polynomial P, then(a) P(c)=________.(b) x c is a___________ of P(x) .(c) c is…

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Q: Show that if p is a positive integer, then the binomialseries for f(x) = (1 + x) p is a polynomial.

A: The binomial expansion is given as 1+xn=1+nC1x+nC2x2+nC3x3+.....nCnxn

Q: If the polynomial f(x)=2x³ +3x²+3x+2 is expressed as f(x)=a,Po(x)+a,P1 (x)+a2P2(x)+23P3(x), then a1…

A: Since you have posted multiple questions i can do first question as per our company guidelines…

Q: For a second degree polynomial P such that P(1) = 3, P'(1) = 4, and P"(1) = 6. P(x) =

A: we have to find the second degree polynomial P such that P(1)=3, P'(1)=4 and P''(1)=6. let the…

Q: (a) Determine whether the polynomials 1+ 3x + 3x?, x + 4x?, 5 + 6x + 3x², 7+ 2x – x² linearly…

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

Q: 32. If p(x) is a polynomial in Z,[x] with no multiple zeros, show that p(x) divides xP" . x for some…

A: Let F be a field, and suppose p(x) ∈ F[x]. hp(x)i is the set of all multiples (by polynomials) of…

Q: Find a second-degree polynomial P such that  P(2) = 3, P'(2) = 7,  and  P''(2) = 6.   P(x) =

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Q: Let F(a) be Z2. Show that a2 and a2+a are zeros of x3+x+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: Find a polynomial of degree 3 (here ao, a1, a2, a3 are real numbers) such that p(x) = ao + a₁x +…

A: This question can be solved using the concept of solving system of linear equation using matrix.

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 a second-degree polynomial P such that  P(1) = 1, P'(1) = 2,  and  P''(1) = 4.

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Q: Determine the zero(s) of m(x) = 7 – 3x. x =

A: The equation is given by mx = 7-3x                                       ........1 To evaluate: The…

Q: If a, Band y are the zeroes of the polynomial p (x) = ax'+ 3bx² + 3cx+ d and having relation 2B =…

A: Option (c) is correct.

Q: Find all zeros of p(x). Include any multiplicities greater than 1.   p(x) = x3 − 3x2 + 9x − 27

A: We have to find all zeros of p(x). Include any multiplicities greater than 1. p(x) = x3 − 3x2 + 9x −…

Q: Given (xi , xi2022 i ) ∀i = 0, ..., n, where x0 = −1, x1 = 0, x2 = 1, x3 = 2, ..., xn = n − 1. Write…

A: Given xi,xi2022∀i=0,1,2,...,n  Where x0=-1, x1=0, x2=1, x3=2,..., xn=n-1

Q: Construct the divided differences table that led to the Hermite polynomial p(x) = 2 – (z – 1) + (z –…

A: We have to construct the divided differences table that led to the Hermite polynomial…

Q: Let p1(x) = x - 2020x² + b,x +c, and p2(x) =x' - 2021x + b,x + cz be polynomials having two common…

A: Given- p1(x)=x3-2020x2+b1x+c1and p2(x)=x3-2021x2+b2x+c2  are having two common roots α and β. There…

Q: The interpolating polynomial p(x) of degree 3 of f(x) = 6x4 – 2a3 + 9 is p(x) = -23 + 9. Select one:…

A: From interpolation formula we know that every polynomial f(x)of order n is approximated by a…

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

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Q: Using the nodes (xo=2, x;=2.5 and x; =4), to find the second interpolation polynomial for f(x) =

A: According to the given information, it is required to find the second interpolation polynomial for…

Q: Find the smallest integer p so that f(x) = −2x3+8x+2 must have a zero on [0, p] 2,1,5,4, or 3

A: Note: We are not given any particular method. So, we will be solving by the method of graph for…

Q: Determine the value of k such that (x^3-2kx+1)/(x-k ) has a remainder of 0.

A: f(x)=(x^3-2kx+1)

Q: Find a second degree polynomial P such that P(2) = 7, P'(2)=5, and P"(2) = 2.

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Q: 7. Show that each polynomial f(x) such that f(x) is irreducible in Z, (a) 7x + 6x? + 4x + 6 (b)

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Q: show that the polynomial p(x) = x^3 - 8x + 6 has exactly two roots on [0,3]

A: The given polynomial is p(x)=x3-8x+6. The roots of this polynomial are the values of x, where its…

Q: Find a polynomial p such that p(−1) = 0, p(4) = 0, and p(2) = 3

A: solution  let consider the desired polynomial is p(x) where p(x)=ax2 +bx+c…

Q: Find all values of k such that f (x) is divisible by the given linear polynomial .

A: Given: fx=kx3+x2+k2x+3k2+11, gx=x+2 We know that if given fx is divisible by gx then x=-2 will be a…

Q: the polynomials t3+ t² + 1, t? – t + 2, t3 + 2, -t³ +t? – 5t + 2 span P3

A: