(a) Determine whether the polynomials 1+ 3x + 3x?, x + 4x?, 5 + 6x + 3x², 7+ 2x – x² linearly dependent or linearly independent in P2

Q: . Do the polynomials x? + 2, 2x? – x + 1,x + 2, and x² + x + 4 span P2?

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

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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: Find a second-degree polynomial P such that P(1) = 6, P'(1) = 5, and P''(1) = 8.

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Q: Which of the following Polynomial satisfies the condition that all of its roots is a root of…

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Q: Express the polynomial a+ 3 +a2 + 2r as a product of its irreducible factors in Z,lr).

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Q: 11. Write a polynomial to express the total value, T(x), of (x + 3) 20-peso bills, (x - 2) 50-peso…

A: Note:  We'll answer the first question. Since the exact question isn't mentioned. Please re-submit…

Q: 4. Write a polynomial to express the total value, T(x), of (x + 4) 20-Peso bills (x – 3) 50-Peso…

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

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Q: 2. Is the following set of polynomials linearly independent in P2? Explain your work. (-1+2t + 3t?),…

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Q: Determine which of the polynomials below is (are) irreducible over Q.a. x5 + 9x4 + 12x2 + 6b. x4 + x…

A: According to Eisenstein's criterion, it provides a suitable condition for polynomial be irreducible…

Q: In Z4{x}, find two distinct polynomials that determine the same function for Z4 to Z4.

A: we have to find two polynomial Z4 {n} which determine the same function for Z4 to Z4. Let us…

Q: For each p(x), q(x), r(x) below, determine whether {p(x), q(x), r(x); is is linearly independent or…

A: To determine whethe {p(x),q(x),r(x)} is linearly independent or linearly dependent.

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 a polynomial cannot be written as the product of two other polynomials (excluding 1 and - 1),…

A: We know that a prime number have only 1 and that number as its factors.

Q: If b is a zero of x2 + x + 2 over Z5, find the other zero.

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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: Find (a) ⟨ p, q⟩, (b) ||p||, (c) ||q||, and (d) d( p, q) for the polynomials in P2 using the inner…

A: We have given that; p(x) = 1 − 3x + x2, q(x) = −x + 2x2

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

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Q: Which of the following polynomials go to negative infinity as x approaches infinity but go to…

A: Checking option b because  Option a,c,d is positive when x approaches to infinity and negative when…

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: Form the polynomial function that has the following real ZEROS: 3,-1, 2, 4 Section 3.1 O z4 + 2z3 -…

A: Given point is; x = -3,-1,2,4   1) x4+2x3-14x2-13x+24  we find the factor of this function and then…

Q: Show that the given values for a and b are lower and upper bounds for the real zeros of the…

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

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Q: Test the sets of polynomials for linear inde­pendence. For those that are linearly dependent,…

A: We have to determine whether or not we can find real numbers r, s,t, which are not all zero, Such…

Q: 9 Find a, b and c, given that the following pairs of polynomials are identically equal. а ax? + bx +…

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Q: The polynomial p(x) = x°- 21x – 20 has a known factor of (x- 5). %3D | Rewrite p(x) as a product of…

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Q: I. Use the Factor Theorem to determine whether the given binomial of the form x - cis a factor of…

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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: 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: 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…

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A: Given{x,2x-x2,3x+2x2}To check dependentIf they are dependent , need to express  one function in…

Q: -2x) +(4x +9x)(4x -9x+1) (b) 8 (a) 7 (c) 9 (d) 10 (e) 11 15. Determine which of the following…

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Q: 2. Determine whether the polynomial 1+ x + ? is a linear combination of P, = 2+x+x², p, = 1-x, p, =…

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Q: Show that the given values for a and b are lower and upper bounds for the real zeros of the…

A: Here we have ;  P(x)=3x4-5x3-2x2+x-1

Q: A. Classify the following polynomials as to degree. 1. 4x2 + 3x - 7x + 8 6. 3 + 3x? 2. x + 2x - 5 7.…

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Q: P (x) = 2x +4x² - 2x+ 12

A: Determine whether x+3 is a divisor of p(x)=2x3+4x2-2x+12

Q: In the characteristic polynomial (3 - A)(4 – 1)³( – 3 – )(3 – 1), what is the multiplicity of A = 4?

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Q: Use Lagrange Interpolation to find an interpolating polynomial of the form…

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Q: Show that the given values for a and b are lower and upper bounds for the real zeros of the…

A: Given: Show that the given values for a and b are lower and upper bounds for the real zeros of the…

Q: Which of the following polynomials has zeros at x = -6 (which is a bounce point), and x = 4 (which…

A: Given data  Polynomial has zeros at x=-6(bounce point) Polynomial has zeros at x=4(cross point)…

Q: Find (a) ⟨ p, q⟩, (b) ||p||, (c) ||q||, and (d) d( p, q) for the polynomials in P2 using the inner…

A: p(x) = 1 − x + 3x2, q(x) = x − x2 We need to Find (a) ⟨ p, q⟩, (b) ||p||, (c) ||q||, and (d) d( p,…

Q: 1. Identify the degree, end behavior and roots (with multiplicities) of the following polynomials.…

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Q: Show that the given values for a and b are lower and upper bounds for the real zeros of the…

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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: -4 -11 25 16. Without formally showing that there is a nontrivial linear combination of x-2x, x++x³…

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Q: 9 Find a, b and c, given that the following pairs of polynomials are identically equa a ax? + bx + c…

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Q: It is possible to write æ* as a linear combination of Chebyshev's polynomials, for example a² =…

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