Find necessary and sufficient conditions on the numbers a, b, c E R under which the polynomial xª + ax? + bx +cis divisible by x² + x + 1.

Q: 2. Decide if the following polynomials are reducible in Q[x]. (a) 2x3 – 3x + 3; (b) 4x3 — 5х + 2.

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

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Q: The polynomial f(x)=x^4+6x^3+ax^2-54x+c has four real roots r1, r2, r3, r4 such that r1+r2 = 0 and…

A: The polynomial f(x)=x^4+6x^3+ax^2-54x+c has four real roots r1, r2, r3, r4 such that r1+r2 = 0 and…

Q: Find a, b, c E R such that the polynomial f(x) = x³ + ax? + bx + c is divisible by g1(x) = x + 2 and…

A: The remainder theorem says that when a polynomial px is divided by x-a. The remainder is obtained as…

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

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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: 1. Show whether the given polynomials 1+4x + x², 2+ 5x + 3x2, 3x 5x, and 3+ 15x - 8x spans p,.…

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Q: Given that π is not the zero of a nonzero polynomial with rationalcoefficients, prove that π 2…

A: Let us consider a function  f(x)= x2 - ax - b Where a and b are rational  So, f(x) a polynomial with…

Q: Find the zeros of the polynomials f (u) =4u2+8u ,and verify the relationship between the zeroes and…

A: Sum of zeros = -(b/a)  product of zeros = (c/a)

Q: Find a, b, c E R such that the polynomial f(x) = x³ + ax? + bx + c is divisible by g1 (x) = x + 2…

A: The given question is related with polynomials. Given that,                  fx = x3 + ax2 + bx + c…

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

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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, P(x)=x4-2x3-9x2+2x+8, a=-3 and b=5       Show that a=-3 is the lower bound and b=5 is the…

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

A: Let u1 = x2+2, u2 = 2x2-x+1, u3 = x+2, and u4 = x2+x+4. Notice that u1 + u3 = u4. So, u1, u2, u3,…

Q: 3) Show that (a + b +c)² = a² + b² + c² + 2ab + 2ac + 2bc is a polynomial identity.

A: To show: a+b+c2=a2+b2+c2+2ab+2ac+2bc

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

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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: Based on the Fundamental Theorem of Algebra, how many zeros does the polynomial z - 3z-5z +15r? -…

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Q: Consider the homogenous polynomial ai + 10r1r2 + x². Find a suitable substitution s = Question 3 azi…

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Q: The number of zeros of the polynomial x² + 2x + 3 in Z, is: O 2 None

A: Number of zeros in the polynomial equation is termed as the whole polynomial equation is zero for…

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: (a) What is the degree of Q(7, V3) over Q ? Justify your answer. Is the polynomial X - 5 € QIX)…

A: Degree of extension- Let E be an extension field of a field F i.e. F is a subfield of E, also E is a…

Q: The polynomial f(x)=x^4+6x^3+ax^2-54x+c has four real roots r1, r2, r3, r4 such that r1+r2 = 0 and…

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Q: The number of zeros of the polynomial x² + 2x + 3 in Z, is: O None O 1 O 2 4

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Q: Obtain the Legendre polynomial P4(z) from Rodrigue's formula Pn (2): = 1 d 2"n! dan [(2²-1)"]

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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: 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: 8. Construct a polynomial with Z coefficents with a root of the given C 3+ V5 (a) (b) 2+ 3і (c) 1-…

A: We have to solve the given problem.

Q: Prove that if a > 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 > 0 and n is any positive integer. Claim:…

Q: Assume that the polynomial 7? + bx +c = 0 in which b, cE Z, has two integer solutions. Which one of…

A: We have to equation as - x2 + bx + c = 0.......(1) The above equation (1) is a quadratic equation…

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: 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: By applying divisionalgorithm prove that the polynomial g(x) = x² +3x +1 is a factor of the…

A: Here we have to apply a division algorithm to prove that the polynomial g(x)=x2+3x+1 is a factor of…

Q: P (x) = 2x +4x² - 2x+ 12

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

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: 4. If p(z) (4x – 5)(4x + 5), what are the zeros of the polynomial? = - and x = : O B. x = - and x =…

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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: 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: Construct a polynomial with Z coefficents with a root of the given C 2+ V7 (d) 3 (e) V3+ v2

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Q: Express f (x)) =2x+5 in term of legendres polynomials . P1 (x) +P,(x)., 2p, (x) +5p,(x) » .b O 2p,…

A: Given,  f(x) = 2x +5

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: It can be proven that the expression (a +b) a - ab +b-) is equivalent to which of the following…

A: Given,            a + b a2 -ab + b2

Q: determine the degree of the given polynomials. -x2y + 2xy2 - xy

A: Given the polynomials  -x2y+2xy2-xy

Q: Find a, b, c e R such that the polynomial f(x) = x° + ax? + bx + c is divisible by g1(x) = x +2 and…

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