Question 6. Prove by using elementary argument that any polynomial with real coefficients of degree 5 can be represented as a product of linear terms (z - a), where a is a real (constant) number and quadratic terms (22- bz + c), where b, c are also real constants. [Hint: you may assume that roots of polynomials with real coefficients are either real or come in conjugate pairs, i.e. if p(x + iy) = 0, then p(x — iy) = 0.

Q: 5) If you know the roots for a polynomial are x = -3, 1, and 2   which of the following could be the…

A: Given: If you know the roots for a polynomial are x = -3, 1, and 2 which of the following could be…

Q: 6. Find all the roots of the following polynomials: a) p(r) = * - 3.r" +?+ 7x – 30 and b) p (r) = r*…

A: Note: Since we are allowed to solve one question at a time, we have solved the first one. Kindly…

Q: 4. If x=-4 and x=1 are the roots of x* + 3x' - 3x +3x – 4 = 0, find the other two roots of the…

A:

Q: Also, our definition of the first 4 Legendre polynomials is: 1 Po(x) V2 P1 (x) = V5/2 (1 – 312)…

A:

Q: Find all roots of each of the following quartic polynomials: (a) z+ V10z + using factorization into…

A: The roots of a function are the points at which the value of the function is 0. The factorization of…

Q: 11. Use the quadratic formula to explain why a quadratic polynomial function f(x) = ax? + bx +c is…

A: fx=3x2+2x+6

Q: Ex.5 - Prove that the polynomial 2ª + z + 1 has exactly two zeros in the right half-plane.

A: Solution: We will be using the method of Routh array For a polynomial Qz=anzn+an-1zn-1+...+a1s+a0…

Q: Find that a Polynomial function given Zeros. 5, 77, -8.

A: Given zeros,  0, 7, -8

Q: 3. y = 2x + 9x² + 7x - 6 Determine the first factor by utilizing the fact if substituting x = c, (c…

A:

Q: Which of the following cannot be the number of real zeros of a polynomial of degree 5 with real…

A: We are required to find   Which of the following cannot be the number of real zeros of a…

Q: Sketch a higher degree polynomial with the following roots: -2, 3, -3, 6. f(x) 9,

A: It is given that the roots are -2, 3, -3 and 6. That is, x+2=0, x-3=0, x+3=0 and x-6=0. The…

Q: 08. z, and 22 are the roots of the equation 422 + (1-1)2 + (3+2)=0. Without finding z, and z₂, find:…

A:

Q: After seeing that line AB was perpendicular to the given line, why were each of the polynomials…

A: Given:- Line x=3t+1,y=-t+3,z=6t and point A-6,7,8

Q: 14) A family of cubic polynomials has roots of -2, -3 and -5. Find the member of this family that…

A: A family of cubic polynomials has roots of -2, -3, and -5. To find the member of this family that…

Q: 2(×+2)^2-18=0 is the equation resulting from setting a quadratic fuction f(×) equal 0 and completing…

A: A parabola having axis of symmetry along y-axis or a line parallel to y-axis is given by…

Q: 12) Given i is a zero of the polynomial P(x) = x² - 4x³ + 4x² - 4x +3, determine all other zeros…

A:

Q: a) Show that x = 1+2i is a root of p(x) where i = surds(-1). b) By using the fact that p(x) is a…

A: We have to solve given problem:

Q: (a) Prove that f(x) = x³ + x² + 2 has no roots in Q.

A: