1. Find the greatest common divisor of p(x) = x³ + 6x² + 6x + 5 and g(x) =2³x²-x-2. Does your answer depend on if we consider these polynomials inthe ring Q[r], R[r], or C[x]? Explain.

To find the gcd of p(x) and q(x), we can use the Euclidean algorithm: First, p(x) /q(x) and we get:…

Answered: 1. Find the greatest common divisor of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(1) For each member of your group record their response to: “ Give an example of a polynomial and an expression involving x that is not a polynomial." (2) A box with no top is to be fashioned from an 18 inch x 24 inch piece of cardboard by cutting out congruent squares from each corner of the cardboard and then folding the resulting tabs. Let x denote the length of the side of the square which is removed from each corner. Determine a polynomial which gives the vol- ume of the box as a function of x. Use a plot of this polynomial to determine the optimal value of x.
Q. 5) please show all work
Which of the following polynomials is reducible over Q: x3 +2 x4 + 3 x5 + 1 x2 + 3 I thought it was x4 + 3 but that was wrong. Is it x5 + 1?
Thank you
8. Find all real zeros of the polynomial f(x) = x* + 5x3 + 6x? and determine the multiplicity of each 9. Use long division to divide. a. (x3 + 5x2 + 10x + 80) + (x + 5) b. (x3 + 8) + (x + 2) 10. Use synthetic division to divide. a. (4x3 + x2 - 11x + 6) + (x + 2) b. (24 + 5x3 + 22x – 27x2) + (x - 2) c. (x3 – 27x + 54) + (x – 3) 11. If f(x) = -3x2 - 4x + 1, use synthetic division to evaluate f(4) 12. If x =-1 is a root of x + 4x? -x -4 =0, use synthetic division to factor the polynomial completely and list all real solutions of the equation.
Graph the given polynomial. f(x) = x^4 - 3x^3 - 20x^2 - 24x - 8 1) List possible zeroes and characteristics of them 2) Show work for testing the zeroes until you find all of the zeroes (x-intercepts) and their multiplicity. Give the exact value of the x-intercepts. 3) Show work for finding the y-intercept. 4) Show the end behavior and reasoning for that end behavior.

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1. Find the greatest common divisor of p(x) = x³ + 6x2 + 6x +5 and g(x) = 23x²-x-2. Does your answer depend on if we consider these polynomials in the ring Q[z], R[r], or C[a]? Explain.