Please show all work and follow the steps! Math, Algebra, Other For each of the following 3 polynomials, do all of the following:1. Write the problem and its number on your page. Be sure to have clear problem numbers, answers, and the correspondingwork for EACH problem in order.2. List all of the possible rational roots.3. Then use synthetic division to test the possible rational roots and factor as far as possible.4. Identify complex roots, if any.5. Sketch a careful graph of the function. You must:o correctly show the end behavioro the shape of the graph near x-interceptso (if relevant) anything you can learn by considering symmetry and transformationsUnit 6 Problems2. p(x) =x*+4x³+ 13x²+36x +36

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Description: Please show all work and follow the steps! Math, Algebra, Other For each of the following 3 polynomials, do all of the following:1. Write the problem and its number on your page. Be sure to have clear problem numbers, answers, and the correspondingwo

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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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Please show all work and follow the steps!

Math, Algebra, Other

For each of the following 3 polynomials, do all of the following:
1. Write the problem and its number on your page. Be sure to have clear problem numbers, answers, and the corresponding
work for EACH problem in order.
2. List all of the possible rational roots.
3. Then use synthetic division to test the possible rational roots and factor as far as possible.
4. Identify complex roots, if any.
5. Sketch a careful graph of the function. You must:
o correctly show the end behavior
o the shape of the graph near x-intercepts
o (if relevant) anything you can learn by considering symmetry and transformations
Unit 6 Problems
2. p(x) =x*+4x³+ 13x²+36x +36

Expert Solution

Step 1

2. Given polynomial is $p (x) = x^{4} + 4 x^{3} + 13 x^{2} + 36 x + 36$ .

We use trail and error method to find the root of equation.

We use synthetic division method to reduce the biquadratic equation to cubic equation.

Then again we use trail and error method to find the roots of the cubic equation and reduce it to quadratic equation.

We know the formula to find the roots of quadratic equation.

Let $x = 0$ then $p (0) = 36$ . x=0 is not root of given equation.

Let $x = - 1$ then $\begin{array}{rcl} p (- 1) & = & - 1^{4} + 4 {(- 1)}^{3} + 13 {(- 1)}^{2} + 36 (- 1) + 36 \\ = & 1 - 4 + 13 - 36 + 36 \\ = & 10 \end{array}$ . $x = - 1$ is not root of the given equation.

Let $x = - 2$ then $\begin{array}{rcl} p (- 2) & = & {(- 2)}^{4} + 4 {(- 2)}^{3} + 13 {(- 2)}^{2} + 36 (- 2) + 36 \\ = & 16 - 32 + 52 - 72 + 36 \\ = & - 16 - 20 + 36 \\ = & - 36 + 36 \\ = & 0 \end{array}$ . Implies $x = - 2$ is root of the given equation.

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