Analyze the function g(x)=2x – 5x* +9x³ –18x² +4x+8. Follow the steps below; show a a. Describe the end behavior (prediction for y as x→ too) of the graph. b. Use Descartes' Rule of Signs. How many positive and negative real zeros might| the function have? c. Apply the Rational Root Theorem. Write a list of potential rational roots. d. Use synthetic division to test the list of roots.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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3. Analyze the function \( g(x) = 2x^5 - 5x^4 + 9x^3 - 18x^2 + 4x + 8 \). Follow the steps below; show all work.

a. Describe the end behavior (prediction for \( y \) as \( x \to \pm\infty \)) of the graph.

b. Use Descartes’ Rule of Signs. How many positive and negative real zeros might the function have?

c. Apply the Rational Root Theorem. Write a list of potential rational roots.

d. Use synthetic division to test the list of roots.

e. Express the function rule as a product of linear factors. Each factor should be in the form \( (x-a) \) or \( (x-(a+bi)) \).

f. Identify the \( x \)-intercepts and describe the function’s behavior at each intercept (does it cross the axis or bounce off?).

g. Identify the \( y \)-intercept.

h. Create (by hand) a rough sketch of the graph of \( y = g(x) \).
Transcribed Image Text:3. Analyze the function \( g(x) = 2x^5 - 5x^4 + 9x^3 - 18x^2 + 4x + 8 \). Follow the steps below; show all work. a. Describe the end behavior (prediction for \( y \) as \( x \to \pm\infty \)) of the graph. b. Use Descartes’ Rule of Signs. How many positive and negative real zeros might the function have? c. Apply the Rational Root Theorem. Write a list of potential rational roots. d. Use synthetic division to test the list of roots. e. Express the function rule as a product of linear factors. Each factor should be in the form \( (x-a) \) or \( (x-(a+bi)) \). f. Identify the \( x \)-intercepts and describe the function’s behavior at each intercept (does it cross the axis or bounce off?). g. Identify the \( y \)-intercept. h. Create (by hand) a rough sketch of the graph of \( y = g(x) \).
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