Show all work **Multiplicity** – If the factor $ x - c $ occurs $ k $ times in the complete factorization of the polynomial $ P(x) $, then $ c $ is called a root of $ P(x) = 0 $ with **multiplicity** $ k $. If $ c $ is a zero of even multiplicity, then the graph **touches** the x-axis and turns around at $ c $. If $ c $ is a zero of odd multiplicity, then the graph **crosses** the x-axis at $ c $.Find the zeros of each function and give the multiplicity of each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero. Show your results on the graph.$$ f(x) = (x+1)(2x-3)^2 $$$$ g(x) = x^2(x+1)(x-1)^3 $$### Graph Explanation- **Graph for $ f(x) = (x+1)(2x-3)^2 $**: - The zero at $ x = -1 $ has a multiplicity of 1, so the graph will cross the x-axis at $ x = -1 $. - The zero at $ x = \frac{3}{2} $ has a multiplicity of 2, so the graph will touch and turn around at the x-axis at $ x = \frac{3}{2} $.- **Graph for $ g(x) = x^2(x+1)(x-1)^3 $**: - The zero at $ x = 0 $ has a multiplicity of 2, so the graph will touch and turn around at the x-axis at $ x = 0 $. - The zero at $ x = -1 $ has a multiplicity of 1, so the graph will cross the x-axis at $ x = -1 $. - The zero at $ x = 1 $ has a multiplicity of 3, so the graph will cross the x-axis at $ x = 1 $.

The multiplicity of the zeroes of the function is the number of times the factor repeated in the…

Find the zeros of each function and give the multiplicity of each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero. Show your results on the graph. f(x)= (x+1)(2x–3) g(x) =x*(x+1)(x-1)

Find the zeros of each function and give the multiplicity of each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero. Show your results on the graph. f(x)= (x+1)(2x–3) g(x) =x*(x+1)(x-1)

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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**Multiplicity** – If the factor $ x - c $ occurs $ k $ times in the complete factorization of the polynomial $ P(x) $, then $ c $ is called a root of $ P(x) = 0 $ with **multiplicity** $ k $. If $ c $ is a zero of even multiplicity, then the graph **touches** the x-axis and turns around at $ c $. If $ c $ is a zero of odd multiplicity, then the graph **crosses** the x-axis at $ c $.

Find the zeros of each function and give the multiplicity of each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero. Show your results on the graph.

$$ f(x) = (x+1)(2x-3)^2 $$

$$ g(x) = x^2(x+1)(x-1)^3 $$

### Graph Explanation

- **Graph for $ f(x) = (x+1)(2x-3)^2 $**:

- The zero at $ x = -1 $ has a multiplicity of 1, so the graph will cross the x-axis at $ x = -1 $.
- The zero at $ x = \frac{3}{2} $ has a multiplicity of 2, so the graph will touch and turn around at the x-axis at $ x = \frac{3}{2} $.

- **Graph for $ g(x) = x^2(x+1)(x-1)^3 $**:

- The zero at $ x = 0 $ has a multiplicity of 2, so the graph will touch and turn around at the x-axis at $ x = 0 $.
- The zero at $ x = -1 $ has a multiplicity of 1, so the graph will cross the x-axis at $ x = -1 $.
- The zero at $ x = 1 $ has a multiplicity of 3, so the graph will cross the x-axis at $ x = 1 $.

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