### Finding the Roots of a Polynomial Function using Synthetic Division#### Problem StatementGiven the graph of the polynomial function \[ f(x) = x^3 - 15x^2 + 73x - 119 \]below, find the real root of the function. Then use synthetic division to find the remaining root(s). Show all work.![Graph of \(f(x) = x^3 - 15x^2 + 73x - 119\)](image-url)#### Graph AnalysisThe graph provided appears to be the plot of the polynomial function \( f(x) = x^3 - 15x^2 + 73x - 119 \). Based on the visual representation, we will identify the real roots by analyzing the points where the function intersects the x-axis (where \( f(x) = 0 \)).#### Steps to Solve1. **Identify the Real Root:** - First, we observe the x-intercepts of the graph. These are the points where the curve crosses the x-axis. From the graph, it looks like one of the roots can be determined approximately by visual inspection. Suppose we identify the real root as \( x = 7 \).2. **Use Synthetic Division:** - To formalize the root found, we will use synthetic division to divide \( f(x) \) by \( (x - 7) \).3. **Perform Synthetic Division:** - Write the coefficients of the polynomial \( 1, -15, 73, -119 \). - Follow the steps of synthetic division to find the quotient and remainder. \[ \begin{array}{r|rrrrr} 7 & 1 & -15 & 73 & -119 & \\ & & 7 & -56 & 119 & \\ \hline & 1 & -8 & 17 & 0 & \\ \end{array} \]4. **Interpret the Results:** - The quotient from synthetic division is \( x^2 - 8x + 17 \), and the remainder is 0, verifying \( x = 7 \) is a root.5. **Find Remaining Roots:** - Solve the quadratic equation \( x^2 - 8x + 17 \) using the quadratic formula: \[ x = \frac{-b \

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Answered: Given the graph of f(x) = x³ - 15x² +…

Given the graph of f(x) = x³ - 15x² + 73x - 119 below, find the real root of the function. Then use synthetic division to find the remaining root(s). Show all work. 1

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

### Finding the Roots of a Polynomial Function using Synthetic Division

#### Problem Statement
Given the graph of the polynomial function
\[ f(x) = x^3 - 15x^2 + 73x - 119 \]
below, find the real root of the function. Then use synthetic division to find the remaining root(s). Show all work.

![Graph of \(f(x) = x^3 - 15x^2 + 73x - 119\)](image-url)

#### Graph Analysis
The graph provided appears to be the plot of the polynomial function \( f(x) = x^3 - 15x^2 + 73x - 119 \). Based on the visual representation, we will identify the real roots by analyzing the points where the function intersects the x-axis (where \( f(x) = 0 \)).

#### Steps to Solve

1. **Identify the Real Root:**
- First, we observe the x-intercepts of the graph. These are the points where the curve crosses the x-axis. From the graph, it looks like one of the roots can be determined approximately by visual inspection. Suppose we identify the real root as \( x = 7 \).

2. **Use Synthetic Division:**
- To formalize the root found, we will use synthetic division to divide \( f(x) \) by \( (x - 7) \).

3. **Perform Synthetic Division:**
- Write the coefficients of the polynomial \( 1, -15, 73, -119 \).
- Follow the steps of synthetic division to find the quotient and remainder.

\[
\begin{array}{r|rrrrr}
7 & 1 & -15 & 73 & -119 & \\
& & 7 & -56 & 119 & \\
\hline
& 1 & -8 & 17 & 0 & \\
\end{array}
\]

4. **Interpret the Results:**
- The quotient from synthetic division is \( x^2 - 8x + 17 \), and the remainder is 0, verifying \( x = 7 \) is a root.

5. **Find Remaining Roots:**
- Solve the quadratic equation \( x^2 - 8x + 17 \) using the quadratic formula:
\[ x = \frac{-b \

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