/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 The zero(s) of the function are ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 24

The zero(s) of the function are given. Verify the zero(s) both algebraically and graphically. \(Function\) \(f(x)=x^{3}-9 x^{2}+18 x\) \(Z e r o(s)\) \(x=0,3,6\)

Short Answer

Expert verified
The given x-values \(x=0,3,6\) are indeed the zeros of the function \(f(x) = x^{3}-9x^{2}+18x\), as verified algebraically by substituting the values into the function, and graphically by observing the function intersection points on the x-axis.

Step by step solution

01

Algebraic Verification

To verify the zeros algebraically, substitute each given x-value into the function and ensure it evaluates to 0. For x = 0: \(f(0) = (0)^{3} - 9(0)^{2} + 18(0) = 0\). For x = 3: \(f(3) = (3)^{3} - 9(3)^{2} + 18(3) = 0\). Lastly, for x = 6: \(f(6) = (6)^{3} - 9(6)^{2} + 18(6) = 0\). All values lead to the function returning 0, which verifies algebraically that given x-values are zeros of the function.
02

Graphic Verification

To verify the zeros graphically, plot the function \(f(x) = x^{3}-9x^{2}+18x\) and mark the x-values: 0, 3, and 6. Observe the points at which the function intersects the x-axis. If these points correspond to the given x-values, then you can confirm that these are indeed the zeros of the function. On plotting the function, the function intersects the x-axis at x=0, x=3 and x=6, thereby verifying the zeros graphically.
03

Conclusion

After analyzing both algebraically and graphically, it can be concluded that \(x=0,3,6\) are indeed zeros of the function \(f(x) = x^{3}-9x^{2}+18x\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphical Verification
Graphical verification involves plotting a function to see where its graph meets the x-axis. This is important because any point where the graph of a function intersects the x-axis represents a zero of the function.

To verify zeros graphically, follow these steps:
  • First, use a graphing tool or graph paper to draw the function. The given function is cubic: \(f(x)=x^{3}-9x^{2}+18x\).
  • Next, check the graph for points where it crosses the x-axis. These intersections confirm the zeros of the function.
  • In this case, plotting \(f(x)\) shows intersections at \(x=0\), \(x=3\), and \(x=6\). Thus, these values are confirmed as zeros graphically.
Graphical verification is a visually intuitive way to support algebraic findings. It helps in understanding root behavior and function changes across the x-axis.
Polynomial Zeros
Polynomial zeros, often called roots, are the values of \(x\) for which the polynomial evaluates to zero. For the function \(f(x)=x^{3}-9x^{2}+18x\), the zeros are \(x=0\), \(x=3\), and \(x=6\).

Determining polynomial zeros involves solving the equation \(f(x) = 0\). In our function:
  • Factor the expression: \(x(x-3)(x-6) = 0\).
  • Set each factor equal to zero: \(x=0\), \(x-3 = 0\), and \(x-6 = 0\).
  • Solve these simple equations to find the zeros.

Algebraically finding zeros provides exact values critical for various applications, such as calculus or solving real-world problems.
Function Intersections
Function intersections describe the points where a function meets specific axes or other functions. In the context of verifying zeros, we're looking at the function's intersection with the x-axis.

To find where a polynomial function like \(f(x)\) meets the x-axis:
  • We solve for \(f(x)=0\) to find these intersection points. This is equivalent to finding the zeros.
  • The function \(f(x)=x^{3}-9x^{2}+18x\) intersects the x-axis at points \(x=0\), \(x=3\), and \(x=6\).
  • These intersections reveal the input values (or x-values) where the output of the function is zero.
Function intersections are significant because they indicate where the graph of the function changes direction or crosses below and above the x-axis. This helps in predicting function behavior and analyzing graphical data.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Operations with Rational Expressions Simplify the expression. $$\frac{2}{x^{2}-4}-\frac{1}{x^{2}-3 x+2}$$

Find the inverse function. $$y=5 x+8$$

Use a graphing utility to graph the equation and graphically approximate the values of \(x\) that satisfy the specified inequalities. Then solve each inequality algebraically. Equation \(y=x^{3}-x^{2}-16 x+16\) Inequalities (a) \(y \leq 0\) (b) \(y \geq 36\)

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$4 x^{2}-4 x+1>0$$

Match the equation with a method you would use to solve it. Explain your reasoning. (Use each method once and do not solve the equations.) (a) \(3 x^{2}+5 x-11=0 \quad\) (i) Factoring (b) \(x^{2}+10 x=3 \quad\) (ii) Extracting square roots (c) \(x^{2}-16 x+64=0 \quad\) (iii) Completing the square (d) \(x^{2}-15=0 \quad\) (iv) Quadratic Formula
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.