/*! 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 104 Find the \(x\) -intercepts and t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 104

Find the \(x\) -intercepts and the zeros of the function. $$h(x)=x^{3}-3 x^{2}+3 x-1[4.3]$$

Short Answer

Expert verified
The x-intercept and zero of the function is x = 1.

Step by step solution

01

Understand the Function

The function given is a polynomial function in the form of a cubic equation: \[ h(x) = x^3 - 3x^2 + 3x - 1 \]
02

Set the Function Equal to Zero

To find the zeros (or the x-intercepts) of the function, set the function equal to zero: \[ x^3 - 3x^2 + 3x - 1 = 0 \]
03

Factor the Polynomial

Attempt to factor the polynomial. Notice that the polynomial is a special form of \[(a - b)^3 \], which can be rewritten as: \[ (x - 1)^3 = 0 \]
04

Solve for x

Since \[(x - 1)^3 = 0 \], the solution for x is: \[ x - 1 = 0 \] Therefore, \[ x = 1 \]
05

Verify the Solution

Substitute \[ x = 1 \] back into the original function to ensure it satisfies the equation. \[ h(1) = 1^3 - 3(1)^2 + 3(1) - 1 = 0 \] So, \[ x = 1 \] is indeed a zero of the function.

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.

x-intercepts
The concept of x-intercepts is crucial in understanding where a function crosses the x-axis. To find the x-intercepts, we need to determine the points where the graph of the equation touches or crosses the x-axis. These points occur where the y-value, or the output of the function, is zero. For our cubic polynomial function, we set it equal to zero and solve for x:
  • Given the polynomial function: \( h(x) = x^3 - 3x^2 + 3x - 1 \)
  • Setting it equal to zero:

    \[x^3 - 3x^2 + 3x -1 = 0\]
This step lets us determine the x-values for which the function's output is zero, finding points where the graph intersects the x-axis.
zeros of the function
Zeros of the function are another way to describe the x-intercepts. They are the points where the function equals zero. In other words, these are the values of x that make the equation true when the entire expression is set to zero. For our function, we are looking for the zeros of:
\[x^3 - 3x^2 + 3x -1 = 0\]
To find these zeros, we can factor the cubic equation. Finding factors that satisfy the polynomial equation will give the values of x that are the function's zeros. In this case, the equation can be factored as:
  • \breviscore:
    \[(x - 1)^3 = 0\].
  • Solving for x, we get:
    \[x - 1 = 0\]
    \[x = 1\].
    • Thus, the zero of this function is x = 1, meaning the polynomial touches the x-axis at this point.
factoring cubic equations
Factoring cubic equations involves breaking down the polynomial into simpler components, making it easier to find the solutions. A cubic polynomial is generally in the form of: \[ax^3 + bx^2 + cx + d = 0\]. In our example, the polynomial is already provided in a simplified form: \[h(x) = x^3 - 3x^2 + 3x - 1\]. Noticing common patterns or using techniques like synthetic division can help. Here, the polynomial resembles a special cubic binomial: \[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \].
  • Rewriting: \[ (x-1)^3 = x^3 - 3x^2 + 3x - 1 \].
  • Setting it to zero: \[ (x - 1)^3 = 0 \].
  • Solving: \[ x - 1 = 0 \implies x = 1 \].
    By recognizing the special form and factoring accordingly, we easily find that the root for this polynomial is x = 1.

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

Express as a single logarithm and, if possible, simplify. $$\log _{a}\left(a^{10}-b^{10}\right)-\log _{a}(a+b)$$

E-Cigarette SE-Cigarette Sales. The electronic cigarette was launched in 2007 , and since then sales have increased from about \(\$ 20\) million in 2008 to about \(\$ 500\) millionales. The electronic cigarette was launched in 2007 , and since then sales have increased from about \(\$ 20\) million in 2008 to about \(\$ 500\) million in 2012 (Sources: UBS; forbes, \(\mathrm{com}\) ). The exponential function $$ S(x)=20.913(2.236)^{x} $$ where \(x\) is the number of years after \(2008,\) models the sales, in millions of dollars. Use this function to estimate the sales of e-cigarettes in 2011 and in 2015 . Round to the nearest million dollars.

Solve using any method. $$\left|2^{x^{2}}-8\right|=3$$

Bachelor's Degrees Earned. The exponential function $$ D(t)=347(1.024)^{t} $$ gives the number of bachelor's degrees, in thousands, earned in the United States \(t\) years after 1970 (Sources: National Center for Educational Statistics; U.S. Department of Education). Find the number of bachelor's degrees earned in \(1985,\) in \(2000,\) and in \(2014 .\) Then estimate the number of bachelor's degrees that will be earned in \(2020 .\) Round to the nearest thousand degrees.

Use a graphing calculator to find the point \((s)\) of intersection of the graphs of each of the following pairs of equations. $$y=\left|1-3^{x}\right|, y=4+3^{-x^{2}}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.