/*! 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 18 Solve each equation by factoring... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 18

Solve each equation by factoring. [Hint for Exer cises 19-22: First factor out a fractional power.] $$ 5 x^{4}=20 x^{3} $$

Short Answer

Expert verified
The solutions are \(x = 0\) and \(x = 4\).

Step by step solution

01

Move All Terms to One Side

Start by moving all terms to one side of the equation to set the equation to 0. So, subtract \(20x^3\) from both sides: \[5x^4 - 20x^3 = 0\]
02

Factor Out the Greatest Common Factor

Identify and factor out the greatest common factor from the terms on the left side. In this case, both terms have \(5x^3\) as a common factor:\[5x^3(x - 4) = 0\]
03

Set Each Factor to Zero

The equation \(5x^3(x - 4) = 0\) can be solved by setting each factor equal to zero separately. This gives two equations:\[5x^3 = 0 \]\[x - 4 = 0\]
04

Solve Each Equation for x

Solve both equations to find the solutions. For \(5x^3 = 0\), divide both sides by 5:\[x^3 = 0\]Take the cube root of both sides:\[x = 0\]For \(x - 4 = 0\), add 4 to both sides:\[x = 4\]
05

Write the Final Solution

The solutions to the equation are the values of \(x\) that make the equation true. These values are:\[x = 0 \quad \text{or} \quad x = 4\]
06

Conclusion

The original equation \(5x^4 = 20x^3\) has the solutions \(x = 0\) and \(x = 4\).

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.

Solving Equations
Solving equations is a fundamental process in algebra that involves finding the value or values of the variable that make the equation true. In the context of the given problem, solving an equation often involves restructuring the equation so that it equals zero. This is done by moving all terms to one side of the equation.
For example, we had the equation \(5x^4 = 20x^3\). By subtracting \(20x^3\) from both sides, it becomes \(5x^4 - 20x^3 = 0\).
Setting an equation to zero is beneficial because it allows us to use factoring to simplify and solve the equation. Once factored, we can apply the zero-product property, which states that if the product of two factors is zero, at least one of the factors must be zero. This concept is crucial in solving polynomial equations through factoring.
Greatest Common Factor
The greatest common factor (GCF) is a key tool when simplifying expressions or equations. It's the largest factor shared among all terms under consideration.
Using the GCF greatly simplifies the process of factoring, as it allows us to take out the common element and simplify what's left.
In our example, we started with the expression \(5x^4 - 20x^3\). Both terms share a common factor, \(5x^3\), which can be factored out:
  • The term \(5x^4\) is composed of \(5x^3 \cdot x\).
  • The term \(20x^3\) is \(5x^3 \cdot 4\).
Factoring out \(5x^3\) leaves us with \(5x^3(x - 4) = 0\). Identifying and factoring out the GCF is an essential step for simplifying equations and making them easier to solve.
Polynomial Equations
Polynomial equations involve expressions that include variables raised to positive integer powers. Solving these equations can range from simple arithmetic to more advanced techniques like factoring.
In our problem, the polynomial equation started as \(5x^4 = 20x^3\), which is a fourth-degree equation due to the highest power of \(x\). By setting the equation to zero and factoring it, we simplified the degree of the polynomial.
After factoring, we use the zero-product property to solve the polynomial equations, revealing each possible solution.
  • The first factor \(5x^3\) when set to zero gives the solution \(x = 0\).
  • The second factor \(x - 4\) when set to zero gives \(x = 4\).
Polynomial equations often have multiple solutions, which are the values that satisfy the equation when substituted back into the original polynomial. Understanding these basics is essential for mastering polynomial equations in algebra.

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

63-64. BUSINESS: Straight-Line Depreciation Straight-line depreciation is a method for estimating the value of an asset (such as a piece of machinery) as it loses value ( \({ }^{\prime \prime}\) depreciates" \()\) through use. Given the original price of an asset, its useful lifetime, and its scrap value (its value at the end of its useful lifetime), the value of the asset after \(t\) years is given by the formula: $$ \begin{aligned} \text { Value }=(\text { Price })-&\left(\frac{(\text { Price })-(\text { Scrap value })}{(\text { Useful lifetime })}\right) \cdot t \\ & \text { for } 0 \leq t \leq(\text { Useful lifetime }) \end{aligned} $$ a. A farmer buys a harvester for $$\$ 50,000$$ and estimates its useful life to be 20 years, after which its scrap value will be $$\$ 6000$$. Use the formula above to find a formula for the value \(V\) of the harvester after \(t\) years, for \(0 \leq t \leq 20\). b. Use your formula to find the value of the harvester after 5 years. c. Graph the function found in part (a) on a graphing calculator on the window \([0,20]\) by \([0,50,000]\). [Hint: Use \(x\) instead of \(t\).]

SOCIAL SCIENCE: Health Club Attendance A recent study analyzed how the number of visits a person makes to a health club varies with the monthly membership price. It found that the number of visits per year is given approximately by \(v(x)=-0.004 x^{2}+0.56 x+42\), where \(x\) is the monthly membership price. What monthly price maximizes the number of visits?

The Apocryphal Manufacturing Company makes blivets out of widgets. If a linear function \(f(x)=m x+b\) gives the number of widgets that can be made from \(x\) blivets, what are the units of the slope \(m\) (widgets per blivet or blivets per widget)?

world population (in millions) since the year 1700 is approximated by the exponential function \(P(x)=522(1.0053)^{x}\), where \(x\) is the number of years since 1700 (for \(0 \leq x \leq 200\) ). Using a calculator, esti mate the world population in the year: 1800

BUSINESS: Salary A sales clerk's weekly salary is \(\$ 300\) plus \(2 \%\) of her total week's sales. Find a function \(P(x)\) for her pay for a week in which she sold \(x\) dollars of merchandise.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

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.