Problem 68 Solve each equation. $$(x-1)(x... [FREE SOLUTION]

Chapter 5: Problem 68

Solve each equation. $$(x-1)(x+3)(x-9)=0$$

Short Answer

Expert verified

The solutions are $x = 1, $ $x = -3, $ and $x = 9$.

Step by step solution

- Set each factor equal to zero

To solve the equation $(x-1)(x+3)(x-9) = 0$, recognize that the product of three factors equals zero if and only if at least one of the factors is zero. Therefore, set each factor equal to zero: $x-1 = 0$, $x+3 = 0$, and $x-9 = 0$.

- Solve each individual equation

Solve each equation separately: \ 1. For $x-1 = 0$, solve for $x$ to get $x = 1$. \ 2. For $x+3 = 0$, solve for $x$ to get $x = -3$. \ 3. For $x-9 = 0$, solve for $x$ to get $x = 9$.

- Write the final solutions

Combine all solutions from Step 2. The solutions to the equation $(x-1)(x+3)(x-9) = 0$ are $x = 1, -3, \text{ or } 9$.

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.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler components called factors. For instance, instead of dealing with a complex expression, you split it into simpler ones that, when multiplied, give you the original polynomial.

To solve polynomial equations like $(x-1)(x+3)(x-9)=0$, we use factoring. This process makes it easier to solve equations by making them simpler and more manageable. If you factor correctly, you reduce a complicated polynomial into a product of simpler ones, as done in the example where $(x-1)(x+3)(x-9)$ factors the polynomial completely.

Zero Product Property

The Zero Product Property states that if the product of two or more factors is zero, at least one of the factors must be zero. This concept is pivotal when solving polynomial equations. For instance, given an equation $(x-1)(x+3)(x-9)=0)$, it implies that either $(x-1)=0)$, $(x+3)=0)$, or $(x-9)=0)$.

Here's how it works:

For $(x-1)=0)$, solving gives us $x=1)$.
For $(x+3)=0)$, solving gives us $x=-3) $.
For $(x-9)=0)$, solving gives us $x=9)$.

By applying the Zero Product Property, we're able to solve each simpler equation individually and find all possible values of x.

Solving Linear Equations

Solving linear equations is essential, especially when dealing with factored polynomials. A linear equation is any equation that can be written in the form of $ax + b = 0$, where a and b are constants.

When we factor polynomials, we often end up with linear equations like $(x-1)=0)$ or $x+3=0)$. Solving these is straightforward:

For $(x-1)=0)$, add 1 to both sides to get $x=1)$.
For $x+3=0)$, subtract 3 from both sides to get $x=-3)$.
Similarly, for $(x-9)=0)$, add 9 to both sides to get $x=9)$.

Understanding how to quickly solve these linear equations is crucial, as it directly lets us find the x-values that satisfy our original polynomial equation.

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Solve each equation. $$(x-1)(x+3)(x-9)=0$$

Short Answer

Step by step solution

- Set each factor equal to zero

- Solve each individual equation

- Write the final solutions

Key Concepts

Factoring Polynomials

Zero Product Property

Solving Linear Equations

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Mechanics Maths

Applied Mathematics

Pure Maths

Probability and Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.