Problem 73 Factor. $$ x^{3}-1 $$... [FREE SOLUTION]

Chapter 6: Problem 73

Factor. $$ x^{3}-1 $$

Short Answer

Expert verified

The factored form of $x^3 - 1$ is $(x - 1)(x^2 + x + 1)$.

Step by step solution

Recall Difference of Cubes Formula

To factor expressions like $x^3 - a^3$, we use the difference of cubes formula: $x^3 - a^3 = (x - a)(x^2 + ax + a^2)$. In our expression, $a = 1$, so the difference of cubes formula applies directly.

Identify 'a' in the Expression

Here, we have $x^3 - 1^3$, because $a = 1$. Therefore, the expression is already in the form $x^3 - a^3$.

Apply Difference of Cubes Formula

Using the formula from Step 1, substitute $a = 1$ into $(x - a)(x^2 + ax + a^2)$ to get $(x - 1)(x^2 + 1 \cdot x + 1^2)$. Simplifying further gives $(x - 1)(x^2 + x + 1)$.

Write Final Factored Form

Now, our original expression $x^3 - 1$ is factored as $(x - 1)(x^2 + x + 1)$.

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.

Difference of Cubes

When you're faced with the task of factoring a cubic polynomial expression like $x^3 - 1$, knowing the difference of cubes formula is crucial. This formula helps in breaking down polynomials of the type $x^3 - a^3$ into simpler parts. Here's how it works:

The formula itself is $x^3 - a^3 = (x - a)(x^2 + ax + a^2)$.
This means that any expression that looks like $x^3$ minus some other perfect cube $a^3$ can be split into two factors.

The variable $a$ in this formula is the cube root of the second term in your polynomial.
In the exercise provided, we see $x^3 - 1^3$. Here, $a$ is 1, since 1 is the cube root of 1.

Factoring Techniques

Factoring is a powerful algebraic tool that allows us to simplify expressions and solve equations. One of the techniques is factoring using special formulas, such as the difference of cubes.
These techniques are invaluable for breaking down complex polynomial expressions into simpler, more manageable pieces.

Recognizing patterns in polynomial expressions is key. With practice, identifying which formula to use becomes second nature.
In our case, the formula $x^3 - a^3 = (x - a)(x^2 + ax + a^2)$ simplifies the cubic expression into a binomial and a trinomial.

Starting with an expression like $x^3 - 1$, recognizing it's a difference of cubes allows you to factor it easily, leading to solutions and simplifications.

Polynomial Expressions

Polynomial expressions are sums of terms consisting of variables raised to various powers and multiplied by coefficients. They form a foundation for much of algebra and are something you'll encounter frequently in mathematics.

Each part of a polynomial, called a term, can be a simple number, a variable, or a variable raised to a power.
In our example, $x^3 - 1$, we have a cubic polynomial since the highest power of x is 3.

Understanding how to manipulate and factor these expressions is crucial.
This includes mastering methods for recognizing specific types of polynomials, such as the difference of cubes, and applying the correct factoring technique for them.

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

Factor. $$ x^{3}-1 $$

Short Answer

Step by step solution

Recall Difference of Cubes Formula

Identify 'a' in the Expression

Apply Difference of Cubes Formula

Write Final Factored Form

Key Concepts

Difference of Cubes

Factoring Techniques

Polynomial Expressions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Pure Maths

Theoretical and Mathematical Physics

Geometry

Calculus

Study anywhere. Anytime. Across all devices.