Problem 73 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: Identify 'a' in the Expression

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

Step 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 \((x - 1)(x^2 + x + 1)\).

Step 4: Write Final Factored Form

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

Key Concepts

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.