Problem 63 - Elementary Algebra

Question

Factor completely. $$ x 3-1000 $$

Step-by-Step Solution

Answer

The expression factors as \((x - 10)(x^2 + 10x + 100)\).

Step 1: Identify the Expression Type

We first need to recognize the expression we have: \( x^3 - 1000 \). This is a difference of cubes, as both \( x^3 \) and \( 1000 \) are perfect cubes.

Step 2: Use the Difference of Cubes Formula

The difference of cubes can be factored using the formula: \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \). Here, \( a = x \) and \( b = 10 \) since \( 1000 = 10^3 \).

Step 3: Apply the Formula

Substitute \( a = x \) and \( b = 10 \) into the formula: \[(x - 10)(x^2 + 10x + 100)\]. This gives us the factorization of the original expression.

Step 4: Verify the Factorization

You can expand \((x - 10)(x^2 + 10x + 100)\) to verify it's equivalent to the original expression \(x^3 - 1000\). Expanding, you will get \(x^3 - 10x^2 + 10x^2 - 100x + 100x - 1000 = x^3 - 1000\).

Key Concepts

Difference of Cubes

The difference of cubes is a specific type of polynomial expression that can be simplified using a distinct factorization method. To identify a difference of cubes, recognize the expression in the form of

• \(a^3 - b^3\)

where \(a\) and \(b\) represent two terms that are both perfect cubes.
Perfect cubes are numbers or expressions raised to the power of three, such as \(x^3\), \(8\), or \(1000\).

Once identified, you apply the difference of cubes formula:

• \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)

The formula breaks down a seemingly complex expression into a product of two simpler expressions.
This template applies to all differences of cubes, making it a powerful tool in algebra.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations such as addition and multiplication.
They are fundamental in algebra as they model real-world and abstract mathematical problems.

For example, in the expression \(x^3 - 1000\), you see:

• \(x^3\) as the variable term
• \(1000\) as a constant term
• '-' the operation subtracting 1000 from \(x^3\)

Algebraic expressions can be summarized as an affable mix of mathematics and logic, illustrating changes and relationships.
Breaking them into simpler terms through factoring can clarify these relationships.

This simplification is not just a method for solving equations more efficiently, but also aids in understanding the underlying structure of the problem at hand.

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors.
This is akin to decomposing a complex construct into its building blocks, making it more manageable.

To factor polynomials effectively, one must:

• Identify the type of polynomial (e.g., difference of cubes, perfect square trinomial)
• Use the appropriate factorization formula
• Verify the factorization through expansion

In the example \(x^3 - 1000\), we applied the difference of cubes formula.
By recognizing \(x^3 - 1000\) as a difference of cubes, it becomes expressible as \((x - 10)(x^2 + 10x + 100)\) after employing the formula.
This not only confirms the accuracy of the factorization but also enhances comprehension of the polynomial's characteristics.

Understanding polynomial factorization equips students with the skills to tackle a range of algebraic problems with ease.