Problem 34 - Elementary and Intermediate Algebra

Question

Factor. $$ 125-t^{3} $$

Step-by-Step Solution

Answer

The expression \(125-t^3\) factors to \((5-t)(25 + 5t + t^2)\).

Step 1: Identify as a Difference of Cubes

The expression given is \(125-t^3\). This can be interpreted as a difference of cubes because it can be written in the form \(a^3 - b^3\). Here, \(a^3 = 125\) and \(b^3 = t^3\). Since \(125 = 5^3\), we have \(a = 5\) and \(b = t\).

Step 2: Apply the Difference of Cubes Formula

The formula for factoring a difference of cubes is \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Substituting \(a = 5\) and \(b = t\) into the formula, we have: \((5-t)(5^2 + 5t + t^2)\).

Step 3: Simplify the Expression

Now, simplify the expression inside the parentheses: \(5^2 + 5t + t^2 = 25 + 5t + t^2\). So the expression \(125-t^3\) factors to \((5-t)(25 + 5t + t^2)\).

Key Concepts

Difference of Cubes

The concept of the "Difference of Cubes" in algebra is fundamental when working with polynomial factoring. It refers to an expression that is the difference between two cube numbers, expressed as \(a^3 - b^3\). Breaking down this concept, it means subtracting one cube from another.

To factor such an expression, you use the standard formula:

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

This is particularly useful because it turns a single complex expression into two more manageable factors. Understanding this formula helps in simplifying expressions and solving equations involving cubes.

For example, for the expression \(125 - t^3\), you identify \(a = 5\) since \(5^3 = 125\) and \(b = t\) as \(t^3\) is simply \(t^3\). By substituting \(a\) and \(b\) into the formula, you can derive a fully factored form. This process allows us to handle cubic differences with ease in algebraic computations.

Algebraic Expressions

In algebra, expressions are made up of constants, variables, and the algebraic operations that describe numbers and their relationships. When you see an expression like \(125 - t^3\), you're dealing with an algebraic expression.

The components of algebraic expressions include:

• Variables: These are symbols like \(t\) that represent numbers.
• Coefficients: These are numbers that multiply the variables, such as \(5\) in the expression \(5t\).
• Constants: These are fixed values, such as \(125\) in the expression.
• Operations: Including addition, subtraction, multiplication, and division.

Understanding the elements and their roles in an algebraic expression is crucial for simplifying, factoring, and solving equations. Expressions can be factored, as in our exercise, from a more complicated form into more manageable parts using the principles of difference of cubes.

Polynomial Factoring

Factoring polynomials is a powerful tool in algebra that makes complex expressions much simpler to work with. Polynomial factoring was applied in our exercise by expressing \(125 - t^3\) in a simpler form, thanks to the difference of cubes rule.

When factoring polynomials, we aim to express them as a product of simpler polynomials. It makes solving and simplifying equations far more manageable. For any given polynomial, identifying patterns like the difference of cubes or other forms of binomial and trinomial factors can be extremely beneficial.

In the exercise, we had the polynomial \(125 - t^3\) and used the difference of cubes formula. This resulted in the factors \( (5-t)(25 + 5t + t^2)\). By practicing polynomial factoring, one can better tackle more complex algebraic problems. It not only aids in solving equations but also provides clarity on the inherent structure of polynomial expressions.