Problem 35 - Elementary and Intermediate Algebra

Question

Factor. $$ 8 s^{3}-t^{3} $$

Step-by-Step Solution

Answer

The expression factors to \((2s - t)(4s^2 + 2st + t^2)\).

Step 1: Recognize the Form

The expression given is a difference of cubes: \( 8s^3 - t^3 \). This is in the form of \( a^3 - b^3 \), where \( a = 2s \) and \( b = t \).

Step 2: Use the Difference of Cubes Formula

The difference of cubes can be factored using the formula: \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \).

Step 3: Substitute and Simplify

Substitute \( a = 2s \) and \( b = t \) into the formula. The factorization becomes: \((2s - t)((2s)^2 + (2s)(t) + t^2) \).

Step 4: Compute the Components

Calculate each term in the trinomial: \((2s)^2 = 4s^2\), \(2s \cdot t = 2st\), and \(t^2 = t^2\).

Step 5: Write Final Factorization

Substitute these values back to get the final factorization: \((2s - t)(4s^2 + 2st + t^2)\).

Key Concepts

Factoring Polynomials

Factoring polynomials is like uncovering a puzzle where you look to express a polynomial as a product of simpler polynomials. Essentially, it's a way of "breaking down" an expression. This is useful for simplifying expressions and solving polynomial equations. For a start, it's crucial to identify the type of polynomial you're dealing with.

• Check whether it’s a sum or difference of cubes, a perfect square, or just a quadratic polynomial.
• Each type has its own specific method or formula for factoring.

In the problem we're examining, notice it's the difference of cubes: \( 8s^3 - t^3 \). Recognizing this informs us about the specific formula to use: \[a^3 - b^3 = (a-b)(a^2+ab+b^2) \]This shows that factoring isn’t random; it's systematic and requires recognizing patterns. With practice, spotting these patterns becomes easier, making the process of factoring more intuitive.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations (like addition and multiplication). Recognizing and manipulating these expressions is central to algebra.

• Variables can be represented by letters and can stand for unknown values.
• Operations like addition, subtraction, and multiplication are used to combine terms.

Understanding an algebraic expression involves breaking it down into simpler components, just like we did with the polynomial \( 8s^3 - t^3 \) where each part of the expression can be identified and used in computations. In this exercise, the use of the formula for the difference of cubes is central, as it translates the algebraic expression into a factored form. Generally, each term and operation in an expression carries a rule for transformation which helps in simplifying the overall polynomial.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into its simplest components, or factors. This is a key technique, especially in solving polynomial equations, because it easily allows us to find its roots.One specific strategy is to use known formulas or identities, like the difference of cubes formula applied in this example:\[ (2s)^3 - t^3 \rightarrow (2s - t)(4s^2 + 2st + t^2) \]Steps like these are crucial when approaching polynomial equations:

• Identify the pattern or form of the polynomial.
• Choose the appropriate method or formula for factoring.
• Simplify and solve for the required variable.

Remember, factorization turns complex expressions into a string of multipliers, revealing zeros and simplifying calculations. This skill, when honed, becomes a valuable tool across many areas of mathematics.