Problem 43 - College Algebra

Question

For the following exercises, factor the polynomials. $$ 125 r^{3}+1,728 s^{3} $$

Step-by-Step Solution

Answer

The polynomial is factored as \((5r + 12s)(25r^2 - 60rs + 144s^2)\).

Step 1: Identify the Type of Polynomial

The polynomial given is in the form of \(a^3 + b^3\), which is a sum of cubes. Recognizing this pattern is crucial for factoring.

Step 2: Write the Sum of Cubes Formula

The sum of cubes formula is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). We will use this formula to factor the given polynomial.

Step 3: Determine Values for a and b

To match the sum of cubes pattern, identify \(a\) and \(b\). Since \(125r^3 = (5r)^3\) and \(1728s^3 = (12s)^3\), let \(a = 5r\) and \(b = 12s\).

Step 4: Apply the Formula

Substitute \(a = 5r\) and \(b = 12s\) into the sum of cubes formula:\[(5r + 12s)((5r)^2 - (5r)(12s) + (12s)^2)\]

Step 5: Simplify the Expression

Calculate the components inside the factored terms:- \((5r)^2 = 25r^2\)- \((5r)(12s) = 60rs\)- \((12s)^2 = 144s^2\)Thus, the factored expression becomes:\[(5r + 12s)(25r^2 - 60rs + 144s^2)\]

Step 6: Final Factored Form

The polynomial \(125r^3 + 1728s^3\) is fully factored as:\[(5r + 12s)(25r^2 - 60rs + 144s^2)\]

Key Concepts

Sum of Cubes

In mathematics, particularly in algebra, recognizing patterns is key to solving polynomial equations efficiently. One such pattern is the "sum of cubes," which refers to a special form of polynomial recognized as \(a^3 + b^3\). This specific type of polynomial has its own unique factoring formula that simplifies the process. The sum of cubes formula is expressed as: \[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\] This formula is instrumental because it provides a direct pathway to factor these polynomials without additional trial and error. The presence of cube powers indicates this pattern, distinguishing it from other forms. By identifying expressions that fit the sum of cubes form, algebra becomes a bit less daunting. Let's illustrate this with a real example: **125\(r^3\) + 1728\(s^3\)**. Here, we identify **125\(r^3\)** as \((5r)^3\), and **1728\(s^3\)** as \((12s)^3\). Recognizing these terms is the first step in deploying the sum of cubes approach.

Algebraic Expressions

Algebraic expressions are mathematical phrases consisting of numbers, variables, and operations. These are fundamental in algebra and serve as the basis for forming equations and solving problems. In the context of our problem, expressions like 125\(r^3\) and 1728\(s^3\) are considered algebraic expressions. They are constructed using variables \(r\) and \(s\), coefficients like 125 and 1728, and the operation of cubing.Understanding the structure of algebraic expressions aids in distinguishing their components. This involves:

• Identifying coefficients: numerical factors multiplying the variable terms (e.g., 125 and 1728)
• Recognizing variables: symbols representing unknown values (e.g., \(r\) and \(s\))
• Applying operations: mathematical functions applied to terms such as cubing

Algebraic expressions play a crucial role in translating real-world situations into a format for analysis and solving. They also form the building blocks of more complex algebraic structures, including equations and systems of equations.

Polynomial Factoring Techniques

Polynomial factoring is an essential technique in algebra that involves breaking down complex expressions into simpler components. This process is vital for simplifying expressions and solving equations, much like we did with the sum of cubes.There are several methodologies in polynomial factoring, some of which include:

• Factoring by grouping: Rearranging and grouping terms in a polynomial to factor it more easily
• Using special formulas, such as the difference and sum of cubes
• Factoring quadratics using different methods like the quadratic formula or completing the square

For a polynomial like **125\(r^3\) + 1728\(s^3\)**, recognizing it fits the form for sum of cubes lets us apply the specific formula. This approach quickly decomposes the polynomial into \((5r + 12s)(25r^2 - 60rs + 144s^2)\). Each step taken is aimed at transforming the polynomial into its simplest form, facilitating easier manipulation or evaluation in broader applications.