Problem 43 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The polynomial \(125r^3 + 1728s^3\) factors to \((5r + 12s)(25r^2 - 60rs + 144s^2)\).

Step 1: Identify the polynomial form

The polynomial given is in the form of a sum of cubes. It can be written as \( x^3 + y^3 \) where \( x = 5r \) and \( y = 12s \), since \((5r)^3 = 125r^3\) and \((12s)^3 = 1728s^3\).

Step 2: Apply the sum of cubes formula

The sum of cubes formula is: \( x^3 + y^3 = (x + y)(x^2 - xy + y^2) \).Substitute \( x = 5r \) and \( y = 12s \) into the formula to get: \((5r + 12s)((5r)^2 - (5r)(12s) + (12s)^2) \).

Step 3: Simplify each term in the expression

Calculate each term in the expression:- \((5r)^2 = 25r^2\).- \((5r)(12s) = 60rs\).- \((12s)^2 = 144s^2\). So the expression becomes \((5r + 12s)(25r^2 - 60rs + 144s^2)\).

Step 4: Write the factored form

Combine the term results back into the expression: The factored form of the original polynomial is:\((5r + 12s)(25r^2 - 60rs + 144s^2)\).

Key Concepts

Sum of Cubes

The sum of cubes is a specific form of polynomial expression characterized by the sum of two cubed terms. The general form is given by: \( a^3 + b^3 \). In the provided problem, the expression \( 125r^3 + 1728s^3 \) is identified as a sum of cubes. This is because we can write it as \( (5r)^3 + (12s)^3 \), which aligns with our general formula.
This particular polynomial format can be simplified through a specific factoring formula: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). This formula helps us express the sum of cubes in a factored, more manageable form.
The formula works by creating two groups from the original terms. It's important to remember this structure, since not all polynomials can be factored as a sum of cubes. To apply this, substitute the respective terms into the formula, ensuring that each component matches the original polynomial correctly.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They are the building blocks in higher mathematics, allowing us to generalize rules and solve for unknowns. In this problem, the expressions used are \( 5r \) and \( 12s \), which are variables that stand for quantities.
An algebraic expression could be as simple as \( x + 3 \), or more complex like \( 125r^3 + 1728s^3 \). Each algebraic term within an expression consists of coefficients (numerical part) and variables (letters representing numbers). In this problem, \( 5r \) and \( 12s \) show multiplication as their core operation.
Understanding how to manipulate these expressions through operations like factoring is crucial. The key is to recognize patterns, such as the sum of cubes, and apply appropriate formulas to simplify and solve. Successfully managing algebraic expressions unlocks the potential of solving broader mathematical challenges.

Polynomial Factorization

Polynomial factorization involves breaking down a complex polynomial into simpler "factor" expressions that, when multiplied together, give the original polynomial.
To factor the polynomial \( 125r^3 + 1728s^3 \) using polynomial factorization, we firstly identify the type of polynomial. Here, recognizing it as a sum of cubes is crucial. This leads us to apply a specific formula.
The formula \( x^3 + y^3 = (x + y)(x^2 - xy + y^2) \) helps us break the polynomial into two simpler factors. This technique is hugely beneficial as it often transforms a complicated expression into manageable parts, allowing further mathematical operations or solutions. Factorization requires practice in recognizing polynomial forms and applying corresponding formulas efficiently. It is a foundational skill in algebra, helping tackle problems ranging from simple equations to complex real-world applications.