Problem 82 - A Graphical Approach to College Algebra

Question

Factor by any method. $$216 p^{3}+125 q^{3}$$

Step-by-Step Solution

Answer

The factored form is \((6p+5q)(36p^2 - 30pq + 25q^2)\).

Step 1: Identify the Expression as a Sum of Cubes

The expression given is \(216p^3 + 125q^3\), which matches the form \(a^3 + b^3\). Here, \(a = 6p\) and \(b = 5q\), since \((6p)^3 = 216p^3\) and \((5q)^3 = 125q^3\).

Step 2: Recall the Formula for Sum of Cubes

The formula for factoring the sum of cubes \(a^3 + b^3\) is \((a + b)(a^2 - ab + b^2)\). We'll use it to factor the expression.

Step 3: Substitute \(a\) and \(b\) into the Formula

Substitute \(a = 6p\) and \(b = 5q\) into the sum of cubes factorization formula: \( (6p + 5q) ((6p)^2 - (6p)(5q) + (5q)^2) \).

Step 4: Simplify Each Term

First, simplify the second factor:1. \((6p)^2 = 36p^2\)2. \((6p)(5q) = 30pq\)3. \((5q)^2 = 25q^2\).Thus, the expression becomes: \((6p + 5q)(36p^2 - 30pq + 25q^2)\).

Step 5: Present the Factored Form

The expression \(216p^3 + 125q^3\) has been successfully factored as \((6p+5q)(36p^2 - 30pq + 25q^2)\). This is its factored form using the sum of cubes.

Key Concepts

Sum of Cubes

The sum of cubes is an important algebraic identity used for factoring expressions of the form \(a^3 + b^3\). This type of expression is called a "sum of cubes" because it comprises terms where each component is a number raised to the third power, or cubed.

For example, in the expression \(216p^3 + 125q^3\), both parts represent cubes: \(216p^3\) can be rewritten as \((6p)^3\) and \(125q^3\) as \((5q)^3\). Thus, this expression is a clear candidate for the sum of cubes factorization method.

The formula for factoring a sum of cubes is:

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

In this equation, \(a + b\) is the sum of the cube roots, and \(a^2 - ab + b^2\) is a quadratic trinomial that completes the factorization. By identifying \(a\) as \(6p\) and \(b\) as \(5q\), we can effectively simplify and factor expressions like \(216p^3 + 125q^3\). This formula and approach allow for handling complex polynomial expressions with greater ease.

Polynomial Expressions

Polynomial expressions are combinations of variables and coefficients that involve operations like addition, subtraction, multiplication, and non-negative integer exponents of variables. They form the basis of much of algebra.

In the given exercise, \(216p^3 + 125q^3\) is a polynomial expression of degree 3, meaning it contains terms where the highest power of the variables is three. In this context, each term \(216p^3\) and \(125q^3\) is itself a cube, making the expression amenable to the sum of cubes method for factorization.

To successfully handle polynomial expressions, it's essential to:

• Identify common patterns or formulas that can be applied, such as the sum or difference of cubes.
• Factor by breaking down terms into their component parts (as done by identifying \(6p\) and \(5q\) as the roots of the cubes).
• Simplify the expression after applying the appropriate algebraic identity, which results in a more basic or reduced form of the expression.

Understanding polynomial expressions and their properties is crucial for simplifying complex algebraic problems and forms the foundation for advanced problem-solving skills.

Algebraic Identities

Algebraic identities are equations that hold for any value of the variables involved. They can simplify the process of factoring and solving polynomial equations significantly.

Some of the most commonly used algebraic identities involve special products, such as:

• Sum of squares: \(a^2 + b^2\)
• Difference of squares: \(a^2 - b^2 = (a - b)(a + b)\)
• Sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
• Difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)

These identities provide structured methods for handling various polynomial expressions. For the sum of cubes, like in our exercise with \(216p^3 + 125q^3\), the identity guides us directly to the factored form without having to perform extensive trial and error.

Using algebraic identities promotes efficiency and clarity in solving equations. It allows mathematicians to predictably simplify intricate polynomials into cleaner, more manageable forms, which are often easier to work with for further algebraic manipulation or solving.