Problem 57 - A Graphical Approach to College Algebra

Question

Factor each sum or difference of cubes completely. $$(r+6)^{3}-216$$

Step-by-Step Solution

Answer

The factored form is \( r(r^2 + 18r + 108) \).

Step 1: Identify a Sum or Difference of Cubes Formula

Recognize that the expression \((r+6)^3 - 216\) is a difference of cubes. Recall the formula for factoring a difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). The goal is to determine values for \(a\) and \(b\) that match this form.

Step 2: Express the Expression as a Difference of Cubes

Observe that \((r+6)^3\) is \(a^3\) and that \(216 = 6^3\). Thus, let \(a = (r+6)\) and \(b = 6\). Now the expression is \(a^3 - b^3\) with \(a = (r+6)\) and \(b = 6\).

Step 3: Apply the Difference of Cubes Formula

Substitute \(a = (r+6)\) and \(b = 6\) into the difference of cubes formula: \[(r+6)^3 - 6^3 = ((r+6) - 6)((r+6)^2 + (r+6)(6) + 6^2)\].

Step 4: Simplify the First Factor

Simplify \((r+6) - 6\) to \(r\). Therefore, the first factor becomes \(r\).

Step 5: Simplify the Second Factor

Expand the second factor:- \((r+6)^2 = (r+6)(r+6) = r^2 + 12r + 36\),- \((r+6)(6) = 6r + 36\),- \(6^2 = 36\).Combine these into the expression \(r^2 + 12r + 36 + 6r + 36 + 36\) which simplifies to \(r^2 + 18r + 108\).

Step 6: State the Final Factored Form

Combine the simplified factors to write the final expression: \(r((r+6)^2 + 6(r+6) + 36) = (r)(r^2 + 18r + 108)\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a complex expression into products of simpler expressions. Think of it like deconstructing a big LEGO build into smaller pieces. In algebra, polynomials are expressions consisting of variables and coefficients. To factor a polynomial, you're essentially finding other polynomials that multiply to give you the original. This can help simplify expressions or solve equations. One of the keys to factoring is recognizing patterns or identities that fit certain forms, such as the sum or difference of cubes. This lets you apply specific formulas to make factoring more straightforward. For example, in the expression \[(r+6)^{3}-216\]we identified it as a difference of cubes because it fits the form \(a^3 - b^3\). Factoring made it simpler to handle, like taking apart a complex picture for easier manipulation.

Algebraic Identities

Algebraic identities are equations that hold true for all values of the variables involved. They are powerful tools that can simplify calculations and transformations in algebra. One such identity is the **Difference of Cubes**: \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]This identity shows us how to decompose cubic expressions into a product of two factors, a linear one and a quadratic one. It’s useful when you encounter an expression that fits this form, like \[(r+6)^3 - 6^3\]In algebra, recognizing these identities helps break down problems. It’s like using a map to navigate the roads; knowing which route to take simplifies the journey. By applying the identity, we can quickly switch from a complex cube to a manageable product of expressions.

Cubic Expressions

Cubic expressions are polynomials where the highest degree of the variable is 3, such as \[x^3 + ax^2 + bx + c\]These expressions can be tricky, as they involve a third power of a variable. Recognizing when you're dealing with a cubic expression helps in applying the right techniques. When you see something like \[(r+6)^3 - 216\]First, identify if it’s a sum or difference of cubes. Cubic expressions, due to their form, can often be simplified into products through identities like the sum or difference of cubes. It’s similar to finding a shortcut on a long trail; the journey becomes more efficient when you use known paths. In algebra, handling cubic expressions effectively is essential for solving equations and simplifying expressions, making them crucial for tackling higher-level math problems.