Problem 62 - Algebra A Combined Function

Question

Factor. $$ r^{3}+125 $$

Step-by-Step Solution

Answer

The expression factors to \( (r + 5)(r^2 - 5r + 25) \).

Step 1: Recognize the Sum of Cubes

The expression given is in the form \( r^3 + 125 \). We recognize this as a sum of cubes because it can be written as \( r^3 + 5^3 \). The sum of cubes can be factored using the formula: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \).

Step 2: Apply the Sum of Cubes Formula

Identify \( a = r \) and \( b = 5 \). Substitute these into the sum of cubes formula: \( (r + 5)(r^2 - r \times 5 + 5^2) \).

Step 3: Simplify the Factorization

Simplify the expression inside the second parenthesis from Step 2: \( r^2 - 5r + 25 \). Therefore, the factorization of \( r^3 + 125 \) is \( (r + 5)(r^2 - 5r + 25) \).

Key Concepts

Factoring Polynomials

Factoring polynomials is like finding the building blocks of an algebraic expression. Consider it as the process of breaking down a complex expression into simpler pieces. This makes it easier to solve equations or understand the inherent properties of the expression.
Factoring can involve the following steps:

• Identify the type of polynomial.
• Look for common factors.
• Apply specific formulas based on the polynomial format, like the sum of cubes.

For the sum of cubes, such as with the expression \(r^3 + 5^3\), we use the formula \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) to break it down. Understanding these formulas is key to mastering factoring.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations packed into a compact form. Think of them as the language of algebra that helps in expressing relationships and forming equations.
Every part of an expression has a name:

• **Terms** are the pieces separated by addition or subtraction signs, like \(r^3\) and \(125\) in our example.
• **Coefficients** are the numbers in front of variables; in this case, the coefficient of \(r^3\) is 1.
• **Constants** are standalone numbers like the 125 here.

Working with algebraic expressions involves performing operations and transformations to solve problems or simplify the expression. This includes understanding their structure and applying rules to split or factor them efficiently.

Cubic Equations

Cubic equations are equations of the form \(ax^3 + bx^2 + cx + d = 0\). These can be daunting due to their complex nature, but breaking them down into more manageable components, like factoring, can make solving them easier.
To solve a cubic equation or simplify its expression form:

• Identify if it's a sum or difference of cubes.
• Apply appropriate formulas to transform them.
• Factor step-by-step until you simplify the equation.

In our scenario, \( r^3 + 125 \) is the sum of cubes, where recognizing it allows you to employ the sum of cubes formula. Each step in solving cubic equations involves identifying patterns and being comfortable with algebraic manipulation, gradually making the complex simple.