Problem 55 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 4 m^{5}+500 m^{2} $$

Step-by-Step Solution

Answer

The factored form is \(4m^2(m + 5)(m^2 - 5m + 25)\).

Step 1: Identify the Greatest Common Factor (GCF)

Firstly, we need to determine if there is a greatest common factor (GCF) that can be factored out from the expression. The expression is \(4m^5 + 500m^2\). Both terms have a common factor of \(m^2\), and the coefficients 4 and 500 have a common factor of 4.

Step 2: Factor out the GCF

The GCF of the expression is \(4m^2\). We factor \(4m^2\) out of each term in the expression:\[4m^5 + 500m^2 = 4m^2(m^3 + 125)\]

Step 3: Identify Remaining Expression for Further Factoring

After factoring out \(4m^2\), the remaining expression inside the parentheses is \(m^3 + 125\). We need to determine if this can be factored further.

Step 4: Recognize the Form of a Sum of Cubes

The expression \(m^3 + 125\) is a sum of cubes. It can be rewritten as \(m^3 + 5^3\). A sum of cubes can be factored using the formula:\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]with \(a = m\) and \(b = 5\).

Step 5: Apply the Sum of Cubes Formula

Apply the formula to factor \(m^3 + 5^3\):\[m^3 + 5^3 = (m + 5)(m^2 - 5m + 25)\]Thus, after applying the formula, the complete factorization is:\[4m^2(m + 5)(m^2 - 5m + 25)\]

Step 6: Verify Completeness of the Factorization

To ensure completeness, verify that each expression is factored as much as possible. The expressions \((m^2 - 5m + 25)\) and \((m + 5)\) cannot be factored further using integers. Hence, the factorization is complete.

Key Concepts

Greatest Common Factor

The greatest common factor, or GCF, is the largest factor that divides two or more numbers or terms. It's a building block for simplifying algebraic expressions.
Finding the GCF involves identifying the common factor that appears in all the terms. Here’s how you do it:

• Look at each term of the expression to find any common variables. For example, in the expression given, both terms, \(4m^5\) and \(500m^2\), include the variable \(m^2\).
• Find the greatest common divisor (GCD) of the numerical coefficients. In \(4\) and \(500\), the largest number that divides both is \(4\).

The GCF for \(4m^5 + 500m^2\) is \(4m^2\). By factoring out the GCF, you simplify the original expression. It’s like peeling back a layer to see what’s underneath.
After factoring out \(4m^2\), the expression inside the parentheses is much simpler, allowing you to explore additional factoring possibilities.

Sum of Cubes

A sum of cubes is a specific type of algebraic expression that takes the form \(a^3 + b^3\). Recognizing and factoring these can simplify expressions and reveal their structure.
To factor a sum of cubes, use the formula:

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

This formula is vital for breaking down complex expressions. Let's examine how it applies to \(m^3 + 125\):
First, notice that \(125 = 5^3\). This means you can rewrite \(m^3 + 125\) as a sum of cubes: \(m^3 + 5^3\). Now apply the formula:

• Set \(a = m\) and \(b = 5\).
• Substitute these into the formula to get: \((m + 5)(m^2 - 5m + 25)\).

Recognizing a sum of cubes allows you to go a step further in factoring, often resulting in a more understandable or useful expression.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. Understanding how to manipulate these expressions is fundamental in algebra.
Each term in an algebraic expression consists of numbers (coefficients) and variables raised to powers (exponents). Examining what’s in each term and how terms relate helps you decide the next steps in factoring or simplifying.
Faced with expressions like \(4m^5 + 500m^2\), you can:

• Identify the GCF and factor it out, simplifying the expression.
• Recognize common patterns, such as sums of cubes, which can be factored using well-recognized formulas.

By factoring expressions, you can transform complex problems into simpler, more manageable ones. Practice recognizing different forms, like sums of cubes or perfect squares, to master factoring.