Problem 8 - College Algebra

Question

For the following exercises, simplify the rational expressions. $$ \frac{m-12}{m^{2}-144} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{m+12} \), with domain exclusions at \( m = 12 \) and \( m = -12 \).

Step 1: Factor the Denominator

The denominator is \( m^2 - 144 \). Recognize this as a difference of squares and factor it as \((m - 12)(m + 12)\). So, the expression now looks like: \[ \frac{m-12}{(m-12)(m+12)} \]

Step 2: Cancel Common Factor

In the expression \( \frac{m-12}{(m-12)(m+12)} \), the numerator \( m-12 \) and one \( m-12 \) factor in the denominator can cancel out. Simplify to get: \[ \frac{1}{m+12} \]

Step 3: Determine Domain Restrictions

Note that while simplifying, we should identify values of \( m \) that make the original denominator zero. From \( m^2 - 144 = 0 \), we find \( m = 12 \) and \( m = -12 \) are not included in the domain. So, the domain is all real numbers except \( m = 12 \) and \( m = -12 \).

Key Concepts

Simplifying Expressions

Simplifying rational expressions means reducing them to their most basic form without changing the expression's value. This process involves looking for common factors that exist in both the numerator and the denominator. Once found, they can cancel each other out. The simplified form of the expression is much easier to work with, which is why this step is so important.

Here’s how simplification works in practice: consider the expression \( \frac{m-12}{(m-12)(m+12)} \). In this case, the term \( m-12 \) is present both in the numerator and once in the denominator. By canceling out these common factors, you simplify the expression to \( \frac{1}{m+12} \).

• Identify common factors.
• Cancel out these common elements.
• Rewrite the expression in its simplest form.

In the end, simplicity is key in mathematics because it helps us see the underlying structure and make further calculations more manageable.

Difference of Squares

The difference of squares is a special type of polynomial that has a very specific pattern: it takes the form \( a^2 - b^2 \). This pattern can be factored into \( (a-b)(a+b) \). Recognizing this pattern allows you to factor polynomials quickly and efficiently.

For the expression \( \frac{m-12}{m^2-144} \), the denominator \( m^2-144 \) fits this difference of squares pattern with \( a = m \) and \( b = 12 \), giving us the factors \( (m-12)(m+12) \).

• Recognize the pattern \( a^2 - b^2 \).
• Factor it into \( (a-b)(a+b) \).
• Use the factored form to simplify rational expressions.

This is a useful technique because it simplifies solving equations and helps determine factors quickly, especially in polynomial equations.

Domain Restrictions

Domain restrictions in rational expressions are about finding values that can make the expression undefined. This usually happens if a value causes the denominator to be zero because division by zero is not allowed in mathematics.

To find these restricted values, set the denominator equal to zero and solve for the variable. For example, in our exercise, the denominator is \( m^2 - 144 \). Setting this to zero gives:\[m^2 - 144 = 0\]Solving for \( m \), we get \( m = 12 \) and \( m = -12 \). These are the values where the original expression is undefined.

• Set the denominator equal to zero.
• Solve for the variable to find restricted values.
• Note these values when giving the domain.

The domain will be all real numbers except \( m = 12 \) and \( m = -12 \), ensuring clarity in calculations and graphs.

Factoring Polynomials

Factoring polynomials is breaking down a polynomial into simpler terms (products of polynomials) that, when multiplied together, give back the original polynomial. This process often involves looking for common factors, grouping terms, or using special patterns such as the difference of squares.

In the problem given, the denominator \( m^2 - 144 \) is factored using the difference of squares method into \( (m-12)(m+12) \).

• Look for common factors.
• Apply special factoring techniques like difference of squares.
• Simplify the original polynomial into a product of factors.

Factoring is vital because it makes further mathematical processes like simplification and solving equations straightforward. It is a foundational skill in algebra that aids in understanding the structure of polynomials.