Problem 27 - Algebra 2

Question

Multiply or divide. State any restrictions on the variable. $$ \frac{a+3}{a^{2}+a-12} \div \frac{a^{2}-9}{a^{2}+7 a+12} $$

Step-by-Step Solution

Answer

The result of the division is \(\frac{a+3}{a-3}\). The only restriction on the variable 'a’ is that 'a' cannot be 3.

Step 1: Simplify Algebraic Fractions

The division of two fractions is equivalent to the multiplication of the first fraction by the reciprocal of the second fraction. Therefore, transform the division into multiplication:\[\frac{a+3}{a^{2}+a-12} \times \frac{a^{2}+7a+12}{a^{2}-9}\]Then, simplify the equation. The quadratic expressions in the denominators can be factored: \[\frac{a+3}{(a-3)(a+4)} \times \frac{(a+4)(a+3)}{(a-3)(a+3)}\]

Step 2: Simplify Multiplication

The same factors in the numerator and denominator cancel out. This is a key principle when simplifying algebraic fractions.\[\frac{a+3}{a-3} \]So, this is the simplified result of the multiplication.

Step 3: Find Restrictions on Variable 'a'

'a' cannot take values that make the denominator equal to zero. Therefore, solve the following equation for 'a':\[a-3 = 0 \]So, 'a' should not be 3. This is the only restriction on 'a' in this case because once the remaining terms are cancelled, there are no other terms in the denominator that could cause the denominator to become zero.

Key Concepts

Simplifying Algebraic Fractions

Simplifying algebraic fractions is akin to simplifying numerical fractions; it's all about reducing the expression to its simplest form. Understanding this concept allows for more manageable calculations and solutions. In the given exercise, we began with the expression \( \frac{a+3}{a^2+a-12} \div \frac{a^2-9}{a^2+7a+12} \).

The division of one fraction by another is equivalent to multiplication by the reciprocal of the second fraction. So, the expression was transformed into a multiplication problem, simplifying as:

• First, flip the second fraction to its reciprocal.
• Then multiply the two resulting fractions.

This step lays the foundation for simplification by allowing one to focus on factorizations and cancellations.

Factoring Quadratic Expressions

Factoring quadratic expressions is a crucial step in working with algebraic fractions. This process involves breaking down a quadratic equation into the product of its linear factors. This is how we get a clearer picture of what's going on inside the expressions.

Consider the quadratic terms from our exercise:

• \( a^2 + a - 12 \) factors into \((a-3)(a+4)\).
• \( a^2 + 7a + 12 \) factors into \((a+4)(a+3)\).
• \( a^2 - 9 \) is a difference of squares, factoring into \((a-3)(a+3)\).

Each factorization step breaks down complex expressions into simpler, easier-to-read components. This gives us the opportunity to cancel out any common factors across numerators and denominators in the algebraic fraction. As illustrated, by identifying and removing common terms such as \( (a-3) \) and \( (a+3) \), we are left with a much more simplified expression.

Restrictions on Variables

When working with algebraic fractions, it's vital to identify and state any restrictions on the variables involved. This ensures the mathematical statements remain valid, as division by zero is undefined.

In fractions, we need to watch out for denominators becoming zero, as these scenarios create undefined cases in mathematics. From the simplified expression \( \frac{a+3}{a-3} \), it's evident that we run into trouble when \( a = 3 \). This would make the denominator zero, causing the fraction to be undefined.

• Thus, a key restriction here is \( a eq 3 \).

Before cancelling terms in a problem, such restrictions should be noted from the original expression to avoid unnecessary errors. By understanding these restrictions, we maintain the integrity of our solutions throughout the problem-solving process.