Problem 23 - Algebra 2 and Trigonometry

Question

Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. \(\frac{4 a^{2}-16}{4 a+8}\)

Step-by-Step Solution

Answer

The simplest form is \(\frac{(a - 2)(2a + 4)}{2(a + 2)}\), and it is undefined for \(a = -2\).

Step 1: Factor Numerator

First, we factor the numerator of the rational expression \(4a^2 - 16\). This can be factored as a difference of squares. Notice that \(4a^2\) can be written as \((2a)^2\) and 16 as \(4^2\). Therefore, \(4a^2 - 16 = (2a - 4)(2a + 4)\).

Step 2: Factor Denominator

Next, let's factor the denominator of the rational expression \(4a + 8\). We can factor out the greatest common factor, which is 4. Therefore, \(4a + 8 = 4(a + 2)\).

Step 3: Simplify the Expression

The expression now becomes \(\frac{(2a - 4)(2a + 4)}{4(a + 2)}\). We can further factor \((2a - 4)\) as \(2(a - 2)\). Thus, the expression is \(\frac{2(a - 2)(2a + 4)}{4(a + 2)}\). Simplifying further, we can cancel out a factor of 2 from the numerator and denominator, so it becomes \(\frac{(a - 2)(2a + 4)}{2(a + 2)}\).

Step 4: Determine Undefined Values

The denominator cannot be zero, so we need \(2(a + 2) eq 0\). Solving \(a + 2 = 0\), we find that \(a eq -2\). This is the value for which the original expression is undefined.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in simplifying rational expressions. A polynomial is an expression made up of variables and coefficients, involving operations such as addition, subtraction, and multiplication. Factoring is the process of breaking it down into simpler terms called factors.

For example, consider the numerator in the expression \(4a^2 - 16\). To factor it, recognize it as a difference of squares because it can be written as \((2a)^2 - 4^2\). Using the difference of squares formula \(a^2 - b^2 = (a - b)(a + b)\), we factor it into \((2a - 4)(2a + 4)\).

In contrast, for the denominator \(4a + 8\), factor out the greatest common factor, which in this case is 4, resulting in \(4(a + 2)\).

When you factor both the numerator and the denominator, it becomes easier to see if any common factors can be canceled, which is a critical step in simplifying rational expressions.

Undefined Values in Rational Expressions

Rational expressions can have undefined values. This happens when the denominator is zero, as division by zero is undefined in mathematics.

In our example, the expression \(\frac{(a - 2)(2a + 4)}{2(a + 2)}\) is undefined for values that make the denominator equal to zero. Here, the denominator is \(2(a + 2)\).

• Set \(2(a + 2) = 0\) to find the undefined values.
• Solving, you get \(a + 2 = 0\), which simplifies to \(a = -2\).

Therefore, the rational expression is undefined when \(a = -2\). Identifying undefined values is essential because they define the restrictions on the variable to ensure the expression is valid.

Difference of Squares

The difference of squares is a formula that provides a way to factor certain polynomials. It's a specific kind of polynomial factorization that applies when two terms are perfect squares separated by a subtraction sign.

The general formula for difference of squares is \[ a^2 - b^2 = (a - b)(a + b) \]This helps in quickly breaking down expressions like \(4a^2 - 16\).

• Identify each term as a square: \(4a^2 = (2a)^2\) and \(16 = 4^2\).
• Apply the formula: \((2a)^2 - 4^2 = (2a - 4)(2a + 4)\).

This technique simplifies factoring dramatically, making it easier to handle expressions and contribute towards simplifying rational expressions. Remembering and applying this formula can save you time and reduce errors when solving problems involving polynomials.