Problem 16 - 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. \(\frac{8 a+16}{12 a}\)

Step-by-Step Solution

Answer

The simplest form is \(\frac{2(a + 2)}{3a}\); \(a = 0\) makes it undefined.

Step 1: Factorize the Numerator

The numerator of the rational expression is \(8a + 16\). Notice that both terms have a common factor of 8. Factor out the 8 to get: \[ 8(a + 2). \]

Step 2: Write the Expression with Factored Numerator

Substitute the factored numerator from Step 1 back into the rational expression. It becomes: \[ \frac{8(a + 2)}{12a}. \]

Step 3: Simplify the Expression

Identify any common factors in the numerator and the denominator. The common factor is 4. Divide both the numerator and the denominator by 4: \[ \frac{2(a + 2)}{3a}. \] This is the simplest form of the expression.

Step 4: Determine Undefined Values

A fraction is undefined when its denominator is zero. Set the denominator equal to zero to find these values: \(3a = 0\). Solve for \(a\): \[ a = 0. \] So, the expression is undefined for \(a = 0\).

Key Concepts

Factoring Polynomials

Factoring polynomials is similar to breaking down numbers into their prime components. When you factor a polynomial, you are finding the roots or the simplest parts that multiply together to create the original expression. Let's consider the polynomial in the given exercise: \(8a + 16\).
Have a look at each term in the polynomial to identify a common factor. Here, both terms, 8a, and 16, are divisible by 8.

• Step 1: Divide all terms by the greatest common factor (GCF). Here, it's 8. Divide 8a by 8 to get a, and 16 by 8 to get 2.
• Step 2: Rewrite the polynomial as a product of the GCF and the simplified polynomial. So, \(8a + 16 = 8(a + 2)\).

Factoring helps in simplifying rational expressions by reducing them to easier components. This makes further calculations more straightforward and highlights any cancellations between the numerator and denominator.

Undefined Expressions

Undefined expressions occur primarily in fractions when the denominator is zero. In mathematics, dividing by zero is undefined because it doesn't fit into any logical or consistent system of numbers.
Consider the fraction \(\frac{2(a + 2)}{3a}\). To find when this expression is undefined, check when the denominator, 3a, equals zero.

• Step 1: Set \(3a = 0\).
• Step 2: Solve for \(a\). Divide both sides by 3 to find \(a = 0\).

So, the expression \(\frac{2(a + 2)}{3a}\) is undefined when \(a = 0\). Recognizing when an expression is undefined helps avoid errors in calculations and ensures logical mathematical practices.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form where no common factors remain in the numerator and denominator. This makes working with fractions easier and helps in further operations like addition, subtraction, or comparison.For the fraction \(\frac{8(a + 2)}{12a}\), start simplifying by identifying any common factors between the numerator and denominator.

• Step 1: The common factor here is 4. Divide both the numerator and the denominator by 4. Doing this to \(8(a+2)\) becomes \(2(a+2)\) and \(12a\) becomes \(3a\).
• Step 2: Write the reduced expression. You get \(\frac{2(a+2)}{3a}\), which is the simplest form of the given rational expression.

Simplifying fractions is a crucial skill in algebra, making expressions manageable and accurate for solving and interpreting mathematical problems.