Problem 83 - Algebra A Combined Function

Question

Factor each numerator and denominator. Then simplify if possible. $$ \frac{-4+2 \sqrt{3}}{6} $$

Step-by-Step Solution

Answer

The simplified fraction is \(\frac{-2+\sqrt{3}}{3}\).

Step 1: Identify the Numerator Components

The given expression is \(-4+2\sqrt{3}\). Here, we see it consists of two terms: \(-4\) and \(2\sqrt{3}\). Our goal is to factor this if possible.

Step 2: Factor Out the Greatest Common Factor (GCF)

Look at the terms in the numerator \(-4\) and \(2\sqrt{3}\). Notice that both terms have a common factor of 2. Factoring out the 2, we write: \(-4+2\sqrt{3} = 2(-2+\sqrt{3})\).

Step 3: Write the Entire Fraction

Replace the numerator in the fraction with the factored form: \(\frac{2(-2+\sqrt{3})}{6}\).

Step 4: Simplify the Fraction

Notice that both 2 (in the numerator) and 6 (in the denominator) can be divided by their greatest common factor, which is 2. Simplifying this gives us: \(\frac{1}{3}(-2+\sqrt{3}) = \frac{-2+\sqrt{3}}{3}\).

Step 5: Final Answer

This is the simplified form of the original fraction. Since no further simplification is possible, the final answer is \(\frac{-2+\sqrt{3}}{3}\).

Key Concepts

Factoring Expressions

Factoring expressions is like rewriting or breaking down an expression into products of simpler terms. It's a useful skill in algebra that greatly helps in simplifying more complex expressions and solving equations.
When we talk about factoring, think of it as pulling out a common part or number that each term can share. If you have some terms added or subtracted together, and they share a similar characteristic – such as a number or a variable – we can "factor them out."
In our original exercise, the expression

• "-4" and "2√3" are the terms in the numerator.
• Both terms share a common factor, 2. When factoring, we aim to write the numerator as a product of this common factor and another expression.

By doing so, \[-4 + 2\sqrt{3} = 2(-2 + \sqrt{3})\]This step makes it easier to simplify further, especially when reducing fractions.

Greatest Common Factor

The greatest common factor (GCF) is the largest number that divides two or more numbers without leaving a remainder. Finding the GCF is essential when factoring expressions or simplifying fractions.
To find the GCF of

• "-4" and "2√3", we first look at their coefficients. The common factor here is 2 because it divides both 4 and 2.

This factorization is crucial because it helps us rewrite the fraction in a simpler form.
When we factored out 2 from the numerator in our problem, \[2(-2 + \sqrt{3})\]we could then divide both the numerator and the denominator by this GCF. Doing so brings:\[\frac{2(-2 + \sqrt{3})}{6} \rightarrow \frac{-2 + \sqrt{3}}{3}\]Thus, extracting the GCF makes our algebraic expression cleaner and more manageable.

Simplifying Fractions

Simplifying fractions involves making the numerator and denominator as small as possible, while still having the same value. This process often uses the GCF to achieve the simplest form.
Here’s how we simplify:

• After factoring the numerator, we have \(\frac{2(-2+\sqrt{3})}{6}\).
• The fraction's numerator and denominator share a common factor of 2, which we can cancel out.

After this cancellation:\[\frac{1}{3}(-2 + \sqrt{3})\]The fraction is now simpler: \[\frac{-2 + \sqrt{3}}{3}\]Finding and using common factors during simplification makes the process easier and the expression more straightforward to understand."