Question

Reduce to lowest terms: \(\frac{3 x-12}{x^{2}-16}\)

Short answer

\frac{3}{x + 4}

Step-by-step solution

- Factorize the numerator

The numerator is \(3x - 12\). Factor out the greatest common factor (GCF), which is 3. \[3x - 12 = 3(x - 4)\]

- Factorize the denominator

The denominator is \(x^2 - 16\). Recognize this as a difference of squares and factor accordingly. \[x^2 - 16 = (x - 4)(x + 4)\]

- Substitute the factored forms

Replace the numerator and denominator with their factored forms in the original fraction.\[\frac{3(x - 4)}{(x - 4)(x + 4)}\]

- Cancel common factors

Identify and cancel the common factor \(x - 4\) from both the numerator and the denominator.\[\frac{3 ot{x - 4}}{ot{x - 4}(x + 4)} = \frac{3}{x + 4}\]

Key concepts

Factoring Polynomials

When it comes to algebra, one important skill is factoring polynomials. This often simplifies expressions and is the key to solving equations. Let's start by understanding what a polynomial is. A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients, for example, \(3x - 12\).
To factor a polynomial, we look for the greatest common factor (GCF). The GCF is the highest number or term that divides each term of the polynomial without leaving a remainder. In our example, the GCF of \(3x - 12\) is 3. So, the expression can be factored as follows:
\[3x - 12 = 3(x - 4)\]
This makes the polynomial easier to manage. Always look for the GCF first, as it simplifies the factoring process.

Simplifying Fractions

Simplifying fractions means reducing a fraction to its lowest terms. This is done by canceling out the common factors from the numerator and the denominator.
For example, consider the fraction \(\frac{3(x - 4)}{(x - 4)(x + 4)}\). To simplify, we first factorize both the numerator and the denominator, as seen in the original solution. Remember, our numerator is \(3(x - 4)\) and the denominator is \((x - 4)(x + 4)\).
Next, we identify and cancel the common factor in both the numerator and the denominator, which is \(x - 4\). An important rule is that we can cancel out a term only if it appears in both the numerator and the denominator. After canceling \(x - 4\), we get this simplified fraction:
\[\frac{3}{x + 4}\]
Always ensure you've simplified the fraction as much as possible. Simplifying fractions helps in easier calculations and understanding in further algebraic manipulations.

Difference of Squares

A difference of squares is a specific type of polynomial that takes the form \(a^2 - b^2\). This can always be factored into two binomials:\[a^2 - b^2 = (a - b)(a + b)\]
This is useful in many algebraic problems, including simplifying fractions. For instance, in our given problem, the denominator is \(x^2 - 16\).
First, recognize that \(x^2 - 16\) fits the difference of squares pattern, where \(a = x\) and \(b = 4\). Hence:\[x^2 - 16 = (x - 4)(x + 4)\]
Understanding this pattern allows you to quickly factorize such expressions, making simplification and further calculations more manageable. It’s a powerful tool in your algebra toolkit and frequently appears in polynomial problems.