Question

Simplify the complex fraction. \(\frac{\frac{1}{3 x^2-3}}{\frac{5}{x+1}-\frac{x+4}{x^2-3 x-4}}\)

Short answer

\[\frac{x+4}{3*(x-1)*(-x^2 + 10x + 16)}\]

Step-by-step solution

Simplify the denominator

The denominator of the major fraction contains two terms that we subtract. We have \[\frac{5}{x+1}-\frac{x+4}{x^2-3x-4}\]. In order to perform the subtraction, we need to have a common denominator. We can see that \(x^2-3x-4\) can be factorized into \((x-1)(x+4)\). Now we can rewrite the denominator as \[\frac{5(x+4) - (x+4)(x-1)}{(x+1)(x+4)} = \frac{5x + 20 - x^2 + 5x - 4}{(x+1)(x+4)} = \frac{-x^2 + 10x + 16}{(x+1)(x+4)}\]

Divide by Reciprocal

Dividing any number by a fraction is equivalent to multiplying that number with the reciprocal of the fraction. Consequently, we can rewrite our given complex fraction by flipping the denominator fraction and changing the division to a multiplication: \[\frac{\frac{1}{3x^2 - 3}}{\frac{-x^2 + 10x + 16}{(x+1)(x+4)}} = \frac{1}{3x^2 - 3} * \frac{(x+1)(x+4)}{-x^2 + 10x + 16}\].

Simplification

Now, we can simplify the expression by reducing it to lowest terms: \[\frac{1*(x+1)(x+4)}{(3x^2 - 3)*(-x^2 + 10x + 16)}\]. Factoring out a 3 from \(3x^2 - 3\), this yields \[\frac{(x+1)(x+4)}{3*(x^2 - 1)*(-x^2 + 10x + 16)}\]. We notice that \(x^2 - 1\) can be factored into \((x-1)(x+1)\). So, the expression becomes \[\frac{(x+1)(x+4)}{3*(x-1)(x+1)*(-x^2 + 10x + 16)}\]. The term \((x+1)\) in the numerator and denominator cancel each other out, giving us the final answer of \[\frac{x+4}{3*(x-1)*(-x^2 + 10x + 16)}\] as the simplified form of the original complex fraction.

Key concepts

Simplifying Fractions

Simplifying fractions may seem simple on the surface, but when working with complex fractions, things can get tricky. A complex fraction has a fraction in the numerator, the denominator, or both. To simplify it, you start by managing the numerator and denominator separately.
First, you find a common denominator if necessary, to combine fractions within these two sections. For example, if your problem looks like \(\frac{\frac{a}{b}}{\frac{c}{d}}\), simplifying would require you to manage the fractions on the top and bottom first. After this, the operation becomes easier to handle!
Once you have a single fraction in the numerator and the denominator, you convert the division by a fraction into multiplication by its reciprocal. Then, simply multiply across to complete the simplification process.
Keeping these steps in mind makes simplifying complex fractions easily manageable!

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They can be as simple as \(3x + 1\) or much more complex, such as \(\frac{\frac{5x^2}{x + 4}}{x^2 - x - 6}\).
When it comes to complex fractions involving algebraic expressions, knowing how to handle these can help in simplifying them.

• Variables like \(x\) can represent numbers, and operations such as addition or subtraction are used between them.
• In some cases, like complex fractions, they can include more complicated operations, requiring strategic manipulation.

Understanding how these expressions work allows you to perform operations like finding common denominators or canceling terms efficiently.
This understanding aids in simplifying the overall fraction and more importantly, in reaching an accurate solution.

Factoring Polynomials

Factoring polynomials is a key step in simplifying complex fractions and working with algebraic expressions. A polynomial is an expression consisting of variables that are raised to various powers and often combined with coefficients. For example, \(x^2 - 3x - 4\) is a polynomial.
To factor a polynomial means to express it as a product of its factors. For instance, \(x^2 - 3x - 4\) can be factorized into \((x - 1)(x + 4)\).

• This is useful when simplifying complex fractions because common factors can often be canceled out from the numerator and the denominator.
• Finding these factors requires recognizing patterns and possibly using the quadratic formula or other methods.

By breaking down expressions into their basic parts, you can simplify many algebraic operations, leading to a cleaner solution in complex fractional problems. This skill turns potentially daunting equations into more straightforward calculations.