Question

Simplify the expression, if possible. $$ \frac{3 x^3-3 x^2+7 x-7}{27 x^4-147} $$

Short answer

The simplified expression is \(\frac{3(x-1)(x^2+1)}{(3x^2 + 7)(3x^2 - 7)}\)

Step-by-step solution

Factorize Numerator

Notice that each term in the numerator has common factors. Specifically, each term has a factor of 3 and \(x-1\). Factoring, the numerator becomes: \(3(x-1)(x^2+1)\).

Factorize Denominator

The denominator might not look factorizable at first glance. However, close examination reveals that it can be written as a difference of two squares. This allows for factorization as follows: \(27x^4-147 = 3^3x^4 - 7^2 = (3x^2)^2 - 7^2 = (3x^2 + 7)(3x^2 - 7)\).

Simplify the Fraction

With the numerator and denominator now in factorized form, it is easier to simplify the fraction. In this case, there are no common factors between the numerator and denominator. Thus, the fraction is already in its simplest form.

Key concepts

Factorization

Factorization is a key technique in algebra that involves breaking down a complex expression into a product of simpler factors. It's very similar to finding the 'ingredients' that make up a particular mathematical 'recipe'. This process can greatly simplify algebraic expressions and solve equations.

Let's consider our numerator, where we employ factorization to reveal the not-so-obvious factors hiding within the expression. The terms within the numerator share a common factor of 3 and also the binomial factor \(x-1\). Recognizing these shared factors allows us to re-write the numerator as \(3(x-1)(x^2+1)\).

Factorization of polynomials may involve different techniques such as looking for common factors, employing the distributive property (or reversing it), or using known patterns like the difference of squares, which will be discussed in another section.

Difference of Two Squares

The difference of two squares is a powerful pattern in algebra that comes from the expansion \(a^2 - b^2 = (a+b)(a-b)\). It allows us to factor expressions in the form of one square subtracted from another square. This technique is particularly useful when you have an expression that doesn't seem factorable at a first glance.

Applying this to our denominator, we observe \(27x^4-147\) which can be rewritten as \(3^3x^4 - 7^2\). Now, as both \(3^3x^4\) and \(7^2\) are perfect squares, the expression fits the pattern of a difference of two squares. Consequently, it factors into \(3x^2 + 7)(3x^2 - 7)\), simplifying the original complex expression.

Algebraic Fractions

Algebraic fractions, also known as rational expressions, are fractions that contain polynomials in their numerator and/or denominator. Simplifying algebraic fractions requires one to factorize the polynomials and reduce any common factors between the numerator and the denominator.

Our original expression \(\frac{3 x^3-3 x^2+7 x-7}{27 x^4-147}\) is an algebraic fraction we aim to simplify. After factorizing both the numerator and denominator separately, as in the steps provided, we get a much clearer picture of the fraction's simplified form. In this case, no further simplification through cancellation is possible, and we've reached the simplest form of the algebraic fraction.

Common Factors

Identifying common factors plays a crucial role in simplifying algebraic expressions. Common factors are numbers or variables that are present in each term of an expression. They can be factored out to simplify the expression as a whole.

In the step-by-step solution provided, we started by factoring out a 3 from the numerator, as it was a common factor in each term. We also looked for any common binomial factors between the numerator and the denominator. Unfortunately, in this instance, there were no binomial common factors left to be canceled after factorization.

It's important to remember that when no common factors are found between a numerator and a denominator, the algebraic fraction is simplified as much as possible, just as we concluded in the given exercise.