Problem 33 - Algebra 1: Concepts and Skills

Question

Simplify the expression. If not possible, write already in simplest form. $$\frac{x^{3}-x}{x^{3}+5 x^{2}-6 x}$$

Step-by-Step Solution

Answer

The simplified expression is \((x+1)/(x+6)\).

Step 1: Factorize the numerator

Check if there is a common term in all the terms in the numerator which can be factored. Here, 'x' is a common term. Factoring it out we get: \(x*(x^{2}-1)\)

Step 2: Factorize the denominator

The denominator is a cubic polynomial. The first strategy is to look if there is a common factor. There isn't. The next step is to see if the polynomial can be factored as a product of linear terms or quadratic terms. For the denominator \(x^3+5x^2-6x\), 'x' is a common factor. Factoring 'x' out we get: \(x*(x^{2}+5x-6)\)

Step 3: Factorize the denominator quadratic term

Now, we focus on the quadratic term \(x^2 + 5x -6\). To factorize this, we can find two numbers that multiply to -6 and add up to 5. The numbers 6 and -1 satisfy these conditions. Therefore, the quadratic expression can be factored as \(x^2 + 5x -6 = (x-1)(x+6)\). Thus, the whole denominator expression becomes \(x*(x-1)*(x+6)\)

Step 4: Factorize the numerator quadratic term

Now, let's turn attention to the quadratic term in the numerator which is \(x^2 -1\). This is a difference of squares which can be factored as \((x-1)(x+1)\). Thus, the whole numerator expression becomes \(x*(x-1)*(x+1)\)

Step 5: Simplify the expression

Now that both the numerator and the denominator have been factored and are in terms of products, we can simplify the expression by cancelling out common factors (if any). Here, \(x\) and \((x-1)\) are common factors in both the numerator and the denominator that can be cancelled out. After cancelling out we are left with \((x+1)/(x+6)\)

Key Concepts

Factoring Polynomials

Polynomials are algebraic expressions that consist of variables raised to whole number powers, often with coefficients. Factoring a polynomial means breaking it down into products of simpler polynomials. This process is fundamental in simplifying algebraic expressions and solving polynomial equations. For example, the expression \(x^3 - x\) can be factored by taking out the greatest common factor (GCF), which is \(x\), resulting in \(x(x^2 - 1)\).

Knowing how to factor polynomials is essential because it helps reduce complex expressions to simpler forms, making them easier to work with. To factor effectively, always check for a common factor first, then look for other factoring techniques such as grouping, using special formulas like the difference of squares, or even polynomial division. Factoring simplifies calculations and is a critical skill in algebra.

Difference of Squares

The difference of squares is a special type of polynomial that can be factored using a simple pattern. It takes the form \(a^2 - b^2\) and can be factored as \((a - b)(a + b)\). This formula is useful and often appears in algebra. In our problem's solution, the term \(x^2 - 1\) is a classic example of a difference of squares since \(x^2 - 1\) can be rewritten as \((x - 1)(x + 1)\).

Recognizing a difference of squares can greatly streamline the factoring process. Whenever you see two squared terms separated by subtraction, check to see if you can apply this formula. Factoring in this way not only makes expressions more manageable but also unveils potential simplifications when dividing expressions or solving equations.

Common Factors

Identifying and factoring out common factors is one of the first steps in simplifying rational expressions. In mathematics, a common factor refers to a term that is present in every part of an expression. By extracting it, you reduce the expression to a simpler form.

In the given exercise, both the numerator \(x^3 - x\) and the denominator \(x^3 + 5x^2 - 6x\) have \(x\) as a common factor. By factoring \(x\) from each term, you make the expression more manageable for further simplification. This step not only simplifies the expression but is also vital for the upcoming operations such as cancelling out terms, especially when simplifying fractions.

Polynomial Division

Polynomial division is a tool for simplifying rational expressions by dividing polynomials. It involves techniques like long division or synthetic division, but what makes the process easier is factoring. When expressions are factorable, as in this example, division becomes a simple matter of cancelling out common factors.

In our solved problem, after factoring both the numerator and denominator, polynomial division essentially consisted of cancelling out the common terms \(x\) and \(x-1\). The result was a much simpler expression: \((x+1)/(x+6)\). This illustrates how factoring is not just an independent task but part of a larger strategy to simplify or solve polynomial expressions. When you grasp polynomial division alongside factoring, complex problems become significantly more approachable.