Problem 59 - Elementary and Intermediate Algebra

Question

Divide, and then simplify, if possible. \(\frac{x^{2}-1}{3 x-3} \div(x+1)\)

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{1}{3}\).

Step 1: Rewriting the Division

To divide two fractions, first rewrite the problem as the multiplication of the first fraction by the reciprocal of the second expression. This means turning the division into a multiplication: \[ \frac{x^{2}-1}{3x-3} \times \frac{1}{x+1}. \]

Step 2: Factoring the Numerator

Factor the numerator of the first fraction, \(x^2-1\). This is a difference of squares, which can be factored into \((x+1)(x-1)\).

Step 3: Factoring the Denominator

Now, factor the denominator of the first fraction, \(3x-3\). You can factor out the greatest common factor, 3, to get \(3(x-1)\).

Step 4: Substituting Factored Forms

Substitute the factored forms into the expression: \[ \frac{(x+1)(x-1)}{3(x-1)} \times \frac{1}{x+1}. \]

Step 5: Canceling Common Factors

Identify and cancel the common factors in the numerator and the denominator. The factor \((x+1)\) cancels out, as does \((x-1)\): \[ \frac{\cancel{(x+1)} \cancel{(x-1)}}{3 \cancel{(x-1)}} \times \frac{1}{\cancel{(x+1)}}. \]

Step 6: Final Simplification

After cancelling, you are left with \(\frac{1}{3}\). Therefore, the original expression simplifies to \(\frac{1}{3}\).

Key Concepts

Difference of Squares

A difference of squares is a special algebraic form, where you have the subtraction of two squared terms. The general formula is \( a^2 - b^2 = (a + b)(a - b) \). Recognizing a difference of squares can help you easily factor the expression. For instance, in the given exercise, the expression \( x^2 - 1 \) is a perfect example of a difference of squares because it can be written as \( x^2 - 1^2 \). Thus, it factors into \( (x+1)(x-1) \).

Whenever you spot an expression that seems like a subtraction of squared terms, try applying this method to simplify your work. Whether you're working on homework or taking a test, mastering this concept can save time and lead to quicker solutions.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler components called factors. These factors, when multiplied together, yield the original polynomial. Let's break down the steps you might use:

• Look for common factors in every term and factor them out. In the exercise, the expression \( 3x - 3 \) was simplified by factoring out a common factor of 3, resulting in \( 3(x-1) \).
• Identify patterns such as the difference of squares, trinomial squares, or any other recognizable patterns that simplify factoring.

Remember, by simplifying polynomials through factoring, you can solve equations more easily, either by setting factors to zero or simplifying the expression for further operations like division or integration.

Simplifying Rational Expressions

Simplifying rational expressions, which are fractions where the numerator and the denominator are polynomials, involves reducing them to their simplest form. Similar to numerical fractions, this often involves finding and cancelling common factors.

In the original exercise, first, the polynomials were factored: the numerator \( x^2 - 1 \) became \( (x+1)(x-1) \) and the denominator \( 3x - 3 \) became \( 3(x-1) \). Once factored, common terms in the numerator and the denominator like \( (x+1) \) and \( (x-1) \) were cancelled out.

• Always ensure that you are cancelling valid factors; you can't cancel terms, only common factors.
• This cancellation results in a simplified fraction, making complex algebraic expressions easier to handle.

Simplifying not only reduces complexity but also helps in comparing rational expressions or evaluating them for specific variable values.