Problem 28 - Algebra and Trigonometry

Question

Simplify the rational expression. $$ \frac{1-x^{2}}{x^{3}-1} $$

Step-by-Step Solution

Answer

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

Step 1: Identify Factoring Formulas

To simplify the rational expression \( \frac{1-x^2}{x^3-1} \), first identify any factoring formulas that can be used. Recall the difference of squares formula \( a^2 - b^2 = (a-b)(a+b) \) and the difference of cubes formula \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \).

Step 2: Factor the Numerator

Rewrite the numerator \( 1-x^2 \) using the difference of squares formula: \( 1-x^2 = (1-x)(1+x) \).

Step 3: Factor the Denominator

Rewrite the denominator \( x^3 - 1 \) using the difference of cubes formula: \( x^3 - 1 = (x-1)(x^2 + x + 1) \).

Step 4: Simplify the Rational Expression

Substitute the factored forms into the rational expression: \( \frac{(1-x)(1+x)}{(x-1)(x^2 + x + 1)} \). Notice that \( 1-x \) can be rewritten as \( -(x-1) \), so the expression becomes \( \frac{-(x-1)(1+x)}{(x-1)(x^2 + x + 1)} \).

Step 5: Cancel Common Factors

Cancel the common factor \( (x-1) \) from the numerator and the denominator to get \( \frac{-(1+x)}{x^2 + x + 1} \).

Step 6: Simplified Expression

The simplified form of the given rational expression is \( \frac{-(1+x)}{x^2 + x + 1} \).

Key Concepts

Factoring Formulas

Factoring formulas play a crucial role in simplifying algebraic expressions. They help break down complex expressions into more manageable parts. When it comes to rational expressions, factoring helps to identify common factors that can be canceled out, making the expression simpler.

• Difference of Squares: This is a common factoring formula and is expressed as \( a^2 - b^2 = (a-b)(a+b) \). It is used for expressions where the square of one term is subtracted from the square of another.
• Difference of Cubes: Another useful formula is \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \). This applies to expressions where one cubic term is subtracted from another.

Recognizing these patterns in an expression is key to simplifying it effectively.

Difference of Squares

The difference of squares formula is a simple yet powerful tool. It helps us recognize that when you subtract one square term from another, it can be factored into a product of two binomials. In our problem, the numerator \(1-x^2\) becomes easy to factor using this formula.
Consider the following steps to grasp this concept better:

• Identify terms: Notice the expression fits the pattern \(a^2 - b^2\) where \(a = 1\) and \(b = x\).
• Apply formula: Factor it as \((1-x)(1+x)\). This transformation is straightforward and results from the formula \( (a-b)(a+b)\).

Utilizing the difference of squares simplifies the process significantly and is the first major step in tackling many rational expressions.

Difference of Cubes

The difference of cubes formula is a bit more involved than the difference of squares, but it follows a logical pattern. For our denominator \(x^3 - 1\), this formula proves to be essential.
To effectively use the difference of cubes:

• Identify terms: Recognize \(a = x\) and \(b = 1\), fitting the pattern \(a^3 - b^3\).
• Apply formula: Factor it as \((x-1)(x^2 + x + 1)\). This results from \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\).

Mastering the difference of cubes conversion can simplify expressions, making them easier to solve or further manipulate. Understanding and applying this formula is a valuable skill in algebra.

Canceling Common Factors

Canceling common factors is the final and crucial step in simplifying rational expressions. After factoring both numerator and denominator, we look for terms that are common.
In our expression, noticing and canceling the common factor \((x-1)\) simplifies the expression further:

• Recognize common factors in both numerator and denominator.
• Cancel the \((x-1)\) since it appears in both places. This step does not change the overall value of the expression, just its appearance, simplifying it.
• What remains is \(-(1+x)/(x^2 + x + 1)\).

Be cautious to only cancel terms that are entirely multiplied within the expression. This process of canceling common factors often trims complex expressions into more digestible forms.