Problem 31 - Algebra and Trigonometry

Question

\(21-34\) . Perform the multiplication or division and simplify. $$ \frac{\frac{x^{3}}{x+1}}{\frac{x}{x^{2}+2 x+1}} $$

Step-by-Step Solution

Answer

\(x^3 + x^2\)

Step 1: Identify the Inner Fractions

The complex fraction \( \frac{\frac{x^{3}}{x+1}}{\frac{x}{x^{2}+2x+1}} \) consists of two inner fractions, which you need to simplify separately. The numerator of the complex fraction is \( \frac{x^3}{x+1} \), and the denominator is \( \frac{x}{x^2+2x+1} \).

Step 2: Simplify the Denominator using Factoring

Notice that the denominator of the denominator fraction, \(x^2 + 2x + 1\), can be factored into \((x + 1)^2\). Therefore, the fraction \( \frac{x}{x^2+2x+1} \) simplifies to \( \frac{x}{(x+1)^2} \).

Step 3: Rewrite the Complex Fraction

You can now rewrite the complex fraction as: \[ \frac{\frac{x^3}{x+1}}{\frac{x}{(x+1)^2}} \]

Step 4: Simplify Using Division of Fractions

Divide the numerator by the denominator by multiplying the numerator by the reciprocal of the denominator: \[ \left(\frac{x^3}{x+1}\right) \times \left(\frac{(x+1)^2}{x}\right) \]

Step 5: Simplify the Result

Perform the multiplication in the previous step: \[ \frac{x^3 \cdot (x+1)^2}{x \cdot (x+1)} \] Simplify by canceling common terms: one \(x+1\) cancels from the numerator and the denominator: \[ \frac{x^3 \cdot (x+1)}{x} \]

Step 6: Simplify and Reduce Further

The expression simplifies to: \[ x^2 \cdot (x+1) \] because \( \frac{x^3}{x} = x^2 \). Therefore, the simplified result is \( x^3 + x^2 \).

Key Concepts

Factoring Expressions

Factoring expressions is a fundamental skill in algebra that involves rewriting an expression as the product of its factors. This skill is particularly useful when we need to simplify expressions or solve equations.
To factor an expression, look for common factors or patterns, such as the sum and difference of squares, perfect square trinomials, or common monomial factors.
In our exercise, the expression in the denominator \(x^2 + 2x + 1\) can be factored into \((x + 1)^2\).

• Identify terms that can be grouped together.
• Find the greatest common factor (GCF) for the terms in the group.
• Rewrite the expression using the identified factors.

Factoring can significantly simplify the process of dealing with fractions and can lead to more manageable computations.

Complex Fractions

Complex fractions are fractions where the numerator, the denominator, or both are themselves fractions. Simplifying complex fractions involves several steps which often include finding a common denominator or using multiplication to clear the fraction.
In the provided exercise, the complex fraction was \(\frac{\frac{x^{3}}{x+1}}{\frac{x}{x^{2}+2 x+1}}\).

• First, simplify the individual fractions present in the numerator and denominator.
• Rewrite the complex fraction as a simpler expression.
• Use the concept of dividing by a fraction using multiplication with its reciprocal to further simplify.

Understanding complex fractions is crucial for manipulating and simplifying rational expressions.

Multiplying Fractions

Multiplying fractions is straightforward and involves multiplying the numerators together and denominators together. However, when dealing with algebraic fractions, this process can include additional steps like factoring and canceling common factors.
In our example, after rewriting the complex fraction, we multiplied the fractions: \(\left(\frac{x^3}{x+1}\right) \times \left(\frac{(x+1)^2}{x}\right)\).

• Multiply the numerators:
• \(x^3 \times (x+1)^2\)
• Multiply the denominators:
• \(x \times (x+1)\)

Simplifying the result includes canceling out like terms, reducing the expression to its simplest form.

Rational Expressions

Rational expressions are quotients of two polynomials, similar to fractions in arithmetic. They follow similar rules as fractions, including the ability to simplify, multiply, divide, add, and subtract.
In the exercise, we dealt with the rational expression that resulted from the simplification: \(\frac{x^3 + x^2}{1}\).

• Identify common factors in the numerator and denominator.
• Factor out these common terms.
• Simplify the expression by canceling common factors.

The goal is to see whether the rational expression can be further reduced or simplified for practical use in equations. Recognizing and working with rational expressions is an essential skill in algebraic problem-solving.