Problem 35 - Precalculus Mathematics for Calculus

Question

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

The expression simplifies to \( x^3 + x^2 \).

Step 1: Understand the Expression

We start by understanding the expression: \( \frac{\frac{x^{3}}{x+1}}{\frac{x}{x^2+2x+1}} \). This is a complex fraction (a fraction where the numerator or denominator is also a fraction), and our goal is to simplify it.

Step 2: Simplify the Denominator

Recognize that the denominator of the complex fraction is \( \frac{x}{x^2+2x+1} \). Notice that \( x^2+2x+1 \) is a perfect square trinomial, so it can be factored as \( (x+1)^2 \). The denominator becomes \( \frac{x}{(x+1)^2} \).

Step 3: Rewrite and Multiply Reciprocal

Rewrite the complex fraction as a division problem: \( \frac{x^3}{x+1} \div \frac{x}{(x+1)^2} \). This is equivalent to multiplying by the reciprocal: \( \frac{x^3}{x+1} \times \frac{(x+1)^2}{x} \).

Step 4: Factor and Simplify

Simplify the expression by canceling out common factors. The expression becomes: \( \frac{x^3 \cdot (x+1)^{2}}{x \cdot (x+1)} \). We cancel out \( x+1 \) from the numerator and the denominator, leaving us with \( \frac{x^3 \cdot (x+1)}{x} \). Simplifying further: \( x^2(x+1) \) or \( x^3 + x^2 \).

Step 5: Final Simplified Expression

The simplified form of the original complex fraction is \( x^2(x+1) \). Therefore, the simplest form of the expression is \( x^3 + x^2 \).

Key Concepts

Simplifying Expressions

Simplifying expressions involves making mathematical expressions easier to work with by reducing them to their simplest form. This usually means removing complex fractions, factoring expressions, and canceling out common terms.

To simplify a complex fraction, like the one in the exercise, try to rewrite it in ways that make it easier to manage. Start by eliminating the smaller fractions inside the main fraction. You can do this by converting the complex fraction into a multiplication problem using the reciprocal of the denominator.

• Identify the complex fraction.
• Write the division problem as multiplication using the reciprocal.
• Factor the numerator and denominator if possible.
• Cancel out any common terms to simplify the expression.

These steps lead to a simplified version of the original expression, making it easier to analyze and understand.

Factoring Polynomials

Factoring polynomials is about breaking down a polynomial into simpler components. These components, called factors, can be multiplied together to get the original polynomial. Factoring is essential for solving various algebraic equations, particularly when simplifying complex expressions.

The goal is to express the polynomial as a product of its simplest factors. In context, this means recognizing and factoring expressions like quadratic trinomials into products of binomials. For example, take the expression \(x^2 + 2x + 1\), recognize it as a perfect square trinomial, and factor it into \((x+1)^2\).

• Look for common factors in all terms.
• Recognize special patterns like perfect squares.
• Apply factoring techniques such as grouping or using the quadratic formula when necessary.

Mastering this skill will greatly aid in solving algebra problems by simplifying polynomial expressions efficiently.

Perfect Square Trinomial

A perfect square trinomial is a specific type of polynomial that results from squaring a binomial. It follows the general pattern \(a^2 + 2ab + b^2 = (a+b)^2\). Recognizing these trinomials can simplify many algebraic challenges.

In the exercise, the expression \(x^2 + 2x + 1\) is a perfect square trinomial. This can be quickly identified, as it fits the general formula where \(a = x\) and \(b = 1\). Factoring it gives us \((x + 1)^2\), which is much simpler to work with.

• Identify the trinomial that matches the perfect square pattern.
• Use the structure of the pattern to write it as a squared binomial.
• Simplify further operations by using this factored form.

Recognizing and factoring perfect square trinomials helps in reducing complexities in algebraic equations.

Multiplication of Fractions

The multiplication of fractions is a straightforward process, as it involves multiplying the numerators to find the new numerator, and multiplying the denominators to find the new denominator. In the case of complex fractions, understanding this process is vital.

When simplifying complex expressions like the exercise provided, converting division into multiplication by a reciprocal is crucial. This step means multiplying a fraction by the reciprocal of another fraction, effectively flipping the second fraction.

• Change division into multiplication by using the reciprocal of the divisor.
• Multiply the numerators together.
• Multiply the denominators together.
• Simplify the resulting fraction by canceling common factors.

Understanding these multiplication steps helps simplify complex fractions and manage intricate algebraic expressions smoothly.