Problem 36 - Precalculus Mathematics for Calculus

Question

Perform the multiplication or division and simplify. $$\frac{\frac{2 x^{2}-3 x-2}{x^{2}-1}}{\frac{2 x^{2}+5 x+2}{x^{2}+x-2}}$$

Step-by-Step Solution

Answer

\(\frac{x-2}{x+1}\)

Step 1: Understand the Expression

The problem involves dividing two fractions. We have \( \frac{\frac{2x^{2}-3x-2}{x^{2}-1}}{\frac{2x^{2}+5x+2}{x^{2}+x-2}} \). This is equivalent to multiplying the first fraction by the reciprocal of the second fraction.

Step 2: Reciprocal and Multiplication

Rewrite the expression from step 1:\[\frac{2x^{2}-3x-2}{x^{2}-1} \times \frac{x^{2}+x-2}{2x^{2}+5x+2}\]This change allows us to multiply the numerators together and the denominators together.

Step 3: Factor Numerators

Factor each polynomial in the numerators.- The expression \(2x^2 - 3x - 2\) factors to \((2x + 1)(x - 2)\).- The expression \(x^2 + x - 2\) factors to \((x + 2)(x - 1)\).

Step 4: Factor Denominators

Factor each polynomial in the denominators.- The expression \(x^2 - 1\) factors to \((x - 1)(x + 1)\).- The expression \(2x^2 + 5x + 2\) factors to \((2x + 1)(x + 2)\).

Step 5: Multiply and Simplify

Substitute the factors obtained in steps 3 and 4 back into the multiplication expression:\[\frac{(2x+1)(x-2)}{(x-1)(x+1)} \times \frac{(x+2)(x-1)}{(2x+1)(x+2)}\]Cancel the common terms \((2x+1)\), \((x+2)\), and \((x-1)\) from the numerator and denominator.

Step 6: Final Simplified Form

After canceling the common terms, the expression simplifies to:\[\frac{x-2}{x+1}\]This is the simplest form of the original complex fraction.

Key Concepts

Polynomial Factoring

Polynomial factoring is a crucial skill in algebra. It involves breaking down a complex polynomial into simpler, multiplied factors. This process makes equations easier to handle, especially in operations like addition, subtraction, and multiplication.
Understanding how to factor effectively requires recognizing different polynomial forms and knowing which factoring technique to use. Here are some tips to guide you:

• Identify the greatest common factor (GCF) as the first step. This is the largest factor that divides all terms of the polynomial.
• Look for patterns like the difference of squares, which follows the form \(a^2 - b^2 = (a-b)(a+b)\).
• For quadratic polynomials, such as \(ax^2 + bx + c\), check if it can be factored into two binomials. This often involves looking for two numbers that multiply to \(ac\) and add to \(b\).
• Recognize special products like perfect square trinomials and sum/difference of cubes.

With practice, recognizing these strategies will become almost second nature, making complex algebraic manipulations smoother.

Complex Fractions

Complex fractions are fractions where the numerator, the denominator, or both, are themselves fractions. They can seem quite intimidating at first, but with a clear method, they become much more approachable.
The key to simplifying complex fractions is to transform them into something more familiar. Here's a simple process:

• Identify the main fraction bar. This separates the overall numerator from the denominator.
• Rewrite the expression by taking the reciprocal of the denominator fraction and changing the division to multiplication.
• Continue by multiplying the expressions, as you would with normal fractions.
• Simplify by canceling any common factors, where possible, between the resulting numerator and denominator.

This method works because division of fractions is equivalent to multiplying by the reciprocal. With practice, complex fractions lose their complexity and become straightforward.

Algebraic Simplification

Algebraic simplification is an essential process for solving and understanding equations. The goal is to reduce expressions to their simplest form.
To successfully simplify algebraic expressions, keep these steps in mind:

• First, perform any operations within parentheses or brackets.
• Look out for and execute any multiplications or divisions across terms.
• Combine like terms, which are terms that have the same variables raised to the same powers.
• Cancel out any common factors in fractions to make the expression as concise as possible.

By consistently applying these steps, you can transform complex expressions into simpler, more understandable versions. This makes subsequent operations, such as solving equations or substituting values, a breeze.