Problem 29 - Precalculus Mathematics for Calculus

Question

Perform the multiplication or division and simplify. $$\frac{x^{2}+7 x+12}{x^{2}+3 x+2} \cdot \frac{x^{2}+5 x+6}{x^{2}+6 x+9}$$

Step-by-Step Solution

Answer

\(\frac{x+4}{(x+1)(x+3)}\)

Step 1: Factorize all quadratic expressions

Begin by factorizing each of the quadratic expressions in the numerators and denominators. \(x^2+7x+12\) factors to \((x+3)(x+4)\), \(x^2+3x+2\) factors to \((x+1)(x+2)\), \(x^2+5x+6\) factors to \((x+2)(x+3)\), and \(x^2+6x+9\) factors to \((x+3)(x+3)\). This gives us the expression, \(\frac{(x+3)(x+4)}{(x+1)(x+2)} \cdot \frac{(x+2)(x+3)}{(x+3)(x+3)}\).

Step 2: Combine the fractions

Multiply the numerators and the denominators of the fractions separately to form a single fraction. The numerator becomes \((x+3)(x+4)(x+2)(x+3)\), and the denominator becomes \((x+1)(x+2)(x+3)(x+3)\).

Step 3: Cancel common factors

Identify and cancel the common factors in the numerator and the denominator. The common factors are \((x+3)\) and \((x+2)\). After cancelling, the expression simplifies to \(\frac{(x+4)}{(x+1)(x+3)}\).

Step 4: Simplify the expression

Since there are no more common factors to cancel, and all factors are already simplified, the final simplified form of the expression is \(\frac{x+4}{(x+1)(x+3)}\).

Key Concepts

Quadratic Factorization

Quadratic factorization is the process of breaking down a quadratic expression into a product of its simpler linear factors. This is similar to "unpacking" a quadratic to find the expressions multiplied together to make it. When dealing with quadratics, it often takes the form of \(ax^2 + bx + c\). Factoring these requires finding two numbers that both sum up to the coefficient \(b\) and multiply to the constant term \(c\).

• Example: For \(x^2 + 7x + 12\), we need numbers that multiply to 12 and add up to 7. Those numbers are 3 and 4, hence this factors to \((x + 3)(x + 4)\).
• Similar steps are followed for other quadratics like \(x^2 + 5x + 6\) which factors to \((x + 2)(x + 3)\).

Recognizing these patterns gets easier with practice, aiding the simplification of algebraic fractions.

Simplifying Expressions

Simplifying expressions involves reducing them to their most compact and manageable form without altering their value. This process often involves factorizing, cancelling like terms, or performing operations.
Let's consider simplifying the expression \(\frac{x^2 + 7x + 12}{x^2 + 3x + 2} \cdot \frac{x^2 + 5x + 6}{x^2 + 6x + 9}\). Here, starting by factorizing each part allows us to see common factors easily.

• The initial expression simplifies after factorization to \(\frac{(x+3)(x+4)}{(x+1)(x+2)} \cdot \frac{(x+2)(x+3)}{(x+3)(x+3)}\).
• Next, cancelling the like factors \((x+2)\) and \((x+3)\) in both the numerator and the denominator helps us simplify further to \(\frac{x+4}{(x+1)(x+3)}\).

This results in a more straightforward expression that's easier to work with, especially in larger and more complex equations.

Multiplying Fractions

When multiplying fractions, you multiply the numerators and denominators separately. This process maintains the balance of the fraction, essentially combining both parts into one expression.
In algebraic terms, when multiplying \(\frac{A}{B}\) by \(\frac{C}{D}\), the result is \(\frac{A \cdot C}{B \cdot D}\).

• For our exercise, multiplying \(\frac{(x+3)(x+4)}{(x+1)(x+2)}\) by \(\frac{(x+2)(x+3)}{(x+3)(x+3)}\) leads to \(\frac{(x+3)(x+4)(x+2)(x+3)}{(x+1)(x+2)(x+3)(x+3)}\).
• This multiplication ensures each part of the fraction is appropriately combined and prepared for further simplification, like cancelling common factors.

The beauty of multiplying fractions in algebra is that it turns what seems complex into a graceful exercise of pattern finding and reducing.

Algebraic Fractions

Algebraic fractions are fractions where both the numerator and the denominator are algebraic expressions. They play a crucial role in simplifying complex algebraic problems.
Understanding how to manage algebraic fractions involves not only knowing basic fraction operations but also being comfortable with algebraic techniques.

• For example, handling algebraic fractions often requires factorization, as seen in expressing \(\frac{x^2 + 7x + 12}{x^2 + 3x + 2}\) as \(\frac{(x+3)(x+4)}{(x+1)(x+2)}\).
• Cancelling common terms post multiplication simplifies the fraction to \(\frac{x+4}{(x+1)(x+3)}\).

The elegance of algebraic fractions lies in their ability to reveal deeper relationships and symmetries in expressions through simplification and manipulation.