Problem 20 - College Algebra

Question

Multiply or divide as indicated. $$ \frac{x^{2}+5 x+6}{x^{2}+x-6} \cdot \frac{x^{2}-9}{x^{2}-x-6} $$

Step-by-Step Solution

Answer

The simplified result of the given expression is \(x+2\).

Step 1: Factorize the Polynomials

Factorize each of the polynomial in the fractions where possible. The first one, \(x^{2}+5x+6\) can be factored into \((x+2)(x+3)\), the second one, \(x^{2}+x-6\) can be factored into \((x+3)(x-2)\), and \(x^{2}-9\) can be factored into \((x-3)(x+3)\). Therefore, the equation simplifies down to \[\frac{(x+2)(x+3)}{(x+3)(x-2)} \cdot \frac{(x-3)(x+3)}{(x+3)(x-2)}.\]

Step 2: Cancel Out Common Terms

The terms \((x+3)\) and \((x-2)\) can be found both in the numerator and the denominator, so they can be cancelled. This leaves us with \[\frac{(x+2)(x-3)}{x-2}.\]

Step 3: Simplify the Final Equation

The term \((x-2)\) can be found both in the numerator and the denominator, cancel it down to simplify. The simplified equation is \((x+2)\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial step in simplifying expressions. It involves breaking down a polynomial into simpler terms or "factors" that, when multiplied together, give you the original polynomial. In the exercise provided, the polynomials were factored into pairs of binomials.

• The polynomial \(x^2 + 5x + 6\) was factored into \((x+2)(x+3)\).
• For \(x^2 + x - 6\), the factors are \((x+3)(x-2)\).
• The difference of squares, \(x^2 - 9\), was rewritten as \((x-3)(x+3)\).

Factoring helps to identify common terms in the multiplication or division of polynomials. Recognizing these can sometimes lead to cancellations, simplifying the overall expression. To factor effectively:- Look for common factors among terms.- Use special formulas like the difference of squares.- Break down longer polynomials into products of binomials.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing a fraction consisting of polynomials on the numerator and denominator by factoring and canceling terms whenever possible. In our exercise, after factoring each polynomial, the expressions were simplified by canceling common factors.

Start by rewriting each polynomial in the fraction as a product of its factors. This method helps reveal common terms that might not be obvious at first glance. Once these terms are visible, they can be canceled out if they appear in both the numerator and the denominator. This reduction is essential for simplifying otherwise complex polynomial fractions.

Remember: - Always confirm that terms are properly factored before simplifying. - Canceled terms must appear identically in both the numerator and the denominator. - Simplified expressions should not have any common factors left. This process helps to bring clarity and reduces the expression to its simplest form for easy evaluation.

Cancelling Common Factors

Cancelling common factors is the process where terms that appear both in the numerator and denominator are removed from a rational expression. This makes the expression simpler, equivalent to its original form but easier to manage. In the given example, after factoring, the expression contained common factors:

• \((x+3)\) appeared in all numerators and denominators.
• \((x-2)\) was also common in the fraction components.

By cancelling these terms, you end up with a simpler expression, which was \(\frac{(x+2)(x-3)}{x-2}\). Further reduction by cancelling \((x-2)\), resulted in a final expression of \(x+2\).

Keep in mind:- Cancelling is only valid when terms are multiplied together and appear on both sides.- It’s vital to factor completely before attempting to cancel so that all potential common factors are considered.- While cancelling simplifies the math, it's crucial to ensure that the operation doesn't change the values the expression can take (except for any excluded values due to cancelling in denominators).