Problem 72 - Algebra for College Students

Question

Perform the indicated operations. $$\frac{x^{2}+5 x+6}{x} \cdot \frac{x^{2}}{3 x+6} \cdot \frac{9}{x^{2}-4}$$

Step-by-Step Solution

Answer

\(\frac{3x(x+3)}{(x-2)}\)

Step 1 - Factor polynomials

First, factor each polynomial in the given expression. \(\frac{x^{2}+5x+6}{x}\) factors to \(\frac{(x+2)(x+3)}{x}\). \(\frac{x^{2}}{3x+6}\) becomes \(\frac{x^{2}}{3(x+2)}\). \(\frac{9}{x^{2}-4}\) is \(\frac{9}{(x-2)(x+2)}\), as \(x^{2}-4\) is a difference of squares.

Step 2 - Rewrite the expression with factored forms

Rewrite the original expression using the factored forms: \[\frac{(x+2)(x+3)}{x} \times \frac{x^{2}}{3(x+2)} \times \frac{9}{(x+2)(x-2)}\]

Step 3 - Simplify by canceling common factors

Identify and cancel out common factors in the numerators and denominators. Cancel \((x+2)\) in the first and second fractions, and cancel \((x+2)\) in the second and third fractions. Remaining expression: \(\frac{(x+3)x^{2} \times 9}{x \times 3(x-2) \times (x+2)}\)

Step 4 - Further simplification

Cancel out additional common factors. \(x\) in the numerator of \(\frac{(x+3)x^{2}}{x}\) and denominator of first fraction. Remaining expression: \(\frac{(x+3)x \times 9}{3(x-2)}\).

Step 5 - Combine and simplify

Combine and simplify the remaining terms: \(\frac{9x(x+3)}{3(x-2)}\). Further simplifying, \(\frac{9}{3} = 3\). Final expression: \(\frac{3x(x+3)}{(x-2)}\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a key step in simplifying algebraic fractions. This process involves breaking down a polynomial into simpler polynomials that, when multiplied together, give back the original polynomial. For example, consider the expression \( x^2 + 5x + 6 \). To factor it, we search for two numbers that multiply to 6 (the constant term) and add up to 5 (the coefficient of the middle term, x). These numbers are 2 and 3, hence \( x^2 + 5x + 6 \) can be factored into \((x + 2)(x + 3)\). This technique is invaluable when dealing with rational expressions, as it allows us to simplify them more effectively by canceling common factors.

Canceling Common Factors

Once polynomials are factored, the next step is to simplify the expression by canceling out common factors. This means identifying factors that appear in both the numerator and the denominator. For instance, in the expression \((x+2)(x+3)/(x) \times x^2/(3(x+2))\), notice that \(x+2\) appears in both, enabling us to cancel it out. This drastically simplifies our calculations and brings us closer to the final simplified form. It's important to only cancel factors, not terms! Understanding the difference between these two is crucial for correctly simplifying expressions.

Rational Expressions

Rational expressions are fractions that consist of polynomials in the numerator and denominator. Simplifying these can often seem complex, but with the right approach, it's manageable. For example, in the given problem, we start with a rational expression: \( x^2 + 5x + 6 \/ x \). Following proper steps, we've transformed it into a simpler expression by factoring out common terms and canceling them. Another example within our problem is \ x^2 \/ (3x + 6). Factoring out a 3 from the denominator gives us \ x^2 \/ 3(x + 2). Being systematic with these steps allows us to simplify the expressions incrementally until we achieve the most simplified form.

Difference of Squares

The difference of squares is a specific factoring technique that is particularly useful. It states that any expression of the form \( a^2 - b^2 \) can be factored into \ (a - b)(a + b) \. In the problem above, we see the expression \ x^2 - 4 \. This fits the pattern of \ a^2 - b^2 \ where \ a = x \ and \ b = 2\. Thus, \ x^2 - 4 \ can be factored into \ (x - 2)(x + 2)\. Recognizing and applying the difference of squares simplifies roadblocks in algebraic fractions, making them easier to manage.