Problem 22 - Algebra and Trigonometry

Question

multiply or divide as indicated. $$ \frac{x^{2}+6 x+9}{x^{3}+27} \cdot \frac{1}{x+3} $$

Step-by-Step Solution

Answer

After simplifying the given expression, the result is \(\frac{1}{x^2 - 3x + 9}\).

Step 1: Factorize the Polynomials

Factorize the polynomials in the numerators and the denominator. The numerator \(x^2+6x+9\) can be factored as \((x+3)^2\) and the denominator \(x^3+27\) as \((x+3)(x^2-3x+9)\). The expression is now \(\frac{(x+3)^2}{(x+3)(x^2-3x+9)} \cdot \frac{1}{x+3}\).

Step 2: Simplify the Fractions

Simplify by canceling out common factors across the numerator and denominator. In this case, \(x+3\) can be cancelled out from the entire fraction. This simplifies the given expression to \(\frac{x + 3}{x^2 - 3x + 9}\).

Step 3: Further Simplification

We can also simplify by factoring when we multiply with \(\frac{1}{x+3}\). Here, \(x+3\) gets cancelled out again and we are left with \(\frac{1}{x^2 - 3x + 9}\).

Key Concepts

Factorization of Polynomials

Factorization is a critical algebraic process used in simplifying expressions and solving equations. In the context of algebraic fractions, it can turn a complex problem into a simpler one. The process involves breaking down a polynomial into a product of simpler polynomials or factors. For example, the quadratic polynomial x^2 + 6x + 9 can be factored into (x + 3)^2. Similarly, factorizing the sum of cubes, such as x^3 + 27, involves identifying and applying specific formulas, leading to (x + 3)(x^2 - 3x + 9).

Understanding these factorizations is not just about applying formulas; it requires recognizing patterns. The quadratic above shows a perfect square trinomial, where a^2 + 2ab + b^2 breaks down to (a + b)^2. For the sum of cubes, use the formula a^3 + b^3 = (a + b)(a^2 - ab + b^2). Mastering factorization can make complex algebraic manipulations seem like a straightforward task, which is exactly why this skill is emphasized in algebra courses.

Algebraic Expression Simplification

Simplification of algebraic expressions makes equations more manageable and solutions clearer. In the case of algebraic fractions, simplification often involves several steps, including expanding expressions, combining like terms, and factorizing. After these preliminary steps, the expression should be in a form that allows for easy identification of common factors that can be cancelled out.

In our exercise, we begin with a complex fraction and simplify it step-by-step. The key is to work methodically, ensuring each action is valid and brings us closer to a more refined form of the expression. Simplifying expressions is not only useful for clarity, but also essential for understanding relationships between variables, interpreting solutions, and solving for unknowns.

Canceling Common Factors

Reduction of Fractions

Cancellation of common factors within algebraic fractions is a powerful tool that can dramatically reduce the complexity of an expression. This technique involves identifying factors that appear in both the numerator and the denominator and eliminating them, as they divide to one. It's crucial to only cancel factors that are multiplied by the entirety of the other terms, not just portions.

When we look back at our exercise, the cancellation simplifies the initial complex fraction down to a much simpler form. Here's a tip: Always factorize completely before cancelling. Partial factorization may leave behind hidden common factors that could further simplify the expression. This process of cancelling does not change the value of the expression—it's simply a form of mathematical 'house-cleaning' that makes our work neater and more comprehensible.