Problem 29 - Introductory Algebra for College Students

Question

Multiply as indicated.$ $$\frac{x^{2}-y^{2}}{x} \cdot \frac{x^{2}+x y}{x+y}$$

Step-by-Step Solution

Answer

\( x - y \)

Step 1: Factor

Factor the numerator of the first expression and the denominator of the second expression. This gives \( \frac{(x-y)(x+y)}{x} \cdot \frac{x(x+y)}{x+y} \)

Step 2: Cancel out common factors

We can see that the factor \( x+y \) is present in both the numerator and the denominator. So it can be removed from both. This leaves us with \( \frac{(x-y)x}{x} \)

Step 3: Simplify

We can cancel the \( x \) term present in both the numerator and the denominator and this gives the final answer as \( x - y \)

Key Concepts

Algebraic Expression Simplification

Simplifying algebraic expressions is a foundational skill in algebra that helps in reducing complexity and understanding the underlying structure of a mathematical problem. When looking at an expression like \(\frac{x^{2}-y^{2}}{x} \cdot \frac{x^{2}+x y}{x+y}\), it may seem intricate at first. Start by identifying opportunities to reduce the expression to a simpler form. Recognize patterns such as the difference of squares \(a^2 - b^2 = (a+b)(a-b)\), which we can apply here to factor \(x^{2}-y^{2}\) into \( (x-y)(x+y) \).

Always assess each part of the expression separately—factor numerators and denominators when possible. By breaking down the expression into simpler parts, the whole problem becomes more approachable. This step-by-step approach is not just about finding the answer; it’s a strategic method to build confidence in handling complex algebraic expressions.

Common Factors Cancellation

The concept of common factors cancellation comes into play when you have the same factor in both the numerator and the denominator of a fraction. After simplifying the given expression, it translates to \(\frac{(x-y)(x+y)}{x} \cdot \frac{x(x+y)}{x+y}\). Notice that \(x+y\) appears in both a numerator and a denominator.

Factors that appear on both sides of the fraction bar can be eliminated because dividing them is equivalent to dividing by 1, simplifying the entire expression. It's essential to be cautious and ensure that the factor you are cancelling is indeed common to both, and not just a similar looking term. Visual inspection is vital, but so is understanding that factor cancellation is a form of division, where any number divided by itself equals one. This skill not only simplifies expressions but also helps in solving equations more efficiently in algebra.

Polynomial Division

Polynomial division is often encountered in algebra when dealing with complex ratios of polynomials. In our example, once we factor and cancel out common terms, we are left with \(\frac{(x-y)x}{x}\). Now it's time to divide, or in simpler terms, see what's left when we 'cut out' common factors.

Since \(x\) is present both in the numerator and the denominator, we divide them out. This is essentially polynomial division on a smaller scale. Recognize that any term divided by itself is 1, hence they eliminate each other. The act of division simplifies the expression further, leaving us with the much simpler \(x - y\).

Understanding polynomial division is crucial as it helps in more complex tasks, such as finding the roots of polynomials and calculus. With practice, the process of recognizing common factors and simplifying becomes almost second nature, enabling students to tackle algebraic expressions and divisions with ease.