Problem 30 - Introductory Algebra for College Students

Question

Multiply as indicated.$ $$\frac{4 x-4 y}{x} \cdot \frac{x^{2}+x y}{x^{2}-y^{2}}$$

Step-by-Step Solution

Answer

The multiplication of these fractions gives a result of \(4x\).

Step 1: Simplify the Expressions

We can simplify the expressions by canceling out common factors in the numerator and denominator. Firstly, factorize the equations in the numerator and denominator. In the first fraction, factor 4 out from the numerator and x out from the denominator. In the second fraction, factor x out from the numerator and \(y(x-y)\) out from the denominator.

Step 2: Cancel out Common Factors

Cancel out the common factors in the numerators and denominators of both fractions. This would result in the following simplified expressions: \(\frac{4(x-y)}{1}\) and \(\frac{x(x+y)}{(x+y)(x-y)}\). The 'x+y' term in the numerator and denominator of the second fraction cancels out, resulting in \(\frac{x}{x-y}\) .

Step 3: Multiplication of Simplified Expressions

Now multiply the simplified expressions, connecting the numerator of the first expression with the numerator of the second, and the denominator of the first with the denominator of the second, to get our final result: \(4x\).

Key Concepts

Fraction Multiplication

Fraction multiplication is simpler than it might seem. When you multiply fractions, you don’t find common denominators like in addition or subtraction. Instead, you directly multiply the numerators together and the denominators together. For example, if you have two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), the result of their multiplication is \( \frac{a \cdot c}{b \cdot d} \). Here is the process in brief:

• Multiply the numerators of both fractions.
• Multiply the denominators of both fractions.
• Write the result as a single fraction.

This straightforward method allows you to focus on the arithmetic without worrying about denominators matching. The key point is to ensure numerators and denominators are multiplied separately, forming a new fraction.

Factoring in Algebra

Factoring is a crucial step in simplifying algebraic expressions, especially in fraction multiplication. We break down complex expressions into simpler parts or factors to uncover common elements. In algebra, to factor means transforming an expression like \(x^2 + xy\) into \(x(x + y)\). This identification of common elements offers tremendous simplification benefits.

In our exercise, each polynomial part was expressed in a factorable form. For instance:

• From \(4x - 4y\), factor out the greatest common factor (GCF), which is 4, giving us \(4(x - y)\).
• From \(x^2 + xy\), factor out the GCF \(x\), resulting in \(x(x + y)\).
• From \(x^2 - y^2\), notice it’s a difference of squares, which factors into \((x + y)(x - y)\).

By identifying these factors, you easily spot elements that can cancel out, vital for simplifying your expression further before multiplying fractions.

Simplifying Expressions

Simplifying expressions means reducing them to their most basic form without changing their value. This process often involves factoring and canceling common terms, making calculations much more manageable.

In our exercise, simplification played a pivotal role. After factoring the original expressions, common terms were canceled out. For example, the common term \((x + y)\) in the second fraction was eliminated, simplifying it further. Here’s how you simplify:

• Identify and cancel common factors from the numerator and denominator.
• Re-express the fraction in its simplest form after cancellation.
• Perform any multiplication necessary post-simplification for a clean result.

By following these steps, the initially complicated expression becomes simpler, paving way for easier multiplication, resulting in a streamlined final expression. Remember, simplifying expressions is about making your math work easier and your answers clearer.