Problem 31 - Introductory Algebra for College Students

Question

Multiply as indicated.$ $$\frac{x^{2}+2 x y+y^{2}}{x^{2}-2 x y+y^{2}} \cdot \frac{4 x-4 y}{3 x+3 y}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(\frac{4(x+y)}{3(x-y)}\)

Step 1: Factorize the Expressions

We begin by factoring the expressions.The first fraction reduces to \(\frac{(x+y)^2}{(x-y)^2}\)The second fraction reduces to \(\frac{4(x-y)}{3(x+y)}\)

Step 2: Multiply the Fractions

Now, Let’s multiply the fractions.\(\frac{(x+y)^2}{(x-y)^2} \cdot \frac{4(x-y)}{3(x+y)}\)

Step 3: Simplify the Equation

On Simplifying, We get \(\frac{4(x+y)(x-y)}{3(x-y)(x-y)}\)

Step 4: Cancel out the common terms

Cancel the common terms\(\frac{4(x+y)}{3(x-y)}\)

Key Concepts

Factoring Expressions

The process of factoring expressions is a way to break down algebraic expressions into simpler components, called 'factors', that when multiplied together give back the original expression. It is an essential skill in algebra because it helps in simplifying complex expressions and solving equations.

Often, you can factor expressions by recognizing patterns, like the difference of squares or perfect square trinomials. In the given exercise, we encountered a perfect square trinomial in both the numerator and denominator of the first fraction, \(x^2 + 2xy + y^2\) and \(x^2 - 2xy + y^2\) respectively. Recognizing these patterns, we factored them into \( (x + y)^2\) and \( (x - y)^2\).

Practicing the identification of common patterns in polynomials can make the process of factoring second nature, greatly simplifying further algebraic manipulations.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing complexity without changing the value of the expression. This is done by combining like terms, performing arithmetic operations, and eliminating any factors common to both the numerator and the denominator when dealing with fractions.

In context, after we factored and multiplied our expressions, we arrived at \(\frac{4(x+y)(x-y)}{3(x-y)(x-y)}\). In this case, simplifying involves canceling out the common terms in the numerator and the denominator. A term \( (x-y) \) is present in both and can be cancelled out. It is important to watch out for terms that appear more than once; \( (x-y)^2 \) means there are two \( (x-y) \) terms multiplied together, which must be accounted for during the simplification process.

Always remember to simplify as much as possible. Simplification is not just about making expressions shorter; it's about making them clearer and easier to work with in future calculations.

Algebraic Fraction Multiplication

Multiplying algebraic fractions follows the same principle as multiplying numerical fractions: multiply numerators across to get the new numerator and denominators across to get the new denominator. However, when it comes to algebraic fractions, you often need to factor and simplify first to make the multiplication easier and the final expression simpler.

In our example, once simplified, the fractions became \(\frac{(x+y)^2}{(x-y)^2}\) and \(\frac{4(x-y)}{3(x+y)}\). By following the rule of multiplication, we then ended up with \(\frac{4(x+y)(x-y)}{3(x-y)(x-y)}\), which we further simplified. This step showcases the importance of simplification before and after multiplying: it can turn an intimidating problem into a manageable one.

Remember, the goal of understanding algebraic fraction multiplication isn't just to get the right answer, but to comprehend the process so that you can tackle more complex problems in the future with confidence.