Problem 86 - Intermediate Algebra

Question

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{3 x^{2}-3 y^{2}}{x^{2}+2 y+2 x+y x} $$

Step-by-Step Solution

Answer

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

Step 1: Factor the Numerator

The numerator of the fraction is \(3x^2 - 3y^2\). Notice that both terms have a common factor of 3, so we can factor this out: \(3(x^2 - y^2)\). We can then use the difference of squares formula \(a^2 - b^2 = (a-b)(a+b)\) to factor this further: \(3(x-y)(x+y)\).

Step 2: Inspect the Denominator for Simplification

The denominator is \(x^2 + 2y + 2x + yx\). We aim to simplify this expression through re-grouping or factoring. We rearrange and simplify to try to make factoring possible: \(x^2 + yx + 2x + 2y\).

Step 3: Group the Denominator Terms for Factoring

Notice the terms \(x^2 + yx\) and \(2x + 2y\). Factor each group separately where possible: \(x(x+y) + 2(x+y)\). The expression can now be seen as a sum of products.

Step 4: Factor the Denominator

Since both terms \(x(x+y)\) and \(2(x+y)\) have a common factor of \((x+y)\), we can factor \((x+y)\) out: \((x+y)(x+2)\).

Step 5: Simplify the Fraction

Now that both the numerator and the denominator are fully factored, we can simplify the original expression by canceling the common factor \((x+y)\) from both: \(\frac{3(x-y)\cancel{(x+y)}}{\cancel{(x+y)}(x+2)}\), leading to \(\frac{3(x-y)}{x+2}\).

Key Concepts

Factoring Expressions

Factoring expressions is like unpacking a complicated suitcase to see what's inside. It involves breaking down a complex expression into simpler factors or components, which can be multiplied together to get the original expression. Think of it as simplifying puzzle pieces to reveal a clearer picture.

To factor an expression, observe each term closely to identify common factors. These common factors could be numbers, variables, or more complex expressions shared across the terms.

• Start with identifying and factoring out the greatest common factor (GCF) from the expression.
• Rewriting the expression with the GCF outside a parenthesis will simplify it.
• This sets the stage for further factoring techniques, like special formulas.

Once factors are identified, make sure to use algebraic identities like the difference of squares for further simplification. This process helps in making complex expressions easier to understand and operate with. Practicing factoring regularly builds confidence in managing different algebraic forms.

Difference of Squares

The difference of squares is a specific pattern in algebra that can be very handy to recognize. An expression following this pattern mirrors the form \(a^2 - b^2\) and is always factored into \((a-b)(a+b)\). This identity simplifies calculations and reveals the structure of the expression instantly.

Identifying the difference of squares involves:

• Recognizing squared terms, ensuring both terms are perfect squares.
• Looking for a subtraction between these squared terms.

For example, in the numerator from our exercise, the expression \(3(x^2 - y^2)\) fits perfectly as we apply the difference of squares formula. Here, \(x^2 - y^2\) becomes \((x-y)(x+y)\), and outside multiplication by 3 simplifies this into \(3(x-y)(x+y)\).

Being able to quickly identify and apply the difference of squares can save time and reduce errors, forming a powerful tool in your algebra toolkit.

Expression Simplification

Expression simplification is like tidying up a messy room. It involves reducing expressions to their simplest form, making them easier to understand or solve. The process involves a few key strategies and can make working with algebraic expressions much more manageable.

• First, focus on both numerator and denominator of any fractional expression.
• Factor each part until you reach the simplest form, looking for common variables or numbers to help narrow it down.
• Once both parts are factored, cancel out common factors shared by the numerator and denominator.

In our example, once the numerator (\(3(x-y)(x+y)\)) and the denominator (\((x+y)(x+2)\)) are factored, the common term \((x+y)\) is canceled, greatly simplifying the expression to \(\frac{3(x-y)}{x+2}\).

This step-by-step process ensures that every part of the expression is as simple as possible, making it easier to work with or evaluate. Expression simplification not only makes calculations more straightforward but also reveals insights about the behavior of algebraic functions.