Problem 63 - Elementary Algebra

Question

Simplify each fraction. You will need to use factoring by grouping. $$\frac{x y-3 x+2 y-6}{x y+5 x+2 y+10}$$

Step-by-Step Solution

Answer

The simplified fraction is \(\frac{y-3}{y+5}\).

Step 1: Identify Groups in Numerator

Look at the numerator, which is \(xy - 3x + 2y - 6\). Group the terms to make two groups: \((xy - 3x)\) and \((2y - 6)\).

Step 2: Factor Each Group in Numerator

Factor out the common factor in each group found in Step 1. For \(xy - 3x\), the common factor is \(x\), so it becomes \(x(y - 3)\). For \(2y - 6\), the common factor is \(2\), so it becomes \(2(y - 3)\).

Step 3: Factor by Grouping in Numerator

You now have \(x(y - 3) + 2(y - 3)\). Notice \(y - 3\) is common, so factor it out: \((y - 3)(x + 2)\).

Step 4: Identify Groups in Denominator

Look at the denominator, which is \(xy + 5x + 2y + 10\). Group the terms to make two groups: \((xy + 5x)\) and \((2y + 10)\).

Step 5: Factor Each Group in Denominator

Factor out the common factor in each group found in Step 4. For \(xy + 5x\), the common factor is \(x\), so it becomes \(x(y + 5)\). For \(2y + 10\), the common factor is \(2\), so it becomes \(2(y + 5)\).

Step 6: Factor by Grouping in Denominator

You now have \(x(y + 5) + 2(y + 5)\). Notice \(y + 5\) is common, so factor it out: \((y + 5)(x + 2)\).

Step 7: Simplify the Fraction

The expression is now \(\frac{(y-3)(x+2)}{(y+5)(x+2)}\). You can cancel the common factor \((x+2)\) from both numerator and denominator. The simplified fraction is \(\frac{y-3}{y+5}\).

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. When simplifying, our goal is to ensure the numerator and the denominator have no common factors other than 1.
To simplify fractions, you can often factor both the numerator and the denominator and eliminate the common factors. This means finding terms that appear in both parts and canceling them out.
For example, if you have a fraction like \(\frac{(y-3)(x+2)}{(y+5)(x+2)}\), you can cancel out the \((x+2)\) term since it appears in both the numerator and the denominator. This makes the fraction simpler, resulting in \(\frac{y-3}{y+5}\).
By doing this, you make calculations simpler and often more insightful. Learning to simplify fractions is a valuable skill for solving algebraic equations more easily.

Polynomial Factoring

Factoring polynomials can seem tricky at first, but it becomes easier with practice. This process involves breaking down a polynomial into a product of simpler polynomials.
Polynomials can often be factored by recognizing patterns or grouping terms. In the exercise, factoring by grouping was used. This means:

• Identify terms that can be grouped.
• Factor out the greatest common factor from each group.
• Recognize and pull out common terms.

As seen in the solution, the polynomial in both the numerator and the denominator was broken up into groups, factored individually, and then combined into factored expressions. In the numerator \(xy - 3x + 2y - 6\), the terms were grouped and resulted in \((y-3)(x+2)\) after factoring.
Factoring is an essential skill in algebra as it simplifies expressions, making them easier to manage and solve.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations such as addition and subtraction. Understanding how to manipulate these expressions is key to mastering various algebraic techniques.
When dealing with algebraic expressions, one often needs to perform operations like factoring, expanding, or simplifying to make calculations easier.
For example, in our exercise, expressions like \(xy - 3x + 2y - 6\) were manipulated to find a simpler form. This involved identifying parts of the expression that could be grouped and factored, demonstrating how versatile and flexible algebraic expressions can be.
Algebraic expressions form the building blocks of more complex algebra problems, and mastering them helps you in various fields such as physics, engineering, and computer science, where they're frequently used to solve real-world problems.