Problem 59 - Algebra A Combined Function

Question

Simplify each expression. Each exercise contains a four-term polynomial that should be factored by grouping. $$ \frac{2 x y+5 x-2 y-5}{3 x y+4 x-3 y-4} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{2y + 5}{3y + 4}\).

Step 1: Identify Terms to Group

We need to identify pairs of terms in both the numerator and the denominator that can be grouped together to factor further. For the numerator \(2xy + 5x - 2y - 5 \) and the denominator \(3xy + 4x - 3y - 4 \), pair them into \((2xy + 5x) + (-2y - 5)\) and \((3xy + 4x) + (-3y - 4)\).

Step 2: Factor by Grouping in the Numerator

In the numerator, factor out the greatest common factor from each grouped pair: \[(2xy + 5x) + (-2y - 5) = x(2y + 5) - 1(2y + 5) = (x - 1)(2y + 5)\].

Step 3: Factor by Grouping in the Denominator

In the denominator, factor out the greatest common factor from each grouped pair:\[(3xy + 4x) + (-3y - 4) = x(3y + 4) - 1(3y + 4) = (x - 1)(3y + 4)\].

Step 4: Simplify the Fraction

The fraction now becomes: \[\frac{(x - 1)(2y + 5)}{(x - 1)(3y + 4)}\]. Since \((x - 1)\) is common in both the numerator and the denominator, we can cancel them out, leaving \[\frac{2y + 5}{3y + 4}\].

Step 5: Final Simplified Expression

After canceling the common factor \((x - 1)\), we are left with the simplified expression: \[\frac{2y + 5}{3y + 4}\].

Key Concepts

Factoring by Grouping

Factoring by grouping is a technique used to simplify polynomial expressions, particularly those with four or more terms. This method works by reorganizing the polynomial into groups of terms that share a common factor, which can then be factored out. In essence, it transforms a complex expression into smaller, more manageable parts.

For example, consider the polynomial: \[ a + b + c + d \]We can apply grouping as follows:

• Group terms: \((a + b) + (c + d)\)
• Factor out common factors from each group.

To see this in action, look at the numerator from the exercise: \(2xy + 5x - 2y - 5 \).

Here’s how you can apply grouping:

• Group as \((2xy + 5x) + (-2y - 5)\)
• Factor each pair: \(x(2y + 5) - 1(2y + 5) \)
• This leads to a simpler expression combining like terms: \((x - 1)(2y + 5)\)

Ultimately, the goal is to identify and factor out terms that make the expression easier to manage, simplifying the process considerably.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions that consist of polynomials, similar to how numerical fractions are simplified. The process often includes finding like terms or common factors and reducing the expression by canceling these factors.

In our exercise, the goal is to simplify:\[\frac{(x - 1)(2y + 5)}{(x - 1)(3y + 4)}\]

• Identify expressions common to both the numerator and the denominator \((x - 1)\).
• Cancel out these common terms to simplify the expression: \[ \frac{2y + 5}{3y + 4} \]

This method of simplification is crucial as it turns a complicated fraction into a much simpler form, making it easier to understand and work with. Whenever simplifying rational expressions, ensure to only cancel factors that appear in both the numerator and the denominator.The essence is that simplification helps reveal the core structure of a polynomial expression, providing insights into its behavior and properties.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides two or more numbers. In the context of polynomial expressions, it is the highest degree of individual terms that can be factored from a polynomial. Finding the GCF is vital to simplifying expressions effectively.

To find the GCF of a polynomial, follow these steps:

• Identify the coefficients and variables involved in each term.
• Determine the highest power of each variable that is present in all terms.
• Identify the largest number that can divide the numerical coefficients in all terms.

For instance, in the step-by-step solution, the GCF is used to simplify grouping:

• For \((2xy + 5x)\), the GCF is \(x\).
• For \((-2y - 5)\), the GCF is \(-1\).
• Applying these factors results in: \(x(2y + 5) - 1(2y + 5)\).

Recognizing the GCF allows you to simplify expressions by removing redundancy and focusing on the core components, streamlining computations and problem-solving.