Problem 57 - Algebra A Combined Function

Question

Simplify each expression. Each exercise contains a four-term polynomial that should be factored by grouping. $$ \frac{5 x+15-x y-3 y}{2 x+6} $$

Step-by-Step Solution

Answer

\(\frac{5 - y}{2}\)

Step 1: Inspect the Numerator

Given the expression \(5x + 15 - xy - 3y\), we need to factor by grouping. Group the terms as follows: \((5x + 15) - (xy + 3y)\).

Step 2: Factor Each Group

Factor out the greatest common factor from each group. From \(5x + 15\), factor out 5: \(5(x + 3)\). From \(-xy - 3y\), factor out \(-y\): \(-y(x + 3)\). This gives us: \(5(x + 3) - y(x + 3)\).

Step 3: Factor Out Common Binomial

Now, both terms share a common binomial \((x + 3)\). Factor this out: \((x + 3)(5 - y)\).

Step 4: Simplify the Denominator

Look at the denominator \(2x + 6\). Factor out 2: \(2(x + 3)\).

Step 5: Cancel Common Factors

The expression now looks like \(\frac{(x + 3)(5 - y)}{2(x + 3)}\). Cancel the common factor \((x + 3)\) from the numerator and the denominator. This leaves \(\frac{5-y}{2}\).

Step 6: Write the Simplified Expression

After cancelling the common factor, the simplified expression is \(\frac{5 - y}{2}\).

Key Concepts

Factoring by Grouping

Factoring by grouping is a handy method for simplifying polynomials, especially when dealing with four-term polynomials. This technique helps in breaking down expressions into simpler parts, making them easier to manage and solve. When you see a polynomial with four terms, such as

• \(5x + 15 - xy - 3y\)

you can split it into two groups of two terms each. Grouping like terms together, in the example above, looks like this:

• \((5x + 15) - (xy + 3y)\)

Notice how we strategically choose the groups to have something in common. The objective is to factor out common factors from each pair. By doing so, we reveal a common binomial that can be factored out, making it easier to simplify the polynomial. Practicing this method will make it much simpler to handle more complex polynomials in the future.

Greatest Common Factor

Finding the greatest common factor (GCF) is a key step in simplifying algebraic expressions. The GCF is the largest number or expression that divides each term in a group without leaving a remainder. To use the GCF in factoring by grouping:

• Identify the common factors in each smaller group of terms.
• For \(5x + 15\), the GCF is 5, so we factor it out: \(5(x + 3)\).
• For \(-xy - 3y\), factor out \(-y\), resulting in \(-y(x + 3)\).

By focusing on the greatest common factor, you simplify the expression into its core components, making it possible to recognize a shared binomial expression to factor further. Understanding and identifying the GCF helps tremendously in reducing expressions effectively.

Simplifying Expressions

Simplifying expressions is about making them as compact and manageable as possible. Once you have factored a polynomial, you should look for opportunities to simplify further by removing common factors. For instance, in

• \(\frac{(x + 3)(5 - y)}{2(x + 3)}\)

we can see that \((x + 3)\) is a common factor in both the numerator and the denominator. By canceling out this factor, the expression simplifies to

• \(\frac{5-y}{2}\)

This step is crucial in expressing the polynomial in the simplest form possible, reducing the complexity and making it easier to understand or solve. Consistently practicing these simplification skills is essential for successfully tackling more intricate algebraic problems.