Problem 41 - Algebra for College Students

Question

Factor completely. $$x(y+2)+3(y+2)$$

Step-by-Step Solution

Answer

The completely factored form is \((y+2)(x+3)\).

Step 1: Identify Common Factor

Look at the expression \( x(y+2) + 3(y+2) \). Notice that \( (y+2) \) is a common factor in both terms \( x(y+2) \) and \( 3(y+2) \).

Step 2: Factor Out the Common Factor

Since \( (y+2) \) is a common factor, factor it out of the entire expression: \( (y+2)(x+3) \).

Step 3: Verify the Result

Distribute the factored expression to check if it returns to the original form: \( (y+2)(x+3) = x(y+2) + 3(y+2) \). The original expression is recovered, confirming the factorization is correct.

Key Concepts

Factorization

Factorization in algebra is a crucial technique where we express a mathematical expression as a product of simpler factors.
For instance, in the expression \( x(y+2) + 3(y+2) \), factorization helps break it down into a simpler form \( (y+2)(x+3) \).
This makes solving equations, finding roots, or simplifying expressions more manageable.

Consider factorization as dismantling a complicated puzzle.

• First, identify the common pieces.
• Reorganize them into a structured layout.
• Finally, verify if reassembling them leads back to the original puzzle.

By mastering this concept, you develop a strong foundation for tackling more complex algebra problems.

Common Factor

A common factor is a term that appears in multiple components of an expression.
Identifying common factors is an essential skill in algebra as it simplifies expressions significantly.

Looking at \( x(y+2) + 3(y+2) \), the expression \((y+2)\) is a common factor shared by both parts of the expression:

• In the first part, \( x(y+2) \), the \((y+2)\) multiplies with \(x\).
• In the second part, \(3(y+2)\), it multiplies with \(3\).

By extracting the common factor \((y+2)\), we can simplify the expression into \((y+2)(x+3)\).
This step not only simplifies calculations but also aids in better understanding the structure of algebraic expressions.

Distribution

Distribution involves spreading out a term across other terms inside parentheses.
It is often performed in reverse during factorization.

In our example, after factoring out the common term \((y+2)\), we ensure correctness by distributing back the terms to verify our factorization process.

• Multiply \((y+2)\) with \(x\) and \(3\) separately to get \( x(y+2) + 3(y+2) \).
• This confirms our factorization \((y+2)(x+3)\) accurately returns the original expression.

Distribution acts as a double-check mechanism.
It is a vital part of not just validating factorization, but also within algebra for expanding expressions.

Polynomial Expression

A polynomial expression consists of variables combined using addition, subtraction, multiplication, and non-negative integer exponents.
This type of expression is essential in algebra and appears frequently.

In the exercise provided, the expression \( x(y+2) + 3(y+2) \) can be seen as a polynomial.

• The terms \(x\) and \(3\) are coefficients connected to the shared variable expression \((y+2)\).
• By factorizing, it was converted into the simpler polynomial expression \((y+2)(x+3)\).

Understanding polynomial expressions is key because they form the core of algebraic manipulations.
Grasping their structure allows for easier identification of patterns and implementation of operations like factorization.