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Chapter 1: Problem 65

Factor out the common factor. $$2 x^{2} y-6 x y^{2}+3 x y$$

Short Answer

Expert verified
The expression factored is \(xy(2x - 6y + 3)\).

Step by step solution

01

Identify the greatest common factor

First, look at each term of the expression: - The first term is \(2x^2y\)- The second term is \(-6xy^2\)- The third term is \(3xy\)Identify the common factor available in all three terms. Each term has factors of \(x\) and \(y\). The numerical greatest common factor is \(1\). Thus, the greatest common factor is \(xy\).
02

Factor out the greatest common factor

Take the greatest common factor \(xy\) out of the entire expression by dividing each term by \(xy\): - \(\frac{2x^2y}{xy} = 2x\)- \(\frac{-6xy^{2}}{xy} = -6y\)- \(\frac{3xy}{xy} = 3\)The expression becomes \(xy(2x - 6y + 3)\).
03

Write the final factored form

After factoring out the greatest common factor \(xy\), the final expression becomes: \[xy(2x - 6y + 3)\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor (GCF)
When working with polynomials, one of the first techniques to master is identifying the greatest common factor, or GCF. It refers to the largest factor that divides each term within a polynomial expression.
Essentially, finding the GCF involves a couple of steps:
  • Look at each term. In our example, these terms are \(2x^2y\), \(-6xy^2\), and \(3xy\).
  • Identify any common variables or numbers. All terms have both \(x\) and \(y\), and the numerical part is divisible by \(1\).
Combining these, the GCF for the expression is \(xy\).
This common factor is crucial for simplifying and restructuring expressions, as we'll see when we factor the expression further.
Polynomial Expressions
A polynomial expression is made up of variables raised to powers and combined using addition or subtraction. The key is recognizing each term's coefficients and variables.
Regarding the expression given:
  • Terms: \(2x^2y\), \(-6xy^2\), and \(3xy\)
  • Coefficients: 2, -6, and 3
  • Variables include \(x\) and \(y\) present in different powers
Understanding how polynomials are structured allows us to break them down elegantly using techniques like factoring.
Factoring polynomials simplifies calculations and helps in finding solutions to algebraic equations more effectively.
Step by Step Algebra
Step-by-step algebra simplifies complex expressions by breaking down the procedure into clear, manageable stages. It enhances comprehension by systematically unraveling each element.
For our polynomial example:
  • Step 1: Identify the GCF. Start by reviewing each term individually and discerning the common factors.
  • Step 2: Divide each term by the GCF, \(xy\). Performing this yields terms like \(2x\), \(-6y\), and 3, effectively restructuring the expression.
  • Step 3: Combine everything back with the GCF. Your expression becomes \(xy(2x - 6y + 3)\).
By following these steps, we can transform a complex polynomial into a simplified form, leveraging fundamental principles to solve or simplify algebraic problems more efficiently.

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