Factorize $x+y-a X-a Y$ Completely.A. $(x-y)(1-a)$ B. $ ( X + Y ) ( 1 + A ) (x+y)(1+a) ( X + Y ) ( 1 + A ) [/tex] C. $(x+y)(1-a)$ D. $(x-y)(1+a)$
Introduction
Factorizing algebraic expressions is a fundamental concept in mathematics that involves expressing an expression as a product of simpler expressions. In this article, we will focus on factorizing the expression $x+y-a x-a y$ completely. We will explore the different methods of factorization and provide a step-by-step guide to help you understand the process.
Understanding the Expression
Before we begin factorizing the expression, let's first understand what it represents. The expression $x+y-a x-a y$ can be rewritten as $(x+y)(1-a)$, where $(x+y)$ is the sum of two variables and $(1-a)$ is a constant term.
Method 1: Factoring by Grouping
One of the methods of factorizing the expression is by grouping. This involves grouping the terms in pairs and then factoring out the common factors.
Step 1: Group the Terms
Group the terms in pairs as follows:
Step 2: Factor Out the Common Factor
Factor out the common factor $(x+y)$ from the grouped terms:
Method 2: Factoring by Distributive Property
Another method of factorizing the expression is by using the distributive property. This involves multiplying the terms inside the parentheses with the terms outside the parentheses.
Step 1: Multiply the Terms
Multiply the terms inside the parentheses with the terms outside the parentheses:
Step 2: Simplify the Expression
Simplify the expression by combining like terms:
Step 3: Factor Out the Common Factor
Factor out the common factor $(x+y)$ from the simplified expression:
Conclusion
In conclusion, the expression $x+y-a x-a y$ can be factorized completely as $(x+y)(1-a)$. We have explored two methods of factorization, namely factoring by grouping and factoring by distributive property. Both methods have led to the same result, which is $(x+y)(1-a)$. This expression can be further simplified by factoring out the common factor $(x+y)$.
Answer
The correct answer is:
- C. $(x+y)(1-a)$
Discussion
Q&A: Factorizing Algebraic Expressions
Q: What is factorizing algebraic expressions?
A: Factorizing algebraic expressions is a fundamental concept in mathematics that involves expressing an expression as a product of simpler expressions. It involves breaking down an expression into its simplest form by identifying the common factors.
Q: Why is factorizing algebraic expressions important?
A: Factorizing algebraic expressions is important because it helps to simplify complex expressions, making it easier to solve algebraic equations. It also helps to identify the common factors, which can be used to solve equations and inequalities.
Q: What are the different methods of factorizing algebraic expressions?
A: There are several methods of factorizing algebraic expressions, including:
- Factoring by grouping
- Factoring by distributive property
- Factoring by difference of squares
- Factoring by sum and difference of cubes
Q: How do I factorize an expression using the distributive property?
A: To factorize an expression using the distributive property, you need to multiply the terms inside the parentheses with the terms outside the parentheses. Then, simplify the expression by combining like terms.
Q: How do I factorize an expression using the difference of squares?
A: To factorize an expression using the difference of squares, you need to identify the two perfect squares in the expression. Then, factor out the common factor from each perfect square.
Q: What is the difference between factoring by grouping and factoring by distributive property?
A: Factoring by grouping involves grouping the terms in pairs and then factoring out the common factors. Factoring by distributive property involves multiplying the terms inside the parentheses with the terms outside the parentheses.
Q: How do I determine which method to use when factorizing an expression?
A: To determine which method to use when factorizing an expression, you need to analyze the expression and identify the common factors. If the expression can be grouped into pairs, you can use the factoring by grouping method. If the expression involves multiplying the terms inside the parentheses with the terms outside the parentheses, you can use the factoring by distributive property method.
Q: What are some common mistakes to avoid when factorizing algebraic expressions?
A: Some common mistakes to avoid when factorizing algebraic expressions include:
- Not identifying the common factors
- Not grouping the terms correctly
- Not simplifying the expression correctly
- Not using the correct method for factorizing the expression
Q: How can I practice factorizing algebraic expressions?
A: You can practice factorizing algebraic expressions by working on exercises and problems. You can also use online resources and tools to help you practice and improve your skills.
Conclusion
In conclusion, factorizing algebraic expressions is a fundamental concept in mathematics that involves expressing an expression as a product of simpler expressions. There are several methods of factorizing algebraic expressions, including factoring by grouping, factoring by distributive property, and factoring by difference of squares. By understanding the different methods and practicing factorizing algebraic expressions, you can improve your skills and become proficient in solving algebraic equations and inequalities.
Answer
The correct answers to the Q&A are:
- Q: What is factorizing algebraic expressions? A: Factorizing algebraic expressions is a fundamental concept in mathematics that involves expressing an expression as a product of simpler expressions.
- Q: Why is factorizing algebraic expressions important? A: Factorizing algebraic expressions is important because it helps to simplify complex expressions, making it easier to solve algebraic equations.
- Q: What are the different methods of factorizing algebraic expressions? A: There are several methods of factorizing algebraic expressions, including factoring by grouping, factoring by distributive property, and factoring by difference of squares.
- Q: How do I factorize an expression using the distributive property? A: To factorize an expression using the distributive property, you need to multiply the terms inside the parentheses with the terms outside the parentheses. Then, simplify the expression by combining like terms.
- Q: How do I factorize an expression using the difference of squares? A: To factorize an expression using the difference of squares, you need to identify the two perfect squares in the expression. Then, factor out the common factor from each perfect square.
- Q: What is the difference between factoring by grouping and factoring by distributive property? A: Factoring by grouping involves grouping the terms in pairs and then factoring out the common factors. Factoring by distributive property involves multiplying the terms inside the parentheses with the terms outside the parentheses.
- Q: How do I determine which method to use when factorizing an expression? A: To determine which method to use when factorizing an expression, you need to analyze the expression and identify the common factors. If the expression can be grouped into pairs, you can use the factoring by grouping method. If the expression involves multiplying the terms inside the parentheses with the terms outside the parentheses, you can use the factoring by distributive property method.
- Q: What are some common mistakes to avoid when factorizing algebraic expressions? A: Some common mistakes to avoid when factorizing algebraic expressions include not identifying the common factors, not grouping the terms correctly, not simplifying the expression correctly, and not using the correct method for factorizing the expression.
- Q: How can I practice factorizing algebraic expressions? A: You can practice factorizing algebraic expressions by working on exercises and problems. You can also use online resources and tools to help you practice and improve your skills.