Factor $a x-2 b x-2 a y+4 b y$.
Answer (2)
Group the terms: ( a x − 2 b x ) + ( − 2 a y + 4 b y ) .
Factor out common factors from each group: x ( a − 2 b ) − 2 y ( a − 2 b ) .
Factor out the common binomial factor: ( a − 2 b ) ( x − 2 y ) .
The factored expression is ( a − 2 b ) ( x − 2 y ) .
Explanation
Understanding the Problem We are given the expression a x − 2 b x − 2 a y + 4 b y and our goal is to factor it. Factoring is like reverse distribution, where we look for common factors in different terms and pull them out to simplify the expression.
Grouping the Terms First, let's group the terms in the expression to make it easier to identify common factors. We can group the terms as follows: ( a x − 2 b x ) + ( − 2 a y + 4 b y ) .
Factoring Each Group Now, let's factor out the common factor from each group. From the first group ( a x − 2 b x ) , we can factor out x , which gives us x ( a − 2 b ) . From the second group ( − 2 a y + 4 b y ) , we can factor out − 2 y , which gives us − 2 y ( a − 2 b ) . So, the expression becomes x ( a − 2 b ) − 2 y ( a − 2 b ) .
Factoring out the Common Term Notice that ( a − 2 b ) is a common factor in both terms. We can factor it out, which gives us ( a − 2 b ) ( x − 2 y ) . Therefore, the factored form of the given expression is ( a − 2 b ) ( x − 2 y ) .
Final Answer So, the factored form of the expression a x − 2 b x − 2 a y + 4 b y is ( a − 2 b ) ( x − 2 y ) .
Examples
Factoring is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or circuits. Similarly, economists use factoring to analyze economic models and predict market behavior. Factoring helps in simplifying expressions, solving equations, and understanding relationships between variables, making it a valuable tool in various fields.
The expression a x − 2 b x − 2 a y + 4 b y can be factored to ( a − 2 b ) ( x − 2 y ) . This is accomplished by grouping the terms, factoring out common factors, and then extracting the common binomial. Therefore, the final factored result is ( a − 2 b ) ( x − 2 y ) .
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