Factorise: $8 x^2 y^2+4 x^2-12 x^2 y$
Answer (2)
Identify the greatest common factor (GCF) of the terms: 4 x 2 .
Factor out the GCF from each term: 8 x 2 y 2 + 4 x 2 − 12 x 2 y = 4 x 2 ( 2 y 2 + 1 − 3 y ) .
Rewrite the expression in factored form: 4 x 2 ( 2 y 2 − 3 y + 1 ) .
The final factored expression is: 4 x 2 ( 2 y 2 − 3 y + 1 ) .
Explanation
Understanding the Problem We are asked to factorise the expression 8 x 2 y 2 + 4 x 2 − 12 x 2 y . Factoring involves finding common factors in each term and extracting them to simplify the expression.
Finding the GCF of Coefficients First, let's identify the greatest common factor (GCF) of the coefficients: 8, 4, and -12. The GCF is 4, as we found using the python tool.
Finding the GCF of Variables Next, let's look at the variables. Each term contains x 2 , so x 2 is a common factor. The first term has y 2 , the second term has no y , and the third term has y . Therefore, the lowest power of y that appears in all terms is y 0 = 1 , so y is not a common factor.
Determining the Overall GCF Combining the GCF of the coefficients and the variables, the greatest common factor (GCF) of the entire expression is 4 x 2 .
Factoring out the GCF Now, we factor out the GCF 4 x 2 from each term:
8 x 2 y 2 = 4 x 2 ( 2 y 2 ) ,
4 x 2 = 4 x 2 ( 1 ) ,
− 12 x 2 y = 4 x 2 ( − 3 y ) .
Writing the Factored Expression So, we can rewrite the expression as:
8 x 2 y 2 + 4 x 2 − 12 x 2 y = 4 x 2 ( 2 y 2 ) + 4 x 2 ( 1 ) + 4 x 2 ( − 3 y ) = 4 x 2 ( 2 y 2 + 1 − 3 y ) .
Final Factored Form Therefore, the factored form of the expression is 4 x 2 ( 2 y 2 − 3 y + 1 ) .
Examples
Factoring is a fundamental concept in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or circuits. In finance, factoring can help in analyzing investment portfolios by breaking down complex financial models into simpler components. Understanding factoring helps in optimizing resources and making informed decisions in various fields.
To factor the expression 8 x 2 y 2 + 4 x 2 − 12 x 2 y , we first identify the greatest common factor as 4 x 2 . We factor this out to get the expression as 4 x 2 ( 2 y 2 − 3 y + 1 ) .
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