Factor completely.
$8 x^9 y^4+12 x^6 y^2-24 x^3 y^6$
Answer (1)
Find the greatest common factor (GCF) of the coefficients: 8, 12, and -24, which is 4.
Find the GCF of the powers of x: x 9 , x 6 , x 3 , which is x 3 .
Find the GCF of the powers of y: y 4 , y 2 , y 6 , which is y 2 .
Factor out the overall GCF, 4 x 3 y 2 , from the expression: 4 x 3 y 2 ( 2 x 6 y 2 + 3 x 3 − 6 y 4 ) .
Explanation
Understanding the Problem We are asked to factor the expression 8 x 9 y 4 + 12 x 6 y 2 − 24 x 3 y 6 completely. This means we want to find the greatest common factor (GCF) of all the terms and factor it out.
Finding the GCF of the Coefficients First, let's find the GCF of the coefficients: 8, 12, and -24. The factors of 8 are 1, 2, 4, and 8. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The greatest common factor of 8, 12, and 24 is 4.
Finding the GCF of the Powers of x Next, let's find the GCF of the powers of x: x 9 , x 6 , x 3 . The smallest power of x is x 3 , so the GCF is x 3 .
Finding the GCF of the Powers of y Now, let's find the GCF of the powers of y: y 4 , y 2 , y 6 . The smallest power of y is y 2 , so the GCF is y 2 .
Factoring out the GCF Therefore, the overall GCF of the expression is 4 x 3 y 2 . Now we factor out the GCF from each term in the expression:
8 x 9 y 4 = 4 x 3 y 2 ( 2 x 6 y 2 ) 12 x 6 y 2 = 4 x 3 y 2 ( 3 x 3 ) − 24 x 3 y 6 = 4 x 3 y 2 ( − 6 y 4 )
Writing the Factored Expression So, we can rewrite the expression as:
8 x 9 y 4 + 12 x 6 y 2 − 24 x 3 y 6 = 4 x 3 y 2 ( 2 x 6 y 2 + 3 x 3 − 6 y 4 )
Checking for Further Factoring Finally, we check if the expression inside the parenthesis, 2 x 6 y 2 + 3 x 3 − 6 y 4 , can be factored further. In this case, it cannot be factored further. Therefore, the completely factored expression is:
4 x 3 y 2 ( 2 x 6 y 2 + 3 x 3 − 6 y 4 )
Examples
Factoring polynomials 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. Imagine you are designing a rectangular garden and want to know the possible dimensions given a certain area. If the area can be expressed as a polynomial, factoring that polynomial can give you the possible lengths and widths of the garden. This helps in optimizing space and resources.
