I am not totally understanding this process.
Factor Completely:
x^5y^2 - x^4y^2 - 2x^3y^2
Complete factoring is the process of breaking the expression down into a multiplicative product of terms that can't be broken down any further. Here, you can simplify the expression you're given quite a bit just by noting that all three terms in it are divisible by both x^3 and y^2, so there are two of your factors already. The whole expression can therefore be expressed as:
(x^3) times (y^2) times (x^2 - x - 2)
Can we do any better than that? Actually yes: that last bracket can be factored into
(x+1) times (x-2)
How did I know that? If I can express (x^2 - x - 2) as the product of two brackets which look like (x+a) times (x+b) for some a and b, then (a times b) must be -2, and (a plus b) must be -1. If a = 1 and b = -2, I'll get the right answer, so I've found a solution that works. I can't see any more simplification I can do, so the final answer should be x^3 times y^2 times (x+1) times (x-2).
Thanks!
To factor the given expression, x^5y^2 - x^4y^2 - 2x^3y^2, we can use the method of common factorization. The idea is to identify any common factors that are present in each term and factor them out.
Step 1: Find the common factors
In this expression, we see that each term contains x^3 and y^2 as common factors. So, we can write the expression as:
x^3y^2 * (x^2 - xy - 2)
Step 2: Factor the remaining quadratic expression
Now, we need to factor the quadratic expression x^2 - xy - 2. Since it cannot be easily factored, we can use the quadratic formula or complete the square method. However, in this case, it seems that the expression cannot be factored any further without the use of complex numbers.
So, the completely factored form of the expression x^5y^2 - x^4y^2 - 2x^3y^2 is:
x^3y^2 * (x^2 - xy - 2)