Factor each of the following

Question

35x^10y^3

-49x^7y^7+63x^10y^10

Answer 1

Original problem:   35x¹⁰y³-49x⁷y⁷+63x¹⁰y¹⁰

 

I'm seeing coefficients (numbers in the front of the terms) of 35, -49, and 63.  Look for the largest value that divides evenly into each of the three.  Some people do prime factor lists, such as:

35: 5*7

-49: -1*7*7

63: 3*3*7

Each of them has 7 as a factor, so factor a single 7 out of each term and leave whatever is left:

7(5x¹⁰y³-7x⁷y⁷+9x¹⁰y¹⁰)    We haven't factored the variables, yet, so we're not finished.  If the prime factorization list helped you, it can be done with variables, too:

x¹⁰: x*x*x*x*x*x*x*x*x*x

y³: y*y*y

We don't usually do that, though. :)  It can quickly become a hard thing to look at.  Usually, we identify which variable will limit how much we can factor out of each term.

7(5x¹⁰y³-7x⁷y⁷+9x¹⁰y¹⁰)   So, factor out x⁷y³ from each term.  Use exponent subtraction: x¹⁰/x⁷=x³

 

7x⁷y³(5x¹⁰y³-7x⁷y⁷+9x¹⁰y¹⁰)

          --------  ------   ----------

            x⁷y³   x⁷y³      x⁷y³

 

7x⁷y³(5x³y⁰-7x⁰y⁴+9x³y⁷)    Anything to the 0 power equals 1, so 5x³y⁰ equals 5x³, -7x⁰y⁴ equals -7y⁴.

 

7x⁷y³(5x³-7y⁴+9x³y⁷)

Answer 2

From what I understood of the problem :

 

7x⁷y³(5x³-7y⁴+9x³y⁷)