Factor the following polynomial completely:

Question

10a^7b^3-810a^3b^3

Answer 1

Start by factoring out everything you can...

 

10a⁷b³ - 810a³b³    Notice there is a 10, and a³ and a b³ in both terms.  Factor this out.

 

10a³b³(a⁴ - 81)   Now, in the parentheses you have the difference of squares.  

 

To factor a difference of squares, you need to memorize that a² - b² - (a - b)(a + b)

 

So  a⁴ - 81 is   [a²]² - [9]²  so in the formula, a = a²  and b = 9

 

a⁴ - 81 = (a² - 9)(a² + 9)

 

All together you have:

 

10a³b³(a² - 9)(a² + 9)   but you are not finished, because a² - 9 is another difference of squares and can also be factored.

 

a² - 9 = [a]² - [3]²  = (a-3)(a+3)

 

So NOW the final, completely factored polynomial is:

 

10a³b³(a - 3)(a + 3)(a² + 9)