factor polynomial

First you'll want to see if each term in the polynomial has any common factors, in this case 63 and 7 both have a common factor of 7, since 63 = 9*7 and 7 = 1*7. So we factor out their greatest common factor, which in this case is 7:

63v^{2}-7 =

7(9v^{2}-1)

Can we factor (9v^{2}-1) any more? Well, we can't factor it by trying to pull out greatest common factors from each term, since 9 and 1 do not share any factors greater than 1. But it can actually be factored in a different way...

9v^{2}-1 is in a special kind of form, that requires a special kind of factoring called "Differences of Squares."

(a^{2}-b^{2}) = (a - b)(a + b)

We can use this formula for 9v^{2}-1 since 9v^{2} is the sqaure of 3v and 1 is the square of 1. So we write:

7(9v^{2}-1) =

7(3v-1)(3v+1)

Which is your final answer since it cannot be factored any further.