Question 813290
{{{216x^3-1}}}
The GCF of this is 1 (which we rarely bother factoring out). Among the factoring patterns there are only two which have two terms with a subtraction between them:<ul><li>{{{a^2-b^2=(a+b)(a-b)}}}</li><li>{{{a^3-b^3 = (a-b)(a^2+ab+b^2)}}}</li></ul>So if our expression is a difference of squares or a difference of cubes we can use the appropriate pattern to factor it.<br>
With the {{{x^3}}} this is not a difference of squares. Is it a difference of cubes? Well, the 1 is the same as 1 cubed so that's good. But what about the 216? With a little investigation of perfect cubes we should find that 216 is 6 cubed. So we can rewrite the expression as a difference of cubes:
{{{(6x)^3-(1)^3}}}
Using '6x' for the 'a' and '1' for the 'b', the pattern gives us:
{{{216x^3-1 = (6x)^3-(1)^3 = ((6x)-(1))((6x)^2+(6x)(1)+(1)^2) = (6x-1)(36x^2+6x+1)}}}
Neither factor will factor further so we are finished.