Question 110148
You first need to realize that the problem is the difference of two cubes, i.e. the cube root of {{{y^3}}} is exactly y, and the cube root of 27 is exactly 3.
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The factorization of the difference of two cubes follows this pattern:
{{{a^3-b^3=(a-b)(a^2+ab+b^2)}}}
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So, substituting:
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{{{y^3-27=(y-3)(y^2+3y+9)}}}
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Which is the final answer because neither of the factors can be factored any further.  How can you tell?  Look at the second factor,{{{(y^2+3y+9)}}}, then calculate {{{b^2-4ac}}} where a, b, and c are the coefficients of the first, second, and third terms -- in this case, they are 1, 3, and 9.  In this case {{{b^2-4ac}}} is a negative number.  Anytime {{{b^2-4ac}}} is negative or not a perfect square, then the expression cannot be factored.
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By the way, the sum of two cubes is factored as follows:
{{{a^3+b^3=(a+b)(a^2-ab+b^2)}}}