Question 683521: factor the trinomial completely 12x*3-27x*2-27x
Answer by RedemptiveMath(80)   (Show Source):
You can put this solution on YOUR website! You would want to start simplifying this expression by looking at each piece and seeing what can be done to the whole. In other words, we want to figure out what can be "factored" out of the expression, but what we factor out has to affect each piece of the polynomial. We can say a piece contains a coefficient, its sign, its variable, and the variable's exponent. The more technical way of expressing a polynomial's pieces would be to separate the coefficients and the variables, but this grouping works fine. Seeing that these pieces share a common variable x, we can factor that out of the expression:
12x^3-27x^2-27x = x(12x^2-27x-27).
Essentially, we are performing division on each of the three pieces of this polynomial. We are dividing each piece by a common factor x. By the law of exponents concerning division, x^3 becomes x^2 and x^2 becomes x when dividing by x. [If we have two different exponents n and m such that (a^n)/(a^m), we have a^(n-m)].
Now that we have done the simpler part, we must look at the coefficients. What do 12, -27, and -27 all have in common? Well, they can all be divisible by 3. So,
x(12x^2-27x-27) = 3x(4x^2-9x-9).
We divide each piece by 3 and place it to the left of the first parenthesis as x. When we factor a second, third, or more quantity out of the expression, we multiply it to anything we have already factored. For this instance, we multiply x by 3 to get 3x on the outside. Can we do anything else to the expression inside of the parentheses? We can't factor anymore variables, and 4 and 9 do not have a common factor. Thus, we must see if the expression can be broken down using the Zero-Product Property. Focusing on the new expression 4x^2-9x-9, we can see that it can be broken down into (4x+3)(x-3). Substituting these two parentheses into the place of the original and multiplying still by 3x, we have the answer 3x(4x+3)(x-3).
In review, start by seeing if all of the numbers (coefficients) share a common variable. Then, focus on the coefficients themselves. Finally, see if the expression can be broken down using the Zero-Product Property. You are able to start with the Zero-Product Property first and get (12x^2+9x)(x-3). Then, you would have to factor 3x out of the first parenthesis. However, starting this way may be more difficult given the larger numbers and the 3rd degree exponent.
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