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Find the factors for
Solution:
Consider the given equation above
If you think that you cannot factor the given equation, then you're right because there's no common factor at each terms. How about if you will expand the given equation and combine similar terms, then we can factor the resulting equation if possible? Let's expand the given equation, we have
The above equation is already arranged according to descending power of x. In order to factor a polynomial using synthetic division, you must know the factors of the last term or coefficient. In this case, 27 is the last term. The factors of 27 are 1, -1, 3, -3, 9, -9, 27, and -27. Unfortunately, we cannot use synthetic division since all factors of 27 will give us a remainder.
Don't worry, we can do something for the above equation in order to get the factors. And so, consider again the above equation
Group the first two terms, we have
Remove x² from the group,
We can make the grouped terms into a perfect trinomial square. Divide the coefficient of the middle term which is 8 by 2 and then square it. In this case, we have to add and subtract 16 at the above equation, as follows
Group the next two terms, we have
Rewrite the first group as a square of a binomial and take out the common factor at the next group,
Did you notice that the resulting equation is a quadratic equation in terms of x(x + 4)? The factors of 27 are 3 and 9. Since the middle term is -12, then the factors of 27 must be -3 and -9. If you add -3 and -9, it will give us -12 which is the coefficient of the middle term. Therefore, the factors of the given equation are
