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Special Products - Factoring, 30

__Category__: Algebra

"Published in Newark, California, USA"

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, 49 is the last term. The factors of 49 are 1, -1, 7, -7, 49, and -49. Unfortunately, we cannot use synthetic division since all
factors of 49 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 2 by 2 and then square it. In
this case, we have to add and subtract 1 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 + 1)? The quadratic equation is a perfect trinomial square since the coefficient of the middle term which is -14 when you divide it by 2 and then square it, it will give us 49 which is the same as the last term. Therefore,
the factors of the given equation are
Since the sign of the middle term is negative, then the square root of the last term must be negative.