## Sunday, November 10, 2013

### 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.