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### Synthetic Division in Factoring

```Date: 04/18/2002 at 01:25:13
From: Brandon
Subject: Factoring completely

Factor completely: 2x^3 + 17x^2 + 58x + 25.

I don't know what to do or where to begin.
```

```
Date: 04/18/2002 at 09:01:23
From: Doctor Rick
Subject: Re: Factoring completely

Hi, Brandon.

This is not an easy type of problem to solve! The approach I would
take is to use synthetic division to search for one root. If you are
not familiar with this method, see the following answer in our Dr.
Math Archives:

Synthetic Division
http://mathforum.org/library/drmath/view/53056.html

Synthetic division can be viewed in two ways: as a way to evaluate a
polynomial in x at a particular value x = a, or as a way to divide
the polynomial by (x-a). Use this method to test possible roots.

If the cubic has any rational roots (and if it doesn't, you won't be
able to find an exact factorization by this method), they must be of
the form of plus or minus a factor of 25 (the constant term) divided
by a factor of 2 (the coefficient of the highest power). Thus the
numerator can be 1, 5, or 25, and the denominator can be 1 or 2, and
the sign can be positive or negative.

Synthetic division has a property that is very useful in searching
for roots: If all the numbers below the line (in the notation used in
the archived explanation) are positive, then you know that there are
no roots greater than the number you tested. On the other hand, if
the numbers below the line alternate positive and negative, then
there are no roots less than the number you tested.

Thus, I suggest that you start by testing x = 0. This will tell you
right away whether the polynomial may have negative roots, positive
roots, or both. Continue in the same way (something like a "binary
search"): each time you choose a possible root to test, choose one
that is in the middle of the range of possible roots that you haven't
already eliminated. It may turn out that this one test eliminates all
possible roots less than the test number, or all possibilities greater
than the test number, thus giving you only half as many possibilities
to test.

Once you find a value of x for which the polynomial evaluates to zero
(in other words, a factor (x-a) of the polynomial), you have also
found the other factor: just use the numbers below the line of your
synthetic division as the coefficients of a polynomial. In your case,
factor this quadratic, if it can be done with real coefficients. (Or,
if you have learned about polynomials with complex coefficients, you
can always factor the quadratic.) You can use familiar methods for
factoring a quadratic, or you can use the sure-fire method of setting
formula.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Basic Algebra
High School Polynomials

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