Factoring in Elementary Algebra

Right now we are trying to learn how to factor in our Elementary Algebra course (Algebra 1).  Many students find this topic to be very difficult... but it is important to get it right because factoring is an essential skill in mathematics. For example,

The general factoring guidelines found in the free Elementary Algebra textbook offered on the Flat World Knowledge website follow:

1. Check for common factors. If the terms have common factors, then factor out the greatest common factor (GCF) and look for the resulting polynomial factors to factor further.
2. Determine the number of terms in the polynomial:

• Four term polynomials are factored by grouping.
• Trinomials (3 terms) are factored using “trial and error” or the AC method.
• Binomials (2 terms) are factored using the following special products.

3. Look for factors that can be factored further.
4. Check by multiplying.


* If a binomial is both difference of squares and cubes, then we should factor it as difference of squares first to obtain a more complete factorization.
* Not all polynomials with integer coefficients factor, in this case, we say that the polynomial is prime.

Solution: This four term polynomial has a GCF of 3x , we factor this out first.

We now factor the resulting four term polynomial by grouping.

The factor x^2 - 1 is a difference of squares and can be factored further.

And therefore we have,

You can check by multiplying.  A similar problem can be found on YouTube:

You will find many more embedded videos in the Elementary Algebra textbook, here are links to the appropriate sections:

6.1 Introduction to Factoring
6.2 Factoring Trinomials x^2 + bx + c
6.3 Factoring Trinomials ax^2 + bx + c
6.4 Factoring Binomials
6.5 General Guidelines for Factoring Polynomials

Learning how to factor takes lots of time and practice.

Adding and Subtracting Rational Expressions

To simplify a rational expression:
Step 1:  Factor all denominators as a means to easily determine the LCD.
Step 2:  Multiply by the appropriate factors to obtain equivalent terms with a common denominator. 
Step 3:  Add or subtract the numerators and place the result over the common denominator. 
Step 4:  Simplify the resulting algebraic fraction.



A more involved example follows:

It is always a good idea to state the restrictions.  Remember that the restrictions are the values that produce zero in the denominator.

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