The difference of perfect squares can be generalized as a factoring technique. By extension, any difference between terms that are raised to an even power like
What about the sum or difference of terms with matching odd powers? How can those be factored?
More Factoring Techniques
Factoring a trinomial of the form
Guess and Check
The educated guess and check method can be time consuming but if the first and last coefficient only have a few factors, there are a finite number of possibilities. Take the expression:
The 6 can be factored into the following four pairs:
The -28 can be factored into the following twelve pairs:
1, -28 or -28, 1
-1, 28 or 28, -1
2, -14 or -14, 2
-2, 14 or 14, -2
4, -7 or -7, 4
-4, -7 or -7, -4
The correctly factored expression will need a pair from the top list and a pair from the bottom list. This is 48 possible combinations to try.
If you try the first pair from each list and multiply out you will see that the first and the last coefficients are correct but the
A systematic approach to every one of the 48 possible combinations is the best way to avoid missing the correct pair. In this case it is:
This method can be extremely long and rely heavily on good guessing which is why other methods are preferable.
Factoring by Grouping
The next factoring technique is factoring by grouping. Suppose you start with an expression already in factored form:
Notice that the first two terms are divisible by both 4 and
To check your work, multiply the binomials and compare it with the original expression.
Usually when you multiply the factored form of a polynomial, two terms can be combined because they are like terms. In this case, there are no like terms that can be combined.
This means that when set equal to zero, this expression is equivalent to
Multiplying by 2 and multiplying by 3 only changes the left hand side of the equation because the right hand side will remain 0. This has the effect of shifting the coefficient from the denominator of the fraction to be in front of the
Another useful and efficient technique is the procedural factoring algorithm. The proof of the algorithm is beyond the scope of this book, but is a reliable technique for getting a handle on tricky factoring questions of the form:
We will factor
First, multiply the first coefficient with the last coefficient and set the first coefficient to 1:
Second, factor as you normally would with
Third, divide the second half of each binomial by the coefficient that was multiplied in step 1:
Fourth, simplify each fraction completely:
Lastly, move the denominator of each fraction to become the coefficient of
When you compare the computational difficulty of the three methods mentioned above, you will see that the factoring algorithm is the most efficient.
Sum or Difference of Matching Odd Powers
The last method of advanced factoring does not involve expressions of the form
This method is shown in the examples below and the pattern is fully explored in the Review.
Earlier, you were asked how the sum and difference of terms with matching odd powers can be factored. The sum or difference of terms with matching odd powers can be factored in a precise pattern because when multiplied out, all intermediate terms cancel each other out.
Notice all the inside terms cancel:
Factor the following expression without using the quadratic formula or trial and error:
Using the factoring algorithm:
Factor each expression completely.
Expand the following expressions. What do you notice?
11. Describe in words the pattern of the signs for factoring the difference of two terms with matching odd powers.
12. Describe in words the pattern of the signs for factoring the sum of two terms with matching odd powers.
Factor each expression completely.
To see the Review answers, open this PDF file and look for section 2.2.