Factoring Polynomials

Free Math Help Online presents Factoring Polynomials.  In this post I will covered factoring out a common factor from all terms, factoring quadratics, and factoring using the basics of the special rules for perfect squares, difference of squares, and sum and difference of cubes.  In my next post, Factoring Polynomials II, I will cover factoring by grouping and recognizing special forms.  Factoring Polynomials III will cover factoring negative and fractional exponents, and putting it all together.  The following post will cover Solving Polynomial Equations.

Factoring is a skill you will need starting in high school algebra and will continue using in just about every other math course you will ever take even past calculus.  The more complicated the math course becomes, the more complicated factoring will be required.

Factors are just terms that multiply together.  For example, 63 can be written as 3·21 or 7·9.  This means 3, 7, 9, and 21 are all factors of 63 (so are 1 and 63, but those are boring factors).

Factoring just means finding factors.  Factoring polynomials means finding polynomial factors.  For example:

x² + 5x + 4 = (x + 1)(x + 4)

This is because multiplying the factors on the right gives you the quadratic on the left.

Now, without further ado, here are some of the common things to know for factoring:

1. Common Factors: This is the easiest thing to do in factoring, but it is often overlooked after more complicated factoring methods are covered.  Let's start with this expression:
   63x - 42
   
   Notice that the coefficient of each term has a common factor of 21.  This can be written as
   63x - 42 = 21·3x - 21·2 = 21(3x - 2)
   
   We have now factored out the 21.  The factors of 63x - 42 are 21 and 3x - 2.
   
   Now consider this expression:
   21x⁵ + 3x⁴ - 6x²
   
   Again, the coefficients each have a common factor (3 this time), but now there is also a common factor of x²
   21x⁵ + 3x⁴ - 6x² = 3x²·7x³ + 3x²·x² - 3x²·2 = 3x²(7x³ + x² - 2)
   
   
2. Factoring Quadratics: The most common type of factoring question in a high school algebra class is to factor a trinomial (a polynomial with three terms).  Polynomials with highest power 2 are called quadratics.  Quadratics with all non-zero coefficients are tinomials.  A very simple quadratic is the following:
   x² + 5x + 4
   
   We already know how to factor this one because it is written in factored form above.  The question is, how do we figure it out when the answer is not already given.  To solve this problem you need to remember how to multiply binomials (polynomials with two terms), often called the FOIL method.  When multiplying two linear binomials (an x term and a constant term), the First terms always give the x² term in the trinomial, the Outside and Inside terms always add to give the x term of the trinomial, and the Last terms always give the constant term of the trinomial.  This tells us the possible terms to try when attempting to factor a trinomial.  For our example, we start with factoring the First terms:
   (x ___)(x ___)
   
   This part was really easy because there is only one way to factor x².  It gets harder when the coefficient of the x² term is not a 1.  Next, we consider the possible factors of 4 for the Last terms.  There are more choices than you might think at first.  Naturally, 4 = 2·2, so 2 and 2 are a possibility, but 4 = 1·4 also.  Not only that, but 4 = (-2)·(-2) and 4 = (-1)·(-4).  These four different factorizations of 4 result in four different trinomials when used to fill in the blanks above, but only one of then will result in the trinomial we want.  The key is to also consider that the Outside and Inside terms always add to give the x term of the trinomial.  Again, when the coefficient of the x² term is a 1 this is easy because all we need to do is add the two factors.  What factors of 4 add to 5? They are 1 and 4!  We now have the answer:
   x² + 5x + 4 = (x + 1)(x + 4)
   
   Here is another easy one, this time without so many words involved.  Factor:
   x² - 4x - 12
   
   What are the factors of -12 that add to -4? They are -6 and 2
   x² - 4x - 12 = (x - 6)(x + 2)
   
   This process takes more trial and error when the problems get harder.  Factor:
   6x² - 23x + 15
   
   First we need factors of 6x² and 15, then we need to find the combination where the Outside and Inside terms add to -23.  Possible factors of 6x² are 3x·2x and x·6x.  Possible factors of 15 are (-3)·(-5) and (-1)·(-15).  Notice that I ignored the positive factors of 15.  This is because the positive factor will never give me a -23.  Next we need to try all the combinations of our 6x² and 15 factors to find the right set:
   3·(-3) + 2·(-5) = -23? NO
   3·(-5) + 2·(-3) = -23? NO
   3·(-1) + 2·(-15) = -23? NO
   3·(-15) + 2·(-1) = -23? NO
   1·(-3) + 6·(-5) = -23? NO
   1·(-5) + 6·(-3) = -23? YES These make up our Outside and Inside terms
   
   So now we know the correct numbers, all that is left is to write them in the correct order.  Notice that the 1x and -5 are NOT together, nor are the 6x and -3, otherwise they would not get multiplied in FOIL:
    6x² - 23x + 15 = (x - 3)(6x - 5)
   
   
3. There are many special forms that will crop up in your homework, quizzes, and tests.  Some of these are:
   a² + 2ab + b² = (a + b)² Perfect Square (sum)
   a² - 2ab + b² = (a - b)² Perfect Square (difference)
   a² - b² = (a + b)(a - b) Difference of Squares
   a³ + b³ = (a + b)(a² - ab + b²) Sum of Cubes
   a³ - b³ = (a - b)(a² + ab + b²) Difference of Cubes
   
   Some examples of these are:
   25x² - 49 = (5x + 7)(5x - 7)
   27t³ + 8 = (3t + 2)(9t² - 6t + 4)
   9z² - 24x + 16 = (3z - 4)²
   
   These can also be more complicated, but I am just sticking to some simple ones today.

This is really just touching the surface of polynomial factoring.  Go to Factoring Polynomials II to see some of the more complicated scenarios commonly found in algebra courses.