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This is one of the pages in the third section about factoring.   It helps you FACTOR TRINOMIALS

In the first page, I presented the perfect square trinomial and the difference of two squares.

In the second page, I presented a method of factoring four-term expressions with pairs of factors.

Pages 3a, 3b, 3c, and 3d give three methods of factoring trinomials:

Page 3a (this page) works by listing the factors of the "a" and "c" coefficients.
Page 3b completes the square.
Page 3c is the "AC Method", which has you multiply the "a" and "c" coefficients, and list all the factors of that product.
Page 3d is the "Simplified AC Method"


6x² + 19x - 20  

Using the F.O.I.L. (First, Outer, Inner, Last) method, you can simplify any of these guesses, and see which, if any, are right.  Notice that the first term of each factor must multiply out to 6x².  Neither the outer, inner, nor last terms have two "x's".  Notice further that the last term of each factor must multiply out to -20.  That's because the first, outer, and inner terms each have an "x".

To summarize what I've said so far:

  1. The trinomial has an x� coefficient, an x-coefficient, and a constant.
  2. Its factors (if it can be factored) both have an x-coefficient and a constant.
  3. The trinomial's x� coefficient is the product of the x-coefficients of its two factors.
  4. The trinomial's constant is the product of the constants in its two factors.

So the way to factor a trinomial is to

  1. list all the pairs of x-coefficients in factors that could multiply out to the x� coefficient of the trinomial,
  2. list all the pairs of constants in factors that could multiply out to the constant of the trinomial.
  3. multiply out all the possible combinations of these factors that you listed, to see which ones add up to the right x-coefficient of the trinomial.

In example 5, which was to factor 6x² + 19x - 20, we will list all the pairs of factors of 6, and the pairs of factors of 20, in a table like this one:

x's \ constants (1,-20) (2,-10) (4,-5) (5,-4) (10,-2) (20,-1)

The x-coefficients are listed down the left column, and the constants are listed across the top.  Inside the table, you should put the sum of the pairwise products of these pairs of numbers.  For example, the sum of the pairwise product of (1,6) and (1,-20) is (1) (1) + (6) (-20), which is 1 - 120, or -119.

x's \ constants (1,-20) (2,-10) (4,-5) (5,-4) (10,-2) (20,-1)
(1,6) -119          

We're looking for a cell that contains 19, which will provide the pairs of coefficients of the factors.  This looks like a mighty big table to fill in, and maybe you should fill it in the first few times just for practice.  But soon, you'll realize there are large numbers of cells you don't need to fill in.  Since 19 is an odd number, it must be the sum of an odd number and an even number, so the column headed (10,-2) needn't be filled in at all.  It can only contain even numbers.

x's \ constants (1,-20) (2,-10) (4,-5) (5,-4) (10,-2) (20,-1)
(1,6) -119 even     even  
(2,3)   even     even  
(3,2)   even     even  
(6,1)   even     even  

Also, the sum of pairwise products (even,odd) with (odd,even) is also even -- this is the sum of even numbers.  For this reason, the following cells can be eliminated as even:

x's \ constants (1,-20) (2,-10) (4,-5) (5,-4) (10,-2) (20,-1)
(1,6) -119 even even   even even
(2,3) even even   even even  
(3,2)   even even   even even
(6,1) even even   even even  

Wow!  With just a little even/odd thinking, we've eliminated two thirds of the possibilities.  Now, just fill in all the cells that haven't already been eliminated:

x's \ constants (1,-20) (2,-10) (4,-5) (5,-4) (10,-2) (20,-1)
(1,6) -119 even even -19 even even
(2,3) even even -7 even even 37
(3,2) -37 even even 7 even even
(6,1) even even
even even 119

Now it's easy to pick out the "19", which matches the middle coefficient of the original trinomial.  That means the x-coefficients of the factors are 6 and 1, and the constants of the factors are 4 and -5.  Putting them together, the factors are (6x-5) and (x+4).  This matches answer C, above.

Was that too hard?  Maybe you'll find it easier to use the method on Page 3b: Complete the Square.  This is another method of factoring trinomials.

Related pages in this website

Next: Completing the Square


The webmaster and author of this Math Help site is Graeme McRae.