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FACTORING TRINOMIALS OF THE FORM   x2 + bx + c  , WHERE  c > 0

Jump right to the exercises!

Before doing this exercise, you may need to study these Basic Concepts Involved in Factoring Trinomials.

Here, you will practice factoring trinomials of the form   x2 + bx + c ,
where   b  and  c  are integers and  c > 0 .
That is, the constant term is positive.

You must first find numbers that multiply to  c  and that add to  b .
Since  c  is positive in this exercise, both numbers will be positive, or both numbers will be negative.
That is, both numbers will have the same sign.

When you add numbers that have the same sign, then in your head you actually do an addition problem.
For example, to mentally add  (-5) + (-3) , in your head you would compute  5 + 3 , and then assign a negative sign to your answer.

The sign of  b  determines the common sign of your numbers.
If  b > 0 , then both numbers will be positive.
If  b < 0 , then both numbers will be negative.

These results are summarized below:

FACTORING TRINOMIALS OF THE FORM   x2 + bx + c ,
WHERE  c  IS POSITIVE
  • Check that the coefficient of the square term is  1 .
  • Check that the constant term (c) is positive.
  • It's easier to do mental computations involving only positive numbers.
    So, if  b  (the coefficient of the  x  term) is negative, you'll initially ignore the minus sign.
    That is, in the next step, notice that you're working with the absolute value of  b , which is positive.
  • Find two numbers that ADD TO  |b|  and MULTIPLY TO  c .
  • Now, you'll use the sign of  b .
    If  b > 0 , then both numbers will be positive.
    If  b < 0 , then both numbers will be negative.
  • Use these two numbers to factor the trinomial, as illustrated in the examples below.
  • Be sure to check your answer using FOIL.

EXAMPLES:
Factor:  x2 + 5x + 6
Thought process:
Find two numbers that add to  5  and multiply to  6 .
 2  and  3  work, since  2 + 3 = 5  and  (2)(3) = 6 .
Since the coefficient of  x  is positive, both numbers will be positive.
The desired numbers are  2  and  3 .
Answer:  x2 + 5x + 6 = (x + 2)(x + 3)
Check: (x+2)(x+3) = x2 + 3x + 2x + 6 = x2 + 5x + 6.

Factor:  x2 - 5x + 6
Thought process:
Find two numbers that add to  5  and multiply to  6 .
(Notice that we initially "throw away" the minus sign on the  5 .)
 2  and  3  work, since  2 + 3 = 5  and  (2)(3) = 6 .
Since the coefficient of  x  is negative, both numbers will be negative.
The desired numbers are  -2  and  -3 .
Answer:  x2 - 5x + 6 = (x - 2)(x - 3)
Check: (x-2)(x-3) = x2 - 3x - 2x + 6 = x2 - 5x + 6.

Factor:   x2 + x + 1
There are no integers that add to  1  and multiply to  1 .
Thus,  x2 + x + 1  is not factorable over the integers.

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.  
Click on "new problem" to get started!


FACTOR: