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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 negative.

You must first find numbers that multiply to  c  and that add to  b .
Since  c  is negative in this exercise, one number will be positive, and the other will be negative.
That is, the numbers will have different signs.

When you add numbers that have different signs, then in your head you actually do a subtraction 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.
Think of it this way:   Start at zero on a number line. Walk  5  units to the left, and  3  units to the right. You end up at  -2 .
You walked farther to the left than you did to the right, so your final answer is negative.

The sign of  b  determines which number will be positive, and which will be negative:
If   b > 0 , then the bigger number (the one farthest from zero) will be positive.
If   b < 0 , then the bigger number (the one farthest from zero) will be negative.

These results are summarized below:

FACTORING TRINOMIALS OF THE FORM   x2 + bx + c ,
WHERE  c  IS NEGATIVE
  • Check that the coefficient of the square term is  1 .
  • Check that the constant term (c) is negative.
  • It's easier to do mental computations involving only positive numbers.
    So, you will initially ignore all minus signs and just work with the numbers  |b|  and  |c| .
  • Find two numbers whose DIFFERENCE is  |b|  and whose PRODUCT is  |c| .
    That is, find two numbers that subtract to give you  |b|  and that multiply to give you  |c| .
  • Now, you'll use the actual sign of  b .
    If  b > 0 , then the bigger of your two numbers is positive; the other is negative.
    If  b < 0 , then the bigger of your two numbers is negative; the other is positive.
  • 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:
Check that the coefficient of the squared term is  1 , and the constant term is negative.
Find two numbers whose difference is  5  and whose product is  6 .
That is, you want two numbers that subtract to give you  5 , and that multiply to give you  6 .
 6  and  1  work, since  6 - 1 = 5  and  (6)(1) = 6 .
Since  b = 5  is positive, the bigger number (6) will be positive; the other will be negative.
The desired numbers are  6  and  -1 .
Answer:  x2 + 5x - 6 = (x + 6)(x - 1)
Check: (x+6)(x-1) = x2 - x + 6x - 6 = x2 + 5x - 6.

Factor:  x2 - 5x - 6
Thought process:
Check that the coefficient of the squared term is  1 , and the constant term is negative.
Find two numbers whose difference is  5  and whose product is  6 .
That is, you want two numbers that subtract to give you  5 , and that multiply to give you  6 .
 6  and  1  work, since  6 - 1 = 5  and  (6)(1) = 6 .
Since  b = -5  is negative, the bigger number (6) will be negative; the other will be positive.
The desired numbers are  -6  and  1 .
Answer:  x2 - 5x - 6 = (x - 6)(x + 1)
Check: (x-6)(x+1) = x2 + x - 6x - 6 = x2 - 5x - 6.

Factor:   x2 + x - 1
There are no integers whose difference and product are both  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!


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