## Factoring Trinomials of the Form x² + bx + c when c is Negative

Read this article. Pay attention to the overview of the steps for these problems. Read the examples and note where factoring is not possible, like in the last example.
After you have reviewed the materials, complete a few practice problems until you feel comfortable. Be sure to check your answers when you are done.

Here, you will practice factoring trinomials of the form , where and are integers, and .

That is, **the constant term is negative**.

Recall that the **integers** are: ..., , , , , , , ...

As discussed in Basic Concepts Involved in Factoring Trinomials, you must first find two numbers that add to and that multiply to , since then:

Since is negative in this exercise, one number will be positive, and the other will be negative.

How can two numbers multiply to give a negative result? One must be positive, and the other 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 , in your head you would compute , and then assign a negative sign to your answer.

Think of it this way: Start at zero on a number line. Walk units to the left, and units to the right. You end up at . You walked farther to the left than you did to the right, so your final answer is negative.

The sign of (the coefficient of the term) determines which number will be positive, and which will be negative:

If , then the bigger number (the one farthest from zero) will be positive.

If , then the bigger number
(the one farthest from zero) will be negative.

In other words, the **biggest** number takes the sign (plus or minus) of .

These results are summarized below:

Factoring Trinomials of the Form ,

- Check that the coefficient of the square term is .
- Check that the constant term is negative.
**It is easier to do mental computations involving only positive numbers.**

So, you will initially ignore all minus signs and just work with the numbers and .- Find two numbers whose DIFFERENCE is and whose PRODUCT is .

That is, find two numbers that**subtract**to give and that**multiply**to give . - Now (and only now), you will use the actual plus-or-minus sign of .

If , then the bigger of your two numbers is positive; the other is negative.

If , then the bigger of your two numbers is negative; the other is positive.

That is, the**biggest**number takes the sign (plus or minus) of . - Use these two numbers to factor the trinomial, as illustrated in the examples below.
- Be sure to check your answer using FOIL.

### Examples

Solution: Thought process: Is the coefficient of the term equal to ? Check!

Is the constant term negative? Check!

Find two numbers whose difference is and whose product is .

That is, find two numbers that subtract

to give and that multiply to give .

The numbers and work, since and .

Since the coefficient of is positive, the bigger number will be positive, and the other will be negative.

The desired numbers are therefore and .

Then

**Solution**: Thought process: Is the coefficient of the term equal to ?Check!

Is the constant term negative? Check!

Find two numbers whose difference is and whose product is .

That is, find two numbers that subtract to give and that multiply to give .

The numbers and work, since and .

Since the coefficient of is negative, the bigger number will be negative, and the other will be positive.

The desired numbers are therefore and .

Then,

**Solution**: There are no integers whose difference and product are both .

Thus, is not factorable over the integers.

Source: Tree of Math, https://www.onemathematicalcat.org/algebra_book/online_problems/fac_tri_one_cneg.htm

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