Think about two numbers whose product equals - 21 and add to - 4.
(x+3)(x-7)
Practice makes perfect
To factor a trinomial with a leading coefficient of one, think of the process as multiplying two binomials in reverse. Let's start by taking a look at the constant term.
x^2-4x-21 ⇔ x^2+(- 4)x+(- 21)
In this case, the constant term is -21. This is a negative number, so for the product of the constant terms in the factors to be negative, these constants must have opposite signs (one positive and one negative.)
Factor Constants
Product of Constants
1 and - 21
- 21
-1 and 21
- 21
3 and - 7
- 21
- 3 and 7
- 21
Next, let's consider the coefficient of the linear term.
x^2+(- 4)x+(- 21)
We need the sum of the factors that produced the constant term to equal the coefficient of the linear term, -4.
Factors
Sum of Factors
1 and - 21
- 20
-1 and 21
20
3 and - 7
-4
-3 and 7
4
We found the numbers whose product is - 21 and whose sum is -4. These numbers are 3 and - 7. We can now rewrite the given expression as a product of two factors.
x^2-4x- 21 ⇔ (x+3)(x-7)
