To factor a trinomial in the form x^2+bx+c, you need to find two integers whose product is c and whose sum is b.
Sometimes
Practice makes perfect
For a factorable trinomial x^2+bx+c, the absolute value of b will sometimes be less than the absolute value of c. To justify our statement, we will consider two examples.
|b|<|c|
Consider the quadratic expression x^2+2x-3. Let's verify that the absolute value of the coefficient of the linear term is less than the absolute value of the constant term.
b= 2 ⇒ |b|= |2| = 2
c=- 3 ⇒ |c|=|- 3|= 3
Since 2< 3, we can state that |b|<|c|. Now we can check that the given is a factorable trinomial. To do so, we need to find two integers whose product is c=- 3 and whose sum is b=2. Those integers are - 1 and 3.
- 1(3)=- 3 and - 1+3=2
We can now rewrite the linear term and factor the expression.
Since we were able to factor the expression, we can state that x^2+2x-3 is a factorable trinomial in which |b|<|c|.
|b|>|c|
Consider the quadratic expression x^2+2x+1. This time, we must verify that the absolute value of the coefficient of the linear term is less than the absolute value of the constant term.
b=2 &⇒ |b|=|2|= 2
c=1 &⇒ |c|=|1|= 1
Since 2> 1, we can state that |b|>|c|. Now we will see that the given is a factorable trinomial. We need to find two integers whose product is c=1 and whose sum is b=2. Those integers are 1 and 1.
1(1)=1 and 1+1=2
We can rewrite the linear term and factor the expression.
