Before we try to factor this expression, we should make sure that the leading coefficient is 1. To do this, we will need to factor out -1 from the expression.
- x^2+3x+40 ⇔ - (x^2-3x-40)
The result of factoring out -1 from the given expression is a trinomial with a leading coefficient of 1.
Factor the Expression
To factor a trinomial with a leading coefficient of 1, think of the process as multiplying two binomials in reverse. Let's start by taking a look at the constant term.
x^2-3x - 40
In this case, we have -40. This is a negative number, so for the product of the constant terms in the factors to be negative, these constants must have the opposite sign (one positive and one negative).
Constants
Product of Constants
1 and -40 or -1 and 40
-40
2 and -20 or -2 and 20
-40
4 and -10 or -4 and 10
-40
5 and -8 or -5 and 8
-40
Next, let's consider the coefficient of the linear term.
x^2 - 3x - 40
For this term, we need the sum of the factors that produced the constant term to equal the coefficient of the linear term, -3. We will be checking each pair until we get this result.
Constants
Sum of Constants
1 and -40
-39
-1 and 40
39
2 and -20
-18
-2 and 20
18
4 and -10
-6
-4 and 10
6
5 and -8
-3
We found the factors whose product is -40 and whose sum is -3.
x^2 - 3x - 40 ⇔ (x+5)(x-8)
Wait! Before we finish, remember that we factored out - 1 from the original expression. To fully complete the factored expression, let's reintroduce the minus sign now.
- x^2+3x+40 ⇔ -(x+5)(x-8)
Checking Our Answer
Check your answer âś“
We can check our answer by applying the Distributive Property and comparing the result with the given expression.
After applying the Distributive Property and simplifying, the result is the same as the given expression. Therefore, we can be sure our solution is correct!
