Which of the following is a factor of x^3+5x^2+6x, given that x is one of the linear factors?
Option #1: x−2
Option #2: x−3
Option #3: x^2+5x+6
(1 point)
Option #2: x−3
To determine which of the given options is a factor of the polynomial x^3 + 5x^2 + 6x, we need to check if substituting the value of x from each option will result in the polynomial evaluating to zero.
Let's go through each option:
Option #1: x−2
For this option to be a factor, we need to verify if the polynomial evaluates to zero when x = 2.
Substituting x = 2 into the polynomial, we get:
(2)^3 + 5(2)^2 + 6(2)
= 8 + 20 + 12
= 40
Since the polynomial does not evaluate to zero when x = 2, Option #1 is not a factor.
Option #2: x−3
For this option to be a factor, we need to verify if the polynomial evaluates to zero when x = 3.
Substituting x = 3 into the polynomial, we get:
(3)^3 + 5(3)^2 + 6(3)
= 27 + 45 + 18
= 90
Since the polynomial does not evaluate to zero when x = 3, Option #2 is not a factor.
Option #3: x^2+5x+6
For this option to be a factor, we need to verify if the polynomial evaluates to zero when x = -2.
Substituting x = -2 into the polynomial, we get:
(-2)^3 + 5(-2)^2 + 6(-2)
= -8 + 20 - 12
= 0
Since the polynomial evaluates to zero when x = -2, Option #3 is a factor.
Therefore, the correct answer is Option #3: x^2+5x+6.