Which of the following is a factor of x3+5x2+6x
, given that x is one of the linear factors?
Option #1: x−2
Option #2: x−3
Option #3: x2+5x+6
(1 point)
Option #1: x−2 is a factor of x^3+5x^2+6x if x=2.
Option #2: x−3 is a factor of x^3+5x^2+6x if x=3.
Option #3: x^2+5x+6 is not a factor of x^3+5x^2+6x.
Therefore, the correct answer is option #2: x−3.
To determine which of the options is a factor of the given polynomial x^3 + 5x^2 + 6x, we can use the Remainder theorem. According to the theorem, if x = a is a factor of a polynomial, then the polynomial will have a remainder of 0 when a is substituted into it.
Let's check each option by substituting the given values of x into the polynomial:
Option #1: x - 2
Substituting x = 2, we get:
(2)^3 + 5(2)^2 + 6(2) = 8 + 20 + 12 = 40
Since the value is not equal to 0, x - 2 is not a factor.
Option #2: x - 3
Substituting x = 3, we get:
(3)^3 + 5(3)^2 + 6(3) = 27 + 45 + 18 = 90
Since the value is not equal to 0, x - 3 is not a factor.
Option #3: x^2 + 5x + 6
Substituting x = -1, we get:
(-1)^3 + 5(-1)^2 + 6(-1) = -1 + 5 + (-6) = -2
Again, the value is not equal to 0, so x^2 + 5x + 6 is not a factor.
Therefore, none of the given options is a factor of x^3 + 5x^2 + 6x.
To determine which of the given options is a factor of x^3 + 5x^2 + 6x, we need to check if substituting the value of x in each option results in the expression becoming zero (i.e., a factor should make the polynomial equal to zero).
Let's go through each option:
Option 1: x - 2
If we substitute x with 2 in x - 2, we get:
2 - 2 = 0, which means the polynomial becomes zero.
Option 2: x - 3
If we substitute x with 3 in x - 3, we get:
3 - 3 = 0, which means the polynomial becomes zero.
Option 3: x^2 + 5x + 6
If we substitute x with any value, the resulting expression will not be zero. So, option #3 is not a factor of the given polynomial.
Therefore, both option #1 (x - 2) and #2 (x - 3) are factors of x^3 + 5x^2 + 6x.