Renee is asked to find a factor of x^3−5x^2+6x, given that x is one of the linear factors. Which of the following is another correct factor?(1 point)
A. x−2
B. x+3
C. x+2
D. x^2+5x+6
To find a factor of x^3-5x^2+6x, we can plug in the given value for x and check if the expression equals zero.
For option A, x-2:
(x-2)^3 - 5(x-2)^2 + 6(x-2) = x^3 - 6x^2 + 12x - 8 - 5(x^2 - 4x + 4) + 6x - 12
= x^3 - 6x^2 + 12x - 8 - 5x^2 + 20x - 20 + 6x - 12
= x^3 - 11x^2 + 38x - 40
Since the expression is not equal to zero, x-2 is not a factor of x^3-5x^2+6x.
For option B, x+3:
(x+3)^3 - 5(x+3)^2 + 6(x+3) = x^3 + 9x^2 + 27x + 27 - 5(x^2 + 6x + 9) + 6x + 18
= x^3 + 9x^2 + 27x + 27 - 5x^2 - 30x - 45 + 6x + 18
= x^3 + 4x^2 - 36x
Since the expression is not equal to zero, x+3 is not a factor of x^3-5x^2+6x.
For option C, x+2:
(x+2)^3 - 5(x+2)^2 + 6(x+2) = x^3 + 6x^2 + 12x + 8 - 5(x^2 + 4x + 4) + 6x + 12
= x^3 + 6x^2 + 12x + 8 - 5x^2 - 20x - 20 + 6x + 12
= x^3 + x^2 - 2x
Since the expression is equal to zero, x+2 is a factor of x^3-5x^2+6x.
For option D, x^2+5x+6:
Plugging x^2+5x+6 into x^3-5x^2+6x gives us a long and complicated expression, which is not convenient to calculate. Hence, we cannot determine if x^2+5x+6 is a factor using direct substitution.
Therefore, the correct answer is C. x+2.
To find a factor of the polynomial x^3 - 5x^2 + 6x, we are given that x is one of the linear factors.
To determine which of the given options is another correct factor, we need to check if it satisfies the polynomial equation when substituted for x.
Let's evaluate each option one by one:
A. x - 2
Substituting x - 2 into the polynomial, we get (x - 2)^3 - 5(x - 2)^2 + 6(x - 2) = x^3 - 6x^2 + 12x - 8 - 5(x^2 - 4x + 4) + 6x - 12 = x^3 - x^2 + 4x - 8 - 5x^2 + 20x - 20 + 6x - 12 = x^3 - 6x^2 + 19x - 40
Since this is not equal to the original polynomial, x - 2 is not a correct factor.
B. x + 3
Substituting x + 3 into the polynomial, we get (x + 3)^3 - 5(x + 3)^2 + 6(x + 3) = x^3 + 9x^2 + 27x + 27 - 5(x^2 + 6x + 9) + 6x + 18 = x^3 + 9x^2 + 27x + 27 - 5x^2 - 30x - 45 + 6x + 18 = x^3 + 4x^2 - 3x = x(x^2 + 4x - 3)
Since this is not equal to the original polynomial, x + 3 is not a correct factor.
C. x + 2
Substituting x + 2 into the polynomial, we get (x + 2)^3 - 5(x + 2)^2 + 6(x + 2) = x^3 + 6x^2 + 12x + 8 - 5(x^2 + 4x + 4) + 6x + 12 = x^3 + 6x^2 + 12x + 8 - 5x^2 - 20x - 20 + 6x + 12 = x^3 + x^2 - 2x = x(x^2 + x - 2)
Since this is not equal to the original polynomial, x + 2 is not a correct factor.
D. x^2 + 5x + 6
Substituting x^2 + 5x + 6 into the polynomial, we get (x^2 + 5x + 6)^3 - 5(x^2 + 5x + 6)^2 + 6(x^2 + 5x + 6) = x^6 + 15x^5 + 81x^4 + 195x^3 + 246x^2 + 162x + 54 - 5(x^4 + 10x^3 + 45x^2 + 60x + 36) + 6(x^2 + 5x + 6) = x^6 + 15x^5 + 81x^4 + 195x^3 + 246x^2 + 162x + 54 - 5x^4 - 50x^3 - 225x^2 - 300x - 180 + 6x^2 + 30x + 36 = x^6 + 10x^5 - 19x^4 - 80x^3 + 37x^2 - 102x + 54
Since this is not equal to the original polynomial, x^2 + 5x + 6 is not a correct factor.
Therefore, none of the given options is another correct factor.