Renee is asked to find a factor of x3−5x2+6x, given that x is one of the linear factors. Which of the following is another correct factor?(1 point)
x−2
x+3
x+2
x^2+5x+6
To find a factor of a polynomial, we can use synthetic division.
Let's use x = 2 as a potential factor.
Starting with the coefficients of our polynomial: 1, -5, 6, 0
Using synthetic division:
2 | 1 -5 6 0
- 2 -6 0
-----------------
1 -3 0 0
The result of synthetic division gives us the remainder 0. This means that (x-2) is a factor of the polynomial x^3 - 5x^2 + 6x.
Therefore, the correct factor from the given options is x-2.
To find the factor of the given polynomial x^3 - 5x^2 + 6x, we can use the Factor Theorem. According to the Factor Theorem, if x is a factor of a polynomial, then the polynomial will be equal to zero when x is substituted into it.
By substituting x = 2 into the polynomial, we can check if x - 2 is indeed a factor.
(2)^3 - 5(2)^2 + 6(2) = 8 - 20 + 12 = 0
Since x - 2 evaluates to zero, x = 2 is a factor of the polynomial.
Therefore, the correct factor from the given options is x - 2. Hence, the correct answer is (1 point) x - 2.
To determine which of the given options is a correct factor of the cubic polynomial x^3 - 5x^2 + 6x, we can use synthetic division or long division to check one by one. However, we can also simplify the process by applying the remainder theorem.
The remainder theorem states that if a polynomial f(x) is divided by x - a, the remainder will be f(a).
Let's apply the remainder theorem to each of the options:
1. x - 2:
Plug in a = 2 into the polynomial:
(2)^3 - 5(2)^2 + 6(2) = 8 - 20 + 12 = 0
The remainder is 0, which means (x - 2) is a factor.
2. x + 3:
Plug in a = -3 into the polynomial:
(-3)^3 - 5(-3)^2 + 6(-3) = -27 - 45 - 18 = -90
The remainder is not equal to 0, so (x + 3) is not a factor.
3. x + 2:
Plug in a = -2 into the polynomial:
(-2)^3 - 5(-2)^2 + 6(-2) = -8 - 20 - 12 = -40
The remainder is not equal to 0, so (x + 2) is not a factor.
4. x^2 + 5x + 6:
This is not a linear factor of the polynomial, so it does not fit the given condition.
Therefore, the correct factor among the given options is (x - 2).