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)
Responses
x2+5x+6
x squared plus 5 x plus 6
x+2
x plus 2
x−2
x minus 2
x+3
x plus 3
To find a factor of x^3 - 5x^2 + 6x when x is already a linear factor, you can divide the polynomial by x and see what remains.
(x^3 - 5x^2 + 6x) ÷ x = x^2 - 5x + 6
The remaining polynomial is x^2 - 5x + 6. Therefore, another correct factor is x^2 + 5x + 6. So, the correct option is:
x^2 + 5x + 6
or
x squared plus 5x plus 6
To find a factor of a polynomial, we need to see if the polynomial is divisible by a linear expression. In this case, we are given that x is one of the linear factors.
To verify if x is a factor, we can use the Remainder Theorem. According to the Remainder Theorem, if a polynomial P(x) is divided by x-a and the remainder is zero, then (x-a) is a factor of P(x).
So, we need to divide the given polynomial x^3 - 5x^2 + 6x by x and see if the remainder is zero.
Dividing x^3 - 5x^2 + 6x by x, we get:
(x^3 - 5x^2 + 6x) / x = x^2 - 5x + 6
Now, we have the quotient x^2 - 5x + 6. We need to check if it can be factored further.
To factor x^2 - 5x + 6, we need to find two numbers whose product is 6 and whose sum is -5. The numbers are -2 and -3.
Therefore, the factored form of x^2 - 5x + 6 is (x - 2)(x - 3).
From the given options, x - 2 is the correct factor of the polynomial x^3 - 5x^2 + 6x. This means that (x - 2) is another correct factor.
So the correct answer from the options is: x - 2.