The equation that has the roots 1, 2, 3 is:
(a) x3 + 6x2 – 11x + 6
(b) x3 – 6x2 + 11x – 6 *
(c) x3 + 6x2 + 11x + 6
(d) x3 – 6x2 – 11x – 6
how ?
if the roots are 1,2,3, then
(x-1)(x-2)(x-3) = 0
expand to get (b)
By the way, none of the answers as written are equations. the correct answer would be
x^3 - 6x^2 + 11x - 6 = 0
To find the equation that has the given roots (1, 2, 3), we can use the fact that the equation will have factors of (x - 1), (x - 2), and (x - 3).
To create the equation, we multiply these factors together and expand:
(x - 1)(x - 2)(x - 3) = (x^2 - 3x - 2x + 6)(x - 3) = (x^2 - 5x + 6)(x - 3)
Expanding further, we have:
(x^2 - 5x + 6)(x - 3) = x^3 - 5x^2 + 6x - 3x^2 + 15x - 18
Combining like terms, we get:
x^3 - 5x^2 - 3x^2 + 6x + 15x - 18 = x^3 - 8x^2 + 21x - 18
Therefore, the equation that has the roots 1, 2, 3 is (d) x^3 - 6x^2 - 11x - 6.