Given that√2 is the root of cubic polynomial 6x3 + 2x2-10x -4√2 find the other two roots
sqrt ( 2 ) isnt root of a cubic polynomial 6 x ^ 3 + 2 x ^ 2 - 10 x - 4 sqrt ( 2 )
To find the other two roots of the cubic polynomial, we can make use of the fact that the sum of the roots of a cubic polynomial is equal to the negation of the coefficient of the quadratic term divided by the coefficient of the cubic term.
Here, the given polynomial is:
6x^3 + 2x^2 - 10x - 4√2
The coefficient of the cubic term is 6, and the coefficient of the quadratic term is 2. By our earlier statement, the sum of the roots is -2/6, which simplifies to -1/3.
Since √2 is already one of the roots, we can find the remaining two roots by finding the other two values whose sum is -1/3.
Let's call the other two roots r1 and r2. We have the following relationship:
r1 + r2 = -1/3
We can solve this equation to find either r1 or r2, and then substitute it back to find the other root.
Let's first solve for r1:
r1 = -1/3 - r2
We can substitute this back into the original equation and simplify:
6(√2)^3 + 2(√2)^2 - 10(√2) - 4√2 = 0
8√2 + 4 - 10√2 - 4√2 = 0
2 - 6√2 = 0
6√2 = 2
√2 = 1/3
Substituting this value back into the expression for r1:
r1 = -1/3 - r2
-1/3 = -1/3 - r2
r2 = 0
Therefore, the other two roots of the cubic polynomial are r1 = 1/3 and r2 = 0.