what are the roots for x^4-6x^3+5x^2+12x
To find the roots of the given polynomial equation, x^4 - 6x^3 + 5x^2 + 12x, we can use the technique of factoring. However, this equation does not appear to be easily factorable.
But don't worry, we can still find the roots by using numerical methods, such as the Newton-Raphson method or polynomial division. Let's use polynomial division in this case.
First, we need to check if there are any rational roots. A rational root is of the form p/q, where p is a factor of the constant term (in this case, 0) and q is a factor of the leading coefficient (in this case, 1).
The factors of 0 are ±1, ±2, ±3, ±4, ±6, ±12.
The factors of 1 are ±1.
Let's test these possible rational roots by substituting them into the equation, x^4 - 6x^3 + 5x^2 + 12x, and see if they yield a value of 0.
Substituting each possible rational root into the equation, we find that the only rational root is x = 0.
Now we can perform polynomial division to divide the given polynomial by (x - 0). This will give us a polynomial of lower degree that we can further factor or solve.
Performing polynomial division, we get:
x^3 - 6x^2 + 5x + 12
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x - 0 | x^4 - 6x^3 + 5x^2 + 12x
The result of the division is x^3 - 6x^2 + 5x + 12.
Now, we can continue factoring the reduced polynomial, x^3 - 6x^2 + 5x + 12, to find its roots.
Using factoring techniques, we find that the polynomial x^3 - 6x^2 + 5x + 12 can be factored as (x - 2)(x + 3)(x - 2).
So, the complete factored form of the equation x^4 - 6x^3 + 5x^2 + 12x is (x - 0)(x - 2)(x + 3)(x - 2).
Therefore, the roots of the given polynomial are x = 0, x = 2, and x = -3.