X^4 - 5x^3 - 4x^2 - 7x + 4 = 0

Предыдущий
вопрос Следующий
вопрос

To find the roots of the given equation, we can use various methods such as factoring, grouping, or the rational root theorem. In this case, the rational root theorem can be used to identify potential rational roots of the equation.

The rational root theorem states that if a polynomial equation has a rational root, then it will be of the form p/q, where p is a factor of the constant term (in this case 4) and q is a factor of the leading coefficient (in this case 1).

The factors of 4 are ±1, ±2, ±4 and the factors of 1 (the leading coefficient) are ±1. Therefore, the potential rational roots are ±1, ±2, ±4.

To find the actual roots, we can use synthetic division or polynomial division to test each potential root. After testing these potential roots, we find that x = 1 and x = -1 are roots of the equation.

Using synthetic division or polynomial division, we can divide the original equation by (x - 1) and (x + 1) to obtain the other factors of the equation.

(x - 1)(x + 1)(x^2 - 6x + 4) = 0

The last factor, x^2 - 6x + 4, can be further factored or solved using the quadratic formula to find the remaining roots.