Question

X^3 + 3x^2 -x - 3 = 0

Answer 1

Ответ:

x(x²+3x-1-3)=0

x=0; x²+3x-4=0

       x+x=-3

        xx=-4

        x=-4

        x=1

Ответ 0,-4,1  

Answer 2

To solve the equation x^3 + 3x^2 - x - 3 = 0, we can use various methods like factoring, synthetic division, or numerical methods. In this case, let's use the rational root theorem to find possible rational solutions and then use synthetic division to check if those solutions are valid.

The rational root theorem states that if the polynomial has a rational root p/q (where p and q are integers with no common factors), then p must be a factor of the constant term (in this case, -3), and q must be a factor of the leading coefficient (in this case, 1).

Possible rational roots (p/q) are therefore combinations of divisors of -3 and 1:

±1, ±3

Now, let's use synthetic division to check if any of these potential roots are valid.

We start with the coefficients of the polynomial: 1, 3, -1, -3, and the potential root we want to check. Let's try x = 1:

1 | 1 3 -1 -3 |__ 1 4 3 1 4 3 0

Since the result is zero in the last row, x = 1 is a root, and (x - 1) is a factor of the polynomial.

Now, we have a quadratic equation left to solve, obtained from the synthetic division result: x^2 + 4x + 3 = 0.

This quadratic equation can be factored as follows:

x^2 + 4x + 3 = (x + 3)(x + 1) = 0

So, the complete factorization of the original polynomial is:

(x - 1)(x + 3)(x + 1) = 0

Now, set each factor to zero and solve for x:

1. x - 1 = 0 --> x = 1
2. x + 3 = 0 --> x = -3
3. x + 1 = 0 --> x = -1

The solutions to the equation x^3 + 3x^2 - x - 3 = 0 are x = 1, x = -3, and x = -1.