Question

X³-3x+2=0find the roots of each polynomial​

Answer 1

Найти корни многочлена x³-3x+2=0.

1) х=1

1³-3·1+2=0

1-3+2=0

     0=0

х₁ = 1

2) (x³-3x+2) : (х-1) = х²+х-2

(Смотри ниже деление в столбик)

3) х²+х-2=0

D=b²-4ac

D=1²-4·1·(-2)=1+8=9=3²

$x_2=\frac{-1-3}{2*1}=\frac{-4}{2}=-2$

$x_3=\frac{-1+3}{2*1}=\frac{2}{2}=1$

х₂= -2

х₃= 1

x₁=x₃=1

Ответ: { -2; 1}

Answer 2

To find the roots of the polynomial x³ - 3x + 2 = 0, we can use various methods such as factoring, synthetic division, or numerical methods like Newton's method. Let's try factoring and see if we can find any rational roots.

The polynomial cannot be factored using simple integer factors. Therefore, we'll proceed to check for rational roots using the rational root theorem. According to the rational root theorem, the possible rational roots can be determined by taking the factors of the constant term (2) divided by the factors of the leading coefficient (1).

The factors of 2 are ±1 and ±2, and the factors of 1 are ±1. So, the possible rational roots are ±1 and ±2.

We can test these values using synthetic division or by substituting them into the polynomial equation to check if any of them satisfy the equation.

When we substitute x = 1 into the equation x³ - 3x + 2 = 0, we get: (1)³ - 3(1) + 2 = 0 1 - 3 + 2 = 0 0 = 0

Therefore, x = 1 is a root of the polynomial.

Next, let's check x = -1: (-1)³ - 3(-1) + 2 = 0 -1 + 3 + 2 = 0 4 = 0

The equation is not satisfied for x = -1.

Next, let's check x = 2: (2)³ - 3(2) + 2 = 0 8 - 6 + 2 = 0 4 = 0

The equation is not satisfied for x = 2.

Lastly, let's check x = -2: (-2)³ - 3(-2) + 2 = 0 -8 + 6 + 2 = 0 0 = 0

Therefore, x = -2 is a root of the polynomial.

We found that x = 1 and x = -2 are roots of the equation x³ - 3x + 2 = 0. To find the remaining root, we can divide the polynomial by the known roots.

Using long division or synthetic division, we divide the polynomial x³ - 3x + 2 by (x - 1) and (x + 2).

(x³ - 3x + 2) / (x - 1) = x² + x - 2

Now, we can solve x² + x - 2 = 0 using factoring or the quadratic formula.

Factoring: (x + 2)(x - 1) = 0

Setting each factor equal to zero: x + 2 = 0, x - 1 = 0

Solving for x: x = -2, x = 1

Therefore, the roots of the polynomial x³ - 3x + 2 = 0 are x = 1, x = -2, and x = -2 (repeated root).