Question

(x-3)^6+(x^2-2x-1)^3=0

Answer 1

${(x - 3)}^{6} + {( {x}^{2} - 2x - 1)}^{3} = 0 \\ \\ {( \: {(x - 3)}^{2} \: ) }^{3} = - {( {x}^{2} - 2x - 1)}^{3} \\ \\ {( \: {(x - 3)}^{2} \: ) }^{3} = {( - ({x}^{2} - 2x - 1))}^{3} \\ \\ {u}^{3} = {v}^{3} \: \: \: \: = > \: \: \: u = v \\ \\ {(x - 3)}^{2} = - {x}^{2} + 2x + 1 \\ \\ {x}^{2} - 6x + 9 = - {x}^{2} + 2x + 1 \\ \\ 2 {x}^{2} - 8x + 8 = 0 \\ \\ {x}^{2} - 4x + 4 = 0 \\ \\ {(x - 2)}^{2} = 0 \\ \\ x = 2$

ОТВЕТ: 2

Answer 2

Ответ:

Объяснение:

Сумма кубов

((x-3)^2)^3 + (x^2-2x-1)^3 = (x^2-6x+9)^3 + (x^2-2x-1)^3 = 0

= (x^2-6x+9+x^2-2x-1)((x^2-6x+9)^2 - (x^2-6x+9)(x^2-2x-1) + (x^2-2x-1)^2) = 0

(2x^2-8x+8)(x^4-12x^3+54x^2-108x+81 - x^4+8x^3-20x^2+22x+9 + x^4-4x^3+2x^2+4x+1) = 0

2(x^2-4x+4)(x^4-8x^3+36x^2-84x+91) =0

2(x-2)^2*(x^4-8x^3+36x^2-84x+91) = 0

x1 = x2 = 2

Уравнение 4 степени корней не имеет. Это можно проверить по схеме Горнера. Смотрите рисунок

Answer 3

To solve the equation $(x-3)^6 + (x^2-2x-1)^3 = 0$, we'll first expand the expressions and then attempt to find the roots.

Let's expand each term:

$(x-3)^6 = x^6 - 18x^5 + 135x^4 - 540x^3 + 1215x^2 - 1458x + 729$

$(x^2-2x-1)^3 = (x^2-2x-1)(x^2-2x-1)^2$

Now, we expand $(x^2-2x-1)^2$ using the FOIL method:

$(x^2-2x-1)^2 = (x^2-2x-1)(x^2-2x-1) = x^4 - 4x^3 + 5x^2 + 4x + 1$

Now we have:

$(x^2-2x-1)^3 = (x^2-2x-1)(x^4 - 4x^3 + 5x^2 + 4x + 1)$

Expand further:

$(x^2-2x-1)^3 = x^6 - 6x^5 + 11x^4 + 6x^3 - 5x^2 - 6x - 1$

Now, our equation becomes:

$x^6 - 18x^5 + 135x^4 - 540x^3 + 1215x^2 - 1458x + 729 + x^6 - 6x^5 + 11x^4 + 6x^3 - 5x^2 - 6x - 1 = 0$

Combine like terms:

$2x^6 - 24x^5 + 146x^4 - 534x^3 + 1210x^2 - 1464x + 728 = 0$

Now, we can try to find the roots of this polynomial equation. However, finding exact solutions for 6th-degree polynomials can be quite complex and often involves numerical methods.

If you need approximate solutions, you can use numerical methods like the Newton-Raphson method, numerical root-finding algorithms, or graphing techniques. Alternatively, you can use software like Wolfram Alpha or numerical computation software like MATLAB or Python libraries (NumPy, SciPy) to find approximate roots.

Keep in mind that there might be multiple solutions (real and/or complex) to this equation.