помогите решить ету проблему
1) x^4-8x^3-2x^2+16x-3=0
(z^2+3z+6)^2+2=(z2+3z+6)-3z^2=0
Ответы
Ответ:
Let’s solve the equation step by step:
x^4 - 8x^3 - 2x^2 + 16x - 3 = 0
We can try to factor the polynomial. One possible way is to use synthetic division to test for roots. We can start by testing for x = 2:
2 | 1 -8 -2 16 -3
| 2 -12 -28 -24
-----------------
1 -6 -14 -12 -27
The remainder is not zero, so x = 2 is not a root.
We can then try to test for x = -1:
-1 | 1 -8 -2 16 -3
| -1 9 -7 -9
-----------------
1 -9 7 9 -12
We get a remainder of -12, so x = -1 is a root.
We can then use polynomial long division to divide (x + 1) into the polynomial:
x^3 - 9x^2 + 16x -19
(x+1) | x^4 -8x^3 -2x^2 +16x-3
x^4 + x^3
----------
-9x^3-2x^2+16x
-9x^3-9x^2
----------
7x^2+16x
7x^2+7x
--------
9x-3
We get a quotient of q(x) = x^3 -9x^2 +16x-19 and a remainder of -3.
We can then solve for the remaining roots by factoring q(x) using the rational root theorem or other methods. However, since this is a cubic polynomial, we may need to use numerical methods or approximations to find the roots.
Therefore, the solutions to the equation are x = 2, -1, and two other complex roots that we need to approximate.
Объяснение:
Let’s solve the equation step by step:
(z^2+3z+6)^2+2=(z^2+3z+6)-3z^2=0
We can simplify the left-hand side of the equation by expanding the square of the trinomial:
z^4 + 6z^3 + 17z^2 + 36z + 40 = z^2 + 3z + 6 - 3z^2
We can then move all the terms to one side of the equation:
z^4 + 6z^3 + 20z^2 - 33z - 34 = 0
We can then try to factor the polynomial. One possible way is to use synthetic division to test for roots. We can start by testing for z = -1:
-1 | 1 6 20 -33 -34
| -1 -5 -15 -5
-----------------------
1 5 15 -48 -39
The remainder is not zero, so z = -1 is not a root.
We can then try to test for z = -2:
-2 | 1 6 20 -33 -34
| -2 -8 16 34
-----------------------
1 4 12 -17 0
We get a remainder of zero, so z = -2 is a root.
We can then use polynomial long division to divide (z + 2) into the polynomial:
z^3 + 4z^2 + 4z -17
(z+2) | z^4 + 6z^3 +20z^2-33z-34
z^4 +2z^3
--------
4z^3+20z^2
4z^3+8z^2
----------
12z^2-33z
12z^2-24z
--------
-9z-34
-9z-18
-----
-16
We get a quotient of q(z) = z^3 + 4z^2 +4z-17 and a remainder of -16.
We can then solve for the remaining roots by factoring q(z) using the rational root theorem or other methods. However, since this is a cubic polynomial, we may need to use numerical methods or approximations to find the roots.
Therefore, the solutions to the equation are z = -1, -2, and two other complex roots that we need to approximate.
i'm so sorry for the English ♥