Question

А)х^5+х^4-3x^2-4x-4=0 б)x^5+x^4-3x^3-3x^2+2x+2=0

Answer 1

Ответ: \frac{4x^4-4x^3+x^2}{-2x^2+5x-2}+ \frac{2x^3-7x^2+5x+1}{x-2}\leqslant0

ОДЗ:

-2x^2+5x-2\neq0\\
2x^2-5x+2\neq0\\
D=25-16=9; \sqrt {D}=3\\\\
x_{1/2}\neq0\\\\
x_1\neq \frac{1}{2}; \ \ x_2\neq2  

-2x^2+5x-2=-(x-2)(2x-1)=(2-x)(2x-1)

\frac{x^2(4x^2-4x+1)}{(2-x)(2x-1)}+ \frac{2x^3-7x^2+5x+1}{x-2}\leqslant0\\\\
 \frac{x^2(2x-1)^2}{(2-x)(2x-1)}+ \frac{2x^3-7x^2+5x+1}{x-2}\leqslant0\\\\
 \frac{2x^3-7x^2+5x+1}{x-2}- \frac{x^2(2x-1)}{x-2}\leqslant0\\\\
 \frac{2x^3-7x^2+5x+1-2x^3+x^2}{x-2}\leqslant0\\\\
-6x^2+5x+1\leqslant0\\\\
6x^2-5x+1=0\\
D=25+24=49; \ \sqrt D=7\\\\
x_{1/2}= \frac{5\pm7}{12}\\\\
x_1=- \frac{1}{6};\ \  x_2=1  

__+__- \frac{1}{6} __-__ \frac{1}{2} __-__1__+__2__-__

Ответ:  x\in [- \frac{1}{6}; \frac{1}{2})\bigcup (\frac{1 }{2};1]\bigcup(2;+\infty)

Объяснение:

Answer 2

a) To solve the equation:

х^5+х^4-3x^2-4x-4=0

We can start by trying to factor it. We can first notice that x = -1 is a root of the equation, as

(-1)^5 + (-1)^4 - 3(-1)^2 - 4(-1) - 4 = 0.

Using synthetic division or polynomial long division, we can then factor out (x+1) from the polynomial and obtain:

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

Now we can focus on solving the second factor:

x^4 + 2x^3 - x^2 - 3x - 4 = 0

We can try to factor this polynomial using various techniques, such as grouping, but in this case, we can notice that x=1 is another root of the equation, as

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

Using polynomial long division or synthetic division, we can then factor out (x-1) and obtain:

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

The remaining cubic polynomial, x^3 + 3x^2 + 2x - 4, can be solved using various methods, such as the rational root theorem, synthetic division, or numerical methods.

b) To solve the equation:

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

We can start by factoring out x^3 from the polynomial:

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

Now we can factor the quadratic expression inside the parentheses:

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

Factorizing this expression further, we obtain:

(x-1)(x^4 + 4x^3 + 9x^2 + 6x - 2) = 0

The quartic polynomial, x^4 + 4x^3 + 9x^2 + 6x - 2, can be solved using various methods, such as numerical methods or the Ferrari method, which involves solving a quadratic equation in a new variable.