Question

Имеется уравнение 243x^4-108x^3+1=0 , так вот, хотелось бы узнать, можно ли решать его вынесением за скобу x^3 ? x^3(243x-108)=-1 , если нет, то почему?

Answer 1

Можно. Только что это дает?
Обозначим 3х=у
3у^4-4y^3+1=0
y=1  сразу видно корень.
3у^4-3y^3-y^3+1=0
(у-1)(3y^3-(y^2+y+1))=0
3y^3-y^2-y-1=0
y=1  корень и этого уравнения
3y^3-3y-y^2+2y-1=0
3y(y-1)(y+1)-(y-1)^2=0
3y(y+1)-y+1=0
3y^2+3y-y+1=0
3y^2+2y+1=0
y^2+2/3y=-1/3
(y+1/3)^2=-1/3+1/9=-2/9
У этого уравнения нет решений.
Стало быть , корень единственный у=1
3х=1
х=1/3
Ответ: х=1/3

Answer 2

Can the equation 243x^4 - 108x^3 + 1 = 0 be solved by factoring out x^3? If not, why?

To determine if the equation 243x^4 - 108x^3 + 1 = 0 can be solved by factoring out x^3, let's analyze the equation.

The equation can be rewritten as x^3(243x - 108) = -1.

To solve this equation by factoring, we would need to find a common factor that can be factored out from both terms. In this case, we have x^3 as a common factor. However, we cannot factor out x^3 from the second term, 243x - 108, because it does not contain x^3 as a factor.

Therefore, we cannot solve the equation 243x^4 - 108x^3 + 1 = 0 by factoring out x^3.

If you have any further questions, please, let me know!