Question

[Nguyễn Hải Dương]

cm pt vô nghiệm
a)x⁴ +x³ +1=0
b)x⁴+x+1=0
 

Answer 1

[Nguyễn Việt Lâm]

a.

\(x^4+x^3+1=\left(\dfrac{x^4}{4}+x^3+x^2\right)+\left(\dfrac{3x^4}{4}-x^2+\dfrac{1}{3}\right)+\dfrac{2}{3}\)

\(=\left(\dfrac{x^2}{2}+x\right)^2+\dfrac{3}{4}\left(x-\dfrac{2}{3}\right)^2+\dfrac{2}{3}>0\) ; \(\forall x\)

\(\Rightarrow x^4+x^3+1=0\) vô nghiệm

b.

\(x^4+x+1=\left(x^4-x^2+\dfrac{1}{4}\right)+\left(x^2+x+\dfrac{1}{4}\right)+\dfrac{1}{2}\)

\(=\left(x^2-\dfrac{1}{2}\right)^2+\left(x+\dfrac{1}{2}\right)^2+\dfrac{1}{2}>0\) ; \(\forall x\)

\(\Rightarrow x^4+x+1=0\) vô nghiệm

Answer 2

[Akai Haruma]

Lời giải:
a. 

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

$=(x^2+x)^2+(x^2-\frac{1}{2})^2+\frac{3}{4}\geq \frac{3}{4}>0$ với mọi $x\in\mathbb{R}$

$\Rightarrow x^4+x^3+1>0, \forall x\in\mathbb{R}$

Do đó pt $x^4+x^3+1=0$ vô nghiệm.

b.

$x^4+x+1=(x^4-x^2+\frac{1}{4})+(x^2+x+\frac{1}{4})+\frac{1}{2}$

$=(x^2-\frac{1}{2})^2+(x+\frac{1}{2})^2+\frac{1}{2}\geq \frac{1}{2}>0$ với mọi $x\in\mathbb{R}$

$\Rightarrow x^4+x+1=0$ vô nghiệm (đpcm).