Ta có: `(a+b)^3=a^3+3a^2b+3ab^2+b^3`
`=a^3+b^3+3ab(a+b)`
$\quad x=\sqrt[3]{20+14\sqrt{2}}+\sqrt[3]{20-14\sqrt{2}}$
`=>`$x^3=(\sqrt[3]{20+14\sqrt{2}}+\sqrt[3]{20+14\sqrt{2}})^3$
`=>`$x^3=20+14\sqrt{2}+20-14\sqrt{2}+3.\sqrt[3]{(20+14\sqrt{2}).(20-14\sqrt{2})}.(\sqrt[3]{20+14\sqrt{2}}+\sqrt[3]{20-14\sqrt{2}})$
`=>`$x^3=40+3.\sqrt[3]{20^2-(14\sqrt{2})^2}.x$
`=>`$x^3=40+3\sqrt[3]{8}.x$
`=>x^3=40+6x`
`=>x^3-6x-40=0`
`=>x^3-4^3-6x+24=0`
`=>(x-4)(x^2+4x+16)-6(x-4)=0`
`=>(x-4)(x^2+4x+10)=0`
`=>(x-4)[(x+2)^2+6]=0`
`=>x-4=0` (vì `(x+2)^2+6\ge 6>0` với mọi `x`)
`=>x=4`
Với `x=4` ta có:
`\qquad x^3-3x^2+x-20`
`=4^3-3. 4^2+4-20`
`=64-48-16=0`
Vậy với $x=\sqrt[3]{20+14\sqrt{2}}+\sqrt[3]{20-14\sqrt{2}}$ thì:
`\qquad x^3-3x^2+x-20=0`
