Question

If `a^3 + 3a^2 + 9a = 1`, then what is the value of `a^3 + (3/a)`?

Answer 1

`28`

Explanation:

`a^3 + 3 a^2 + 9 a + 27 " has root "a = -3.`
`"So we divide away the factor "(a + 3) :`
`a^3 + 3 a^2 + 9 a + 27 = (a + 3)(a^2+9) = 28`

`"Now we try to solve "(a+3)(a^2+9) = 28.`
`"We multiply both sides with "(a-3) : "`

`(a+3)(a-3)(a^2+9) = 28(a-3)`
`=> (a^2-9)(a^2+9) = 28(a-3)`
`=> (a^4 - 81) = 28(a - 3)`
`=> a^4 - 28 a + 3 = 0`

`"Now we divide both sides by a : "`

`=> a^3 + 3/a = 28`