Question

Find the zeros of polynomial functions and solve polynomial equations.
f(x)=8x^3+27

Answer 1

f(x) = 8x³ + 27

To find the zeros equal the equation to 0

8x³ + 27 = 0

8x³ = -27

x³ = -27/8

x =
`\sqrt[3]{(-27)/(8)}`

x =
`(-3)/(2)`

This is the real root, but there are two other irrational roots.

By rule we know:

For f(z) = x³ the three solutions are:

x =
`\sqrt[3]{f(z)}`

x =
`\sqrt[3]{f(z)}.(-1-√(3)i)/(2)`

x =
`\sqrt[3]{f(z)}.(-1+√(3)i)/(2)`

Making some subs.

x =
`(-3)/(2)`

x =
`(-3)/(2).(-1-√(3)i)/(2)`

x =
`(-3)/(2).(-1+√(3)i)/(2)`

So, after all we have:

x =
`(-3)/(2)`

x =
`(3+3√(3)i)/(4)`

x =
`(3-3√(3)i)/(4)`

Answer 2

`image`