(a) Determine a cubic polynomial with integer coefficients which has $\sqrt[3]{2} + \sqrt[3]{4}$ as a root. (b) Prove that $\sqrt[3]{2} + \sqrt[3]{4}$ is irrational.

(a)

(b) Proved

Step-by-step explanation:

Given

--- the root

Solving (a): The polynomial

A cubic function is represented as:

Expand

Rewrite as:

The root is represented as:

By comparison:

So, we have:

Expand

Evaluate like terms

Recall that:

So, we have:

Equate to 0

Rewrite as:

Express as a cubic function

Hence, the cubic polynomial is:

Solving (b): Prove that r is irrational

The constant term of is -6

The divisors of -6 are: -6,-3,-2,-1,1,2,3,6

Calculate f(x) for each of the above values to calculate the remainder when f(x) is divided by any of the above values

For r to be rational;

The divisors of -6 must divide f(x) without remainder

i.e. Any of the above values  must equal 0

Since none equals 0, then r is irrational

step-by-step explanation: