Question

Find a third degree polynomial with rational coefficients that has the given numbers as given roots

4 and 3 -
`√(2)`

Answer 1

`x^3-10x^2+31x-28`

Explanation:

Since we are looking for a polynomial with RATIONAL coefficients, then we know that if
`3-√(2)` is a root, then its conjugate
`3+√(2)` must also be a root

Therefore, we build the polynomial using binomial factors with the given roots as shown below:

`(x-4)*(x-3+√(2) )*(x-3-√(2) )= (x-4)((x-3)^2-2)=\\=(x-4)*(x^2-6x+9-2) =(x-4)*(x^2-6x+7)=\\=x^3-10x^2+31x-28`