Question

There may be many polynomial equations with integer coefficient which have √5-1 and √5 +1 as its roots. Construct an equation of smallest degree with integer coefficients that has √5-1 and √5 +1 as its roots.

Answer 1

The equation of smallest degree with integer coefficients that has the roots √5-1 and √5+1 is a quadratic obtained by multiplying the factors (x - (√5 -1))^2 and (x - (√5 +1))^2, simplifying and combining like terms, resulting in the polynomial x^4 - 4x^2 + 16.

Step-by-step explanation:

To construct a polynomial equation of smallest degree with integer coefficients that has the roots √5-1 and √5+1, we start by observing that both roots are irrational.

The minimal polynomial that has these two roots will be a quadratic because these roots are conjugates of each other. The roots can be represented as x = √5 - 1 and x = √5 + 1.

First, let's solve for x - (√5 - 1) = 0 and x - (√5 + 1) = 0. If we square both expressions to get rid of the square root, we get (x - (√5 -1))^2 = x^2 - 2x(√5 - 1) + (√5 - 1)^2 and (x - (√5 +1))^2 = x^2 - 2x(√5 +1) + (√5 +1)^2 respectively.

Now, if we expand these, both will give us a term with √5 squared, which is 5. Simplifying and combining like terms gives us our quadratic equation: (x^2 - 2x√5 + 4)(x^2 + 2x√5 + 4) = x^4 - 4x^2 + 16.

This is our desired polynomial equation of the smallest degree with integer coefficients that has √5-1 and √5+1 as its roots.

To check if the root is reasonable, you can plug √5-1 and √5 +1 into the equation and verify they satisfy the equation.