Find a polynomial with integer coefficients such that (sqrt3 + sqrt5) is a root of the polynomial

Going back over some previous posts, I noticed you had posted this same question earlier, but for that you had

3 + √5 as a root.

I will answer it as if that was the right question.

If 3+√5 is a root, then its conjugate 3-√5 must also be a root, or else a radical will show up in the expansion.

so the polynomial is
(x - (3+√5))(x- (3-√5))
= (x - 3 - √5)(x - 3 + √5)
= x^2 - 3x + x√5 - 3x + 9 - 3√5 - x√5 + 3√5 - 5
= x^2 - 6x + 4

To find a polynomial with integer coefficients such that (sqrt(3) + sqrt(5)) is a root, we can start by considering the conjugate of (sqrt(3) + sqrt(5)).

The conjugate of (sqrt(3) + sqrt(5)) is (sqrt(3) - sqrt(5)).

To obtain a polynomial with integer coefficients, we can multiply these two conjugate roots together:

(sqrt(3) + sqrt(5)) * (sqrt(3) - sqrt(5))
= (sqrt(3))^2 - (sqrt(5))^2
= 3 - 5
= -2

Therefore, the polynomial with integer coefficients that has (sqrt(3) + sqrt(5)) as a root is:

x + 2 = 0

To find a polynomial with integer coefficients such that (sqrt3 + sqrt5) is a root, we can make use of the concept of conjugate pairs.

Let's start by considering the number (sqrt3 + sqrt5). We can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, which is (sqrt3 - sqrt5):

(sqrt3 + sqrt5) * (sqrt3 - sqrt5) / (sqrt3 - sqrt5)
= (3 - 2sqrt15 + 5) / (sqrt3 - sqrt5)
= (8 - 2sqrt15) / (sqrt3 - sqrt5)

Now, let's simplify the expression further by multiplying the numerator and denominator by the conjugate of the denominator again:

[(8 - 2sqrt15) / (sqrt3 - sqrt5)] * [(sqrt3 + sqrt5) / (sqrt3 + sqrt5)]
= [(8 * sqrt3 + 8 * sqrt5 - 2sqrt15 * sqrt3 - 2sqrt15 * sqrt5) / (3 - 5)]
= [(8sqrt3 + 8sqrt5 - 2sqrt(15 * 3) - 2sqrt(15 * 5)) / (-2)]
= [(8sqrt3 + 8sqrt5 - 2sqrt45 - 2sqrt75) / -2]
= [(-2sqrt45 - 2sqrt75 + 8sqrt3 + 8sqrt5) / -2]
= [(-sqrt45 - sqrt75 + 4sqrt3 + 4sqrt5)]

Therefore, we have found a polynomial with integer coefficients such that (sqrt3 + sqrt5) is a root:

x = -sqrt45 - sqrt75 + 4sqrt3 + 4sqrt5