find a polynomial with integer coefficients such that square root of 3 + square root of 5 is a root of the polynomial
To find a polynomial with integer coefficients such that √3 + √5 is a root, we can use the concept of algebraic conjugates.
When we have an expression with a square root like √3 + √5, we can multiply it by its conjugate to eliminate the square roots. The conjugate of √3 + √5 is √3 - √5.
To create a polynomial with integer coefficients, we can consider the product of (√3 + √5) and its conjugate (√3 - √5). We get:
(√3 + √5)(√3 - √5)
= (√3 * √3) - (√3 * √5) + (√5 * √3) - (√5 * √5)
= 3 - √15 + √15 - 5
= -2
Hence, the product of (√3 + √5) and its conjugate is -2.
Since (√3 + √5) is a root of the polynomial, we know that the polynomial has a factor of (√3 + √5). To obtain the polynomial, we can multiply (√3 + √5) by some factor.
Let's assume the polynomial is of the form (a * (√3 + √5)) where 'a' is the multiplying factor. We know that (√3 + √5) * (√3 - √5) equals -2, so we have:
(a * (√3 + √5)) * (√3 - √5) = -2
(a * (√3 + √5) * √3) - (a * (√3 + √5) * √5) = -2
(a * (√3 * √3)) - (a * (√3 * √5)) + (a * (√5 * √3)) - (a * (√5 * √5)) = -2
(3a - 5a) + √15a - √15a = -2
-2a = -2
a = 1
Thus, the polynomial with integer coefficients such that √3 + √5 is a root is:
(√3 + √5) * 1 = √3 + √5
Therefore, the polynomial is simply (√3 + √5).