One root of the equation 3x ^ 5 - 4x ^ 4 - 42x ^ 3 + 56x ^ 2 + 27x - 36 = 0is * sqrt(2) + sqrt(5)

Find the remaining roots.

We are given that one root of the polynomial equation 

\[
3x^5 - 4x^4 - 42x^3 + 56x^2 + 27x - 36 = 0
\]

is \( \sqrt{2} + \sqrt{5} \). To find the remaining roots, let's proceed step by step.

### Step 1: Verify the given root.
The root \( \sqrt{2} + \sqrt{5} \) satisfies the equation, so substituting \( x = \sqrt{2} + \sqrt{5} \) into the polynomial should give us zero. This helps confirm the correctness of the root before proceeding further.

### Step 2: Form the conjugate root.
Since the coefficients of the polynomial are all real, if \( \sqrt{2} + \sqrt{5} \) is a root, its conjugate \( \sqrt{2} - \sqrt{5} \) must also be a root.

### Step 3: Construct a quadratic factor.
If both \( \sqrt{2} + \sqrt{5} \) and \( \sqrt{2} - \sqrt{5} \) are roots, we can form a quadratic factor based on these roots. The quadratic polynomial formed by these roots is:

\[
(x - (\sqrt{2} + \sqrt{5}))(x - (\sqrt{2} - \sqrt{5}))
\]

This simplifies to:

\[
(x - (\sqrt{2} + \sqrt{5}))(x - (\sqrt{2} - \sqrt{5})) = (x^2 - 2(\sqrt{2})^2 + (\sqrt{5})^2) . equation will be

One root of the equation 3x^5 - 4x^4 - 42x^3 + 56x^2 + 27x - 36 = 0 is sqrt(2) + sqrt(5). Find the remaining roots.

Explore how to find the remaining roots of the polynomial equation 3x^5 - 4x^4 - 42x^3 + 56x^2 + 27x - 36, given that one root is sqrt(2) + sqrt(5). This solution applies advanced algebraic methods, including the Conjugate Root Theorem and quadratic factorization.

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