## Polynomial equation

Let denote the golden ratio. Solve the equation

**Solution**

First of all we note that

We easily note that is one root of the equation, hence using Horner we get that

Hence is a double root and the other root is .

## Similarity implies equality?

Let be similar to . Does hold?

**Solution**

No! Take then . The matrices are similar but not equal.

## On tensors

Let -Vect, , -linear. Prove that

**Solution**

Recall the general definition of the tensor product of linear maps, we have successively:

Thus, the two linear maps are equal when composed with the canonical bilinear map , hence equal (by the universal property).

## Fibonacci series

Let denote the Fibonacci sequence such that and . Prove that

**Solution**

It holds that

It follows from Cassini’s identity that

(1)

Setting back at we get

(2)

Since we get

(3)

Hence,

## Logarithmic integral

Evaluate the integral