## 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