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OEF polynomial --- Introduction ---

This module actually contains 32 exercises on one-variable polynomials (with real or complex coefficients): roots, degrees, composition, euclidean division, ...
Please note that this module is obsolete and it is only kept for backwards compatibility. You should use the module OEF polynomial instead, that contains an updated English version.

Deg gcd with derivative

Let P(x) be a polynomial of degree and with coefficients, having different real roots and different complex roots (not counted with multiplicities). Let P'(x) be the derivative of P(x). What is the degree of gcd(P(x),P'(x)) ?

Min. deg multiple roots

What is the minimum of the degree of a polynomial P(x) with coefficients such that:
• is a root of multiplicity  ;
• is a root of multiplicity  ?
Answer -1 if you think that such polynomial does not exist.

Degree of sum

Let () and () be two polynomials. Complete:

If deg()= and deg()=, then is a polynomial of degree ________.

Difference equation

Find the polynomial () such that

()-() = 2

and that ()=.

Type x^3 for 3, etc.

Find multiple root degree 3

The following polynomial has a multiple root. Find this root.

Find multiple root degree 4

The following polynomial has a multiple root. Find this root.

Find multiple root degree 5

The following polynomial has a multiple root. Find this root.

Find multiple root degree 6

The following polynomial has a multiple root. Find this root.

Given gcd with derivative

Find the polynomial () such that:
• gcd((),()) = ()() , where () is the derivative of ();
• ()= ;
• The degree of is as small as possible.
You may enter your polynomial under any form, developed or factored. Type x^3 for 3, etc.

Given root deg 3

Determine the polynomial

P() = 32 ,

knowing that and are real, and that is one of its roots.

Min. deg gcd with derivative 2

Let P(x) be a polynomial of degree and with coefficients, having different real roots and different complex roots (not counted with multiplicities). Let P''(x) be the second derivative of P(x). What is the minimum of degree of gcd(P(x),P''(x)) ?

Min. deg gcd with derivative n

Let P(x) be a polynomial of degree and with coefficients, having different real roots and different complex roots (not counted with multiplicities). Let P()(x) be the -th derivative of P(x). What is the minimum of degree of gcd(P(x),P()(x)) ?

Multiplicity of a root degree 3

The number is a root of the polynomial below. Compute its multiplicity.

Multiplicity of a root degree 4

The number is a root of the polynomial below. Compute its multiplicity.

Multiplicity of a root degree 5

The number is a root of the polynomial below. Compute its multiplicity.

Multiplicity of a root degree 6

The number is a root of the polynomial below. Compute its multiplicity.

Parametric multiplicity degree 3

Find a value of so that the following polynomial has a multiple root, and find this multiple root.

WARNING. This exercise does not accept approximative replies! There is always an integer solution. Find it.

Parametric multiplicity degree 4

Find a value of so that the following polynomial has a multiple root, and find this multiple root.

WARNING. This exercise does not accept approximative replies! There is always an integer solution. Find it.

Parametrized deg 2

For which real values of the parameter the polynomial

()2 + (2) +

has ? (Under the condition that 0.)

Parametrized deg 2 II

For which real value of the parameter the polynomial

()2 + () + ()

has a root equal to ? (Under the condition that 0.)

Roots complex polynomial deg 2

Compute the two roots of the polynomial

P() = 2 + () + ().

You may enter the two roots , in any order.

Function of roots deg 2

Let , be the two roots of the polynomial

2  ,

where is a real coefficient. What is the value of t = 2+2 ? (This value is a function of .)

Function of roots deg 3

Let , , be the 3 roots of the polynomial

3 2  ,

where is a non-zero real coefficient. What is the value of t =  ? (This value is a function of .)

Re(root) deg 2

Let P() = 2 + be a polynomial with real coefficients, having two conjugate complex roots. What is the real part of a root r?

Count roots with derivative

Let P(x) be a polynomial of degree and with coefficients, and let P'(x) be the derivative of P(x). We know that gcd(P(x),P'(x)) is a polynomial of degree . What is the number of distinct roots of P(x) ? (both real and complex roots)

Root of composed polynomial

Let () be a polynomial, and () = 2 another polynomial. Consider the composed polynomials (()) and (()). Complete:

If is a root of , then .

Real roots deg 2

Find the two roots r1, r2 of the polynomial

2  .

(The roots are real, and the order in which you give the roots has no importance.)

Root multiplicity of sum

Let () and () be two polynomials. Complete:

If is a root of multiplicity of () and also a root of multiplicity of (), then is a root of multiplicity ________ of .

Root status deg 2

What is the type of roots of the following degree 2 polynomial?

2

Factorization of trinomial

Factor .

Step 1. We put the terms of into a complete square:

= ( )2.
We have .

Step 2. Therefore

 - = - 2 .
Therefore

 - =

Step 3. Now we apply the formula ( )( ).

Result: . (You should enter the simplified expressions.)

Triple root deg 3

For which real values of the parameters and the polynomial

P() = 3 + 2 + + (-)

has a triple root?

Triple root deg 3 II

For which real values of the parameters and the polynomial

P() = 3 2 +(++)

has a triple root? (There may be several solutions.)

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