#
OEF polynomial
--- Introduction ---

This module actually contains 32 exercises on one-variable polynomials
(with real or complex coefficients): roots, degrees, composition, euclidean
division, ...

### 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*() = ^{3}^{2} , 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 *r*_{1}, *r*_{2} 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

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.)

Other exercises on:
Polynomials
Roots
complex numbers

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- Description: collection of exercises on polynomials of one variable (real or complex coefficients). Serveur Wims de l'ESPE-Nice-Toulon - Université de Nice - Sophia Antipolis
- Keywords: interactive mathematics, interactive math, server side interactivity, algebra, polynomial,root,euclidean division,gcd,lcm,complex number