Deg gcd with derivative

Let P(x) be a polynomial of degree $val14 and with $val13 coefficients, having $val8 different real roots and $val9 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 $val20 coefficients such that: Answer -1 if you think that such polynomial does not exist.

Degree of sum

Let $val6($val8) and $val7($val8) be two polynomials. Complete:

If deg($val6)=$val9 and deg($val7)=$val11, then $val13 is a polynomial of degree ________.


Difference equation

Find the polynomial $val7($val6) such that

$val7($val6$val17)-$val7($val6$val18) = $val23$val62$val26$val6$val27

and that $val7($val28)=$val29.

Type x^3 for $val63, 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 $val7($val6) such that: You may enter your polynomial under any form, developed or factored. Type x^3 for $val63, etc.

Given root deg 3

Determine the polynomial

P($val6) = $val63$val22$val62$val23$val6$val24 ,

knowing that $val8 and $val9 are real, and that $val13 is one of its roots.


Min. deg gcd with derivative 2

Let P(x) be a polynomial of degree $val14 and with $val13 coefficients, having $val8 different real roots and $val9 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 $val13 and with $val12 coefficients, having $val6 different real roots and $val7 different complex roots (not counted with multiplicities). Let P($val8)(x) be the $val8-th derivative of P(x). What is the minimum of degree of gcd(P(x),P($val8)(x)) ?

Multiplicity of a root degree 3

The number $val8 is a root of the polynomial below. Compute its multiplicity.


Multiplicity of a root degree 4

The number $val8 is a root of the polynomial below. Compute its multiplicity.


Multiplicity of a root degree 5

The number $val8 is a root of the polynomial below. Compute its multiplicity.


Multiplicity of a root degree 6

The number $val8 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 $val7 the polynomial

($val7$val11)$val62 + (2$val7$val12)$val6 + $val7$val13

has $val16? (Under the condition that $val7$val11 $m_ne 0.)


Parametrized deg 2 II

For which real value of the parameter $val7 the polynomial

($val12$val7$val16)$val62 + ($val13$val7$val17)$val6 + ($val14$val7$val18)

has a root equal to $val11? (Under the condition that $val12$val7$val16 $m_ne 0.)


Roots complex polynomial deg 2

Compute the two roots of the polynomial

P($val7) = $val72 + ($val14$val18$val6)$val7 + ($val16$val19$val6).

You may enter the two roots $val8,$val9 in any order.


Function of roots deg 2

Let $val8, $val9 be the two roots of the polynomial

$val62 $val13$val6 $val14 ,

where $val7 is a real coefficient. What is the value of t = $val82+$val92 ? (This value is a function of $val7.)


Function of roots deg 3

Let $val8, $val9, $val10 be the 3 roots of the polynomial

$val63 $val16$val62 $val17$val6 $val18 ,

where $val7 is a non-zero real coefficient. What is the value of t = $val24 ? (This value is a function of $val7.)


Re(root) deg 2

Let P($val6) = $val8$val62 $val10$val6 +$val7 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 $val11 and with $val14 coefficients, and let P'(x) be the derivative of P(x). We know that gcd(P(x),P'(x)) is a polynomial of degree $val10. What is the number of distinct roots of P(x) ? (both real and complex roots)

Root of composed polynomial

Let $val6($val8) be a polynomial, and $val7($val8) = $val82$val11$val8$val16 another polynomial. Consider the composed polynomials $val6($val7($val8)) and $val7($val6($val8)). Complete:

If $val19 is a root of $val18, then $val21.


Real roots deg 2

Find the two roots r1, r2 of the polynomial

$val10$val62 $val15 $val14 .

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

Root multiplicity of sum

Let $val6($val8) and $val7($val8) be two polynomials. Complete:

If $val9 is a root of multiplicity $val10 of $val6($val8) and also a root of multiplicity $val12 of $val7($val8), then $val9 is a root of multiplicity ________ of $val14.


Root status deg 2

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

$val17$val62 $val21$val6 $val22


Factorization of trinomial

Factor .

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

= ( )2.
We have .

Step 2. Therefore

-
= - 2 .
Therefore

- $val15
=

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 $val7 and $val8 the polynomial

P($val6) = $val63 + $val10$val7$val62 + $val8$val6 + ($val9-$val7)

has a triple root?


Triple root deg 3 II

For which real values of the parameters $val7 and $val8 the polynomial

P($val6) = $val63 $val12$val7$val62 $val13$val8$val6 +($val9+$val7+$val8)

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