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: - $val10$val12$val7 is a root of multiplicity $val8 ;
- $val17 is a root of multiplicity $val9 ?
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: - gcd($val7($val6),$val7$val8($val6)) = ($val6$val18)$val25($val6$val19)$val26 , where $val7$val8($val6) is the derivative of $val7($val6);
- $val7($val28)=$val29 ;
- The degree of $val7 is as small as possible.
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
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 $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.)