Prove that if $f(x)$ divides $g(x)$ and $f(x)$ divides $h(x)$, then $f(x)$ divides $s(x)g(x) +t(x)h(x)$. Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)If $3$ divides $a^2 + b^2$, then $3$ divides $a$ and $3$ divides $b$If $gcd(a,b)= 1$ and $a$ divides $bc$ then $a$ divides $c $ [Euclid's Lemma]Show that if $10$ divides into $n^2$ evenly then $10$ divides into $n$ evenlyIf $d$ divides $k$ and $d$ divides $n$, then $d$ divides $(8k - 3n)$How to prove that linear polynomials are irreducible?Prove divisibility: If $j_1mid j_2$ and $j_2mid j_1$, then $j_1 = pm j_2$Prove that if $(a+b)$ divides $a^2$ then $(a+b)$ divides $b^2$prove that 15 divides $n^7+2n^5+4n^3+8n$ for any integer nIf $f(x)$ divides $g(x)$ in $R[x]$ then $f(h(x))$ divides $g(h(x))$ in $R[x]$ (?).Prove that if $a$ divides $b^2$ then $a$ divides $b$
Random body shuffle every night—can we still function?
How do I find out the mythology and history of my Fortress?
How to report t statistic from R
Is CEO the "profession" with the most psychopaths?
Can the Flaming Sphere spell be rammed into multiple Tiny creatures that are in the same 5-foot square?
How does Belgium enforce obligatory attendance in elections?
Do wooden building fires get hotter than 600°C?
How does light 'choose' between wave and particle behaviour?
Co-worker has annoying ringtone
Semigroups with no morphisms between them
How to write capital alpha?
Why can't I install Tomboy in Ubuntu Mate 19.04?
Converted a Scalar function to a TVF function for parallel execution-Still running in Serial mode
Strange behavior of Object.defineProperty() in JavaScript
How can I prevent/balance waiting and turtling as a response to cooldown mechanics
Why are vacuum tubes still used in amateur radios?
Why is it faster to reheat something than it is to cook it?
What does 丫 mean? 丫是什么意思？
What does this say in Elvish?
How does the math work when buying airline miles?
What to do with repeated rejections for phd position
Why do early math courses focus on the cross sections of a cone and not on other 3D objects?
Why do aircraft stall warning systems use angle-of-attack vanes rather than detecting airflow separation directly?
Karn the great creator - 'card from outside the game' in sealed
Prove that if $f(x)$ divides $g(x)$ and $f(x)$ divides $h(x)$, then $f(x)$ divides $s(x)g(x) +t(x)h(x)$.
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)If $3$ divides $a^2 + b^2$, then $3$ divides $a$ and $3$ divides $b$If $gcd(a,b)= 1$ and $a$ divides $bc$ then $a$ divides $c $ [Euclid's Lemma]Show that if $10$ divides into $n^2$ evenly then $10$ divides into $n$ evenlyIf $d$ divides $k$ and $d$ divides $n$, then $d$ divides $(8k - 3n)$How to prove that linear polynomials are irreducible?Prove divisibility: If $j_1mid j_2$ and $j_2mid j_1$, then $j_1 = pm j_2$Prove that if $(a+b)$ divides $a^2$ then $(a+b)$ divides $b^2$prove that 15 divides $n^7+2n^5+4n^3+8n$ for any integer nIf $f(x)$ divides $g(x)$ in $R[x]$ then $f(h(x))$ divides $g(h(x))$ in $R[x]$ (?).Prove that if $a$ divides $b^2$ then $a$ divides $b$
$begingroup$
Let $F$ be a field. Prove that for all polynomials $f(x), g(x), h(x) in F[x]$, if $f(x)$ divides $g(x)$ and $f(x)$ divides $h(x)$, then for all polynomials $s(x), t(x)in F[x]$, $f(x)$ divides $s(x)g(x) +t(x)h(x)$.
How do I prove this question? I know that $f(x)=g(x)q(x)+r(x)$ but I'm not sure if I use that at all in this question.
polynomials divisibility
asked Apr 2 at 1:36
SaniaSania
406
$endgroup$
add a comment |
$begingroup$
Let $F$ be a field. Prove that for all polynomials $f(x), g(x), h(x) in F[x]$, if $f(x)$ divides $g(x)$ and $f(x)$ divides $h(x)$, then for all polynomials $s(x), t(x)in F[x]$, $f(x)$ divides $s(x)g(x) +t(x)h(x)$.
How do I prove this question? I know that $f(x)=g(x)q(x)+r(x)$ but I'm not sure if I use that at all in this question.
polynomials divisibility
asked Apr 2 at 1:36
SaniaSania
406
$endgroup$
$begingroup$
$f(x)$ is a factor of both $g(x)$ and $h(x)$. Consider the case where $f(x)=2$, $g(x)=4$, and $h(x)=20$. Does $f(x)$ divide $4cdot s + 20 cdot t$?
$endgroup$
– Andrew Tawfeek
Apr 2 at 1:49
add a comment |
$begingroup$
Let $F$ be a field. Prove that for all polynomials $f(x), g(x), h(x) in F[x]$, if $f(x)$ divides $g(x)$ and $f(x)$ divides $h(x)$, then for all polynomials $s(x), t(x)in F[x]$, $f(x)$ divides $s(x)g(x) +t(x)h(x)$.
How do I prove this question? I know that $f(x)=g(x)q(x)+r(x)$ but I'm not sure if I use that at all in this question.
polynomials divisibility
asked Apr 2 at 1:36
SaniaSania
406
$endgroup$
Let $F$ be a field. Prove that for all polynomials $f(x), g(x), h(x) in F[x]$, if $f(x)$ divides $g(x)$ and $f(x)$ divides $h(x)$, then for all polynomials $s(x), t(x)in F[x]$, $f(x)$ divides $s(x)g(x) +t(x)h(x)$.
How do I prove this question? I know that $f(x)=g(x)q(x)+r(x)$ but I'm not sure if I use that at all in this question.
polynomials divisibility
polynomials divisibility
asked Apr 2 at 1:36
SaniaSania
406
asked Apr 2 at 1:36
SaniaSania
406
asked Apr 2 at 1:36
SaniaSania
406
asked Apr 2 at 1:36
SaniaSania
406
asked Apr 2 at 1:36
SaniaSania
406
406
$begingroup$
$f(x)$ is a factor of both $g(x)$ and $h(x)$. Consider the case where $f(x)=2$, $g(x)=4$, and $h(x)=20$. Does $f(x)$ divide $4cdot s + 20 cdot t$?
$endgroup$
– Andrew Tawfeek
Apr 2 at 1:49
add a comment |
$begingroup$
$f(x)$ is a factor of both $g(x)$ and $h(x)$. Consider the case where $f(x)=2$, $g(x)=4$, and $h(x)=20$. Does $f(x)$ divide $4cdot s + 20 cdot t$?
$endgroup$
– Andrew Tawfeek
Apr 2 at 1:49
$begingroup$
$f(x)$ is a factor of both $g(x)$ and $h(x)$. Consider the case where $f(x)=2$, $g(x)=4$, and $h(x)=20$. Does $f(x)$ divide $4cdot s + 20 cdot t$?
$endgroup$
– Andrew Tawfeek
Apr 2 at 1:49
$begingroup$
$f(x)$ is a factor of both $g(x)$ and $h(x)$. Consider the case where $f(x)=2$, $g(x)=4$, and $h(x)=20$. Does $f(x)$ divide $4cdot s + 20 cdot t$?
$endgroup$
– Andrew Tawfeek
Apr 2 at 1:49
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Hint: How would you prove an equivalent statment for the integers?
answered Apr 2 at 1:38
Maria MazurMaria Mazur
50.3k1361126
$endgroup$
$begingroup$
Would I use DIC for polynomials?
$endgroup$
– Sania
Apr 2 at 1:40
$begingroup$
What is DIC?...
$endgroup$
– Maria Mazur
Apr 2 at 1:41
$begingroup$
If a divides b and a divides c then for integers x,y a divides bx+cy
$endgroup$
– Sania
Apr 2 at 1:41
$begingroup$
Actualy this is an equivalent statement, you need to remember how do you prove it.
$endgroup$
– Maria Mazur
Apr 2 at 1:43
add a comment |
$begingroup$
If $f(x)$ divides $g(x)$, then $g(x) = f(x)q(x)$ and likewise $h(x) = f(x)r(x)$.
Also note that $f(x)$ divides $s(x)g(x)$, since $s(x)g(x) = s(x) f(x)q(x)$.
answered Apr 2 at 1:40
user458276user458276
7431315
$endgroup$
add a comment |
Your Answer
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3171354%2fprove-that-if-fx-divides-gx-and-fx-divides-hx-then-fx-divid%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: How would you prove an equivalent statment for the integers?
answered Apr 2 at 1:38
Maria MazurMaria Mazur
50.3k1361126
$endgroup$
$begingroup$
Would I use DIC for polynomials?
$endgroup$
– Sania
Apr 2 at 1:40
$begingroup$
What is DIC?...
$endgroup$
– Maria Mazur
Apr 2 at 1:41
$begingroup$
If a divides b and a divides c then for integers x,y a divides bx+cy
$endgroup$
– Sania
Apr 2 at 1:41
$begingroup$
Actualy this is an equivalent statement, you need to remember how do you prove it.
$endgroup$
– Maria Mazur
Apr 2 at 1:43
add a comment |
$begingroup$
Hint: How would you prove an equivalent statment for the integers?
answered Apr 2 at 1:38
Maria MazurMaria Mazur
50.3k1361126
$endgroup$
$begingroup$
Would I use DIC for polynomials?
$endgroup$
– Sania
Apr 2 at 1:40
$begingroup$
What is DIC?...
$endgroup$
– Maria Mazur
Apr 2 at 1:41
$begingroup$
If a divides b and a divides c then for integers x,y a divides bx+cy
$endgroup$
– Sania
Apr 2 at 1:41
$begingroup$
Actualy this is an equivalent statement, you need to remember how do you prove it.
$endgroup$
– Maria Mazur
Apr 2 at 1:43
add a comment |
$begingroup$
Hint: How would you prove an equivalent statment for the integers?
answered Apr 2 at 1:38
Maria MazurMaria Mazur
50.3k1361126
$endgroup$
Hint: How would you prove an equivalent statment for the integers?
answered Apr 2 at 1:38
Maria MazurMaria Mazur
50.3k1361126
answered Apr 2 at 1:38
Maria MazurMaria Mazur
50.3k1361126
answered Apr 2 at 1:38
Maria MazurMaria Mazur
50.3k1361126
answered Apr 2 at 1:38
Maria MazurMaria Mazur
50.3k1361126
50.3k1361126
$begingroup$
Would I use DIC for polynomials?
$endgroup$
– Sania
Apr 2 at 1:40
$begingroup$
What is DIC?...
$endgroup$
– Maria Mazur
Apr 2 at 1:41
$begingroup$
If a divides b and a divides c then for integers x,y a divides bx+cy
$endgroup$
– Sania
Apr 2 at 1:41
$begingroup$
Actualy this is an equivalent statement, you need to remember how do you prove it.
$endgroup$
– Maria Mazur
Apr 2 at 1:43
add a comment |
$begingroup$
Would I use DIC for polynomials?
$endgroup$
– Sania
Apr 2 at 1:40
$begingroup$
What is DIC?...
$endgroup$
– Maria Mazur
Apr 2 at 1:41
$begingroup$
If a divides b and a divides c then for integers x,y a divides bx+cy
$endgroup$
– Sania
Apr 2 at 1:41
$begingroup$
Actualy this is an equivalent statement, you need to remember how do you prove it.
$endgroup$
– Maria Mazur
Apr 2 at 1:43
$begingroup$
Would I use DIC for polynomials?
$endgroup$
– Sania
Apr 2 at 1:40
$begingroup$
Would I use DIC for polynomials?
$endgroup$
– Sania
Apr 2 at 1:40
$begingroup$
What is DIC?...
$endgroup$
– Maria Mazur
Apr 2 at 1:41
$begingroup$
What is DIC?...
$endgroup$
– Maria Mazur
Apr 2 at 1:41
$begingroup$
If a divides b and a divides c then for integers x,y a divides bx+cy
$endgroup$
– Sania
Apr 2 at 1:41
$begingroup$
If a divides b and a divides c then for integers x,y a divides bx+cy
$endgroup$
– Sania
Apr 2 at 1:41
$begingroup$
Actualy this is an equivalent statement, you need to remember how do you prove it.
$endgroup$
– Maria Mazur
Apr 2 at 1:43
$begingroup$
Actualy this is an equivalent statement, you need to remember how do you prove it.
$endgroup$
– Maria Mazur
Apr 2 at 1:43
add a comment |
$begingroup$
If $f(x)$ divides $g(x)$, then $g(x) = f(x)q(x)$ and likewise $h(x) = f(x)r(x)$.
Also note that $f(x)$ divides $s(x)g(x)$, since $s(x)g(x) = s(x) f(x)q(x)$.
answered Apr 2 at 1:40
user458276user458276
7431315
$endgroup$
add a comment |
$begingroup$
If $f(x)$ divides $g(x)$, then $g(x) = f(x)q(x)$ and likewise $h(x) = f(x)r(x)$.
Also note that $f(x)$ divides $s(x)g(x)$, since $s(x)g(x) = s(x) f(x)q(x)$.
answered Apr 2 at 1:40
user458276user458276
7431315
$endgroup$
add a comment |
$begingroup$
If $f(x)$ divides $g(x)$, then $g(x) = f(x)q(x)$ and likewise $h(x) = f(x)r(x)$.
Also note that $f(x)$ divides $s(x)g(x)$, since $s(x)g(x) = s(x) f(x)q(x)$.
answered Apr 2 at 1:40
user458276user458276
7431315
$endgroup$
If $f(x)$ divides $g(x)$, then $g(x) = f(x)q(x)$ and likewise $h(x) = f(x)r(x)$.
Also note that $f(x)$ divides $s(x)g(x)$, since $s(x)g(x) = s(x) f(x)q(x)$.
answered Apr 2 at 1:40
user458276user458276
7431315
answered Apr 2 at 1:40
user458276user458276
7431315
answered Apr 2 at 1:40
user458276user458276
7431315
answered Apr 2 at 1:40
user458276user458276
7431315
7431315
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3171354%2fprove-that-if-fx-divides-gx-and-fx-divides-hx-then-fx-divid%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
$f(x)$ is a factor of both $g(x)$ and $h(x)$. Consider the case where $f(x)=2$, $g(x)=4$, and $h(x)=20$. Does $f(x)$ divide $4cdot s + 20 cdot t$?
$endgroup$
– Andrew Tawfeek
Apr 2 at 1:49