Polynomial and roots problemsUsing Vieta's theorem for cubic equations to derive the cubic discriminantExist another method to solve the problem?Polynomial with real rootsVieta's Theorems ClarificationHow to find area of a polygon built on the roots of a given polynomial?Finding $a$ in quadratic equation $2x^2 - (a+1)x + (a-1)=0$ so that difference of two roots is equal to its productFind $frac1x_1^3 + frac1x_2^3 + frac1x_3^3$ for $ax^3 + bx^2 + cx + d$What are the values of $a$, one of the roots of the equation is greater than $1$ and the other is less than $1$?For $3x^3-5x^2+12x-18=0$ find the value of $(1+fracx_1x_2)(1+fracx_2x_3)(1+fracx_3x_1)$ using Vieta's formulasFind all $a, b in mathbb R$, ($bne0)$, such that the roots of $x^2+ax+a=b$ and $x^2+ax+a=-b$ are 4 consecutive numbers
How might boat designs change in order to allow them to be pulled by dragons?
Cannot overlay, because ListPlot does not draw same X range despite the same PlotRange
Replacing 5 gang light switches that have 3 of them daisy chained together
Why are examinees often not allowed to leave during the start and end of an exam?
Why are symbols not written in words?
Turing Machines: What is the difference between recognizing, deciding, total, accepting, rejecting?
Why will we fail creating a self sustaining off world colony?
How can I know (without going to the station) if RATP is offering the Anti Pollution tickets?
Are the plates of a battery really charged?
What is the function of const specifier in enum types?
How can solar sailed ships be protected from space debris?
Simplify the code
2019 2-letters 33-length list
Disk usage confusion: 10G missing on Linux home partition on SSD
How do I tell my girlfriend she's been buying me books by the wrong author for the last nine months?
How soon after takeoff can you recline your airplane seat?
Are there advantages in writing by hand over typing out a story?
Is this house-rule removing the increased effect of cantrips at higher character levels balanced?
Searching for single buildings in QGIS
How to idiomatically express the idea "if you can cheat without being caught, do it"
GFCI versus circuit breaker
What caused the flashes in the video footage of Chernobyl?
To “Er” Is Human
Russian equivalents of 能骗就骗 (if you can cheat, then cheat)
Polynomial and roots problems
Using Vieta's theorem for cubic equations to derive the cubic discriminantExist another method to solve the problem?Polynomial with real rootsVieta's Theorems ClarificationHow to find area of a polygon built on the roots of a given polynomial?Finding $a$ in quadratic equation $2x^2 - (a+1)x + (a-1)=0$ so that difference of two roots is equal to its productFind $frac1x_1^3 + frac1x_2^3 + frac1x_3^3$ for $ax^3 + bx^2 + cx + d$What are the values of $a$, one of the roots of the equation is greater than $1$ and the other is less than $1$?For $3x^3-5x^2+12x-18=0$ find the value of $(1+fracx_1x_2)(1+fracx_2x_3)(1+fracx_3x_1)$ using Vieta's formulasFind all $a, b in mathbb R$, ($bne0)$, such that the roots of $x^2+ax+a=b$ and $x^2+ax+a=-b$ are 4 consecutive numbers
$begingroup$
Let $x_1$, $x_2$, $x_3$ be the roots of $x^3−3x−15=0$.
Find $x_1^3+3x_2+3x_3$.
I tried solving the problem using formulas from Vieta's theorem, but I was unable to find any plausible ways to calculate the end result. Does anyone know how to do this?
algebra-precalculus
edited 9 hours ago
David G. Stork
13.3k4 gold badges19 silver badges37 bronze badges
asked 9 hours ago
AaronAaron
662 bronze badges
New contributor
Aaron is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Let $x_1$, $x_2$, $x_3$ be the roots of $x^3−3x−15=0$.
Find $x_1^3+3x_2+3x_3$.
I tried solving the problem using formulas from Vieta's theorem, but I was unable to find any plausible ways to calculate the end result. Does anyone know how to do this?
algebra-precalculus
edited 9 hours ago
David G. Stork
13.3k4 gold badges19 silver badges37 bronze badges
asked 9 hours ago
AaronAaron
662 bronze badges
New contributor
Aaron is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
@Monadologie's deleted answer is correct: $15$.
$endgroup$
– David G. Stork
9 hours ago
add a comment |
$begingroup$
Let $x_1$, $x_2$, $x_3$ be the roots of $x^3−3x−15=0$.
Find $x_1^3+3x_2+3x_3$.
I tried solving the problem using formulas from Vieta's theorem, but I was unable to find any plausible ways to calculate the end result. Does anyone know how to do this?
algebra-precalculus
edited 9 hours ago
David G. Stork
13.3k4 gold badges19 silver badges37 bronze badges
asked 9 hours ago
AaronAaron
662 bronze badges
New contributor
Aaron is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Let $x_1$, $x_2$, $x_3$ be the roots of $x^3−3x−15=0$.
Find $x_1^3+3x_2+3x_3$.
I tried solving the problem using formulas from Vieta's theorem, but I was unable to find any plausible ways to calculate the end result. Does anyone know how to do this?
algebra-precalculus
algebra-precalculus
edited 9 hours ago
David G. Stork
13.3k4 gold badges19 silver badges37 bronze badges
asked 9 hours ago
AaronAaron
662 bronze badges
New contributor
Aaron is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 9 hours ago
David G. Stork
13.3k4 gold badges19 silver badges37 bronze badges
asked 9 hours ago
AaronAaron
662 bronze badges
New contributor
Aaron is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 9 hours ago
David G. Stork
13.3k4 gold badges19 silver badges37 bronze badges
edited 9 hours ago
David G. Stork
13.3k4 gold badges19 silver badges37 bronze badges
edited 9 hours ago
David G. Stork
13.3k4 gold badges19 silver badges37 bronze badges
13.3k4 gold badges19 silver badges37 bronze badges
asked 9 hours ago
AaronAaron
662 bronze badges
New contributor
Aaron is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 9 hours ago
AaronAaron
662 bronze badges
asked 9 hours ago
AaronAaron
662 bronze badges
662 bronze badges
New contributor
Aaron is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Aaron is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
@Monadologie's deleted answer is correct: $15$.
$endgroup$
– David G. Stork
9 hours ago
add a comment |
$begingroup$
@Monadologie's deleted answer is correct: $15$.
$endgroup$
– David G. Stork
9 hours ago
$begingroup$
@Monadologie's deleted answer is correct: $15$.
$endgroup$
– David G. Stork
9 hours ago
$begingroup$
@Monadologie's deleted answer is correct: $15$.
$endgroup$
– David G. Stork
9 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
$3(x_1+x_2+x_3)=0$ since the polynomial has no $x^2$ term. Thus,
$$
x_1^3+3x_2+3x_3=x_1^3-3x_1=15.
$$
answered 9 hours ago
pre-kidneypre-kidney
15.6k20 silver badges56 bronze badges
$endgroup$
add a comment |
$begingroup$
Let $a,b,c$ be the roots of
$$
x^3 -3x -15=0
$$
Let
$$
I=a^3 + 3b + 3c
$$
We have
$$
a^3 = 3a + 15
$$
Since $a+b+c=- fraca_2a_3 = 0$ (Vieta's formulas),
$$
I= 3a + 15 + 3b + 3c = 15 + 3(a+b+c)
= 15
$$
answered 9 hours ago
MonadologieMonadologie
3413 bronze badges
$endgroup$
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
);
);
Aaron is a new contributor. Be nice, and check out our Code of Conduct.
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%2f3279088%2fpolynomial-and-roots-problems%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
$begingroup$
$3(x_1+x_2+x_3)=0$ since the polynomial has no $x^2$ term. Thus,
$$
x_1^3+3x_2+3x_3=x_1^3-3x_1=15.
$$
answered 9 hours ago
pre-kidneypre-kidney
15.6k20 silver badges56 bronze badges
$endgroup$
add a comment |
$begingroup$
$3(x_1+x_2+x_3)=0$ since the polynomial has no $x^2$ term. Thus,
$$
x_1^3+3x_2+3x_3=x_1^3-3x_1=15.
$$
answered 9 hours ago
pre-kidneypre-kidney
15.6k20 silver badges56 bronze badges
$endgroup$
add a comment |
$begingroup$
$3(x_1+x_2+x_3)=0$ since the polynomial has no $x^2$ term. Thus,
$$
x_1^3+3x_2+3x_3=x_1^3-3x_1=15.
$$
answered 9 hours ago
pre-kidneypre-kidney
15.6k20 silver badges56 bronze badges
$endgroup$
$3(x_1+x_2+x_3)=0$ since the polynomial has no $x^2$ term. Thus,
$$
x_1^3+3x_2+3x_3=x_1^3-3x_1=15.
$$
answered 9 hours ago
pre-kidneypre-kidney
15.6k20 silver badges56 bronze badges
answered 9 hours ago
pre-kidneypre-kidney
15.6k20 silver badges56 bronze badges
answered 9 hours ago
pre-kidneypre-kidney
15.6k20 silver badges56 bronze badges
answered 9 hours ago
pre-kidneypre-kidney
15.6k20 silver badges56 bronze badges
15.6k20 silver badges56 bronze badges
add a comment |
add a comment |
$begingroup$
Let $a,b,c$ be the roots of
$$
x^3 -3x -15=0
$$
Let
$$
I=a^3 + 3b + 3c
$$
We have
$$
a^3 = 3a + 15
$$
Since $a+b+c=- fraca_2a_3 = 0$ (Vieta's formulas),
$$
I= 3a + 15 + 3b + 3c = 15 + 3(a+b+c)
= 15
$$
answered 9 hours ago
MonadologieMonadologie
3413 bronze badges
$endgroup$
add a comment |
$begingroup$
Let $a,b,c$ be the roots of
$$
x^3 -3x -15=0
$$
Let
$$
I=a^3 + 3b + 3c
$$
We have
$$
a^3 = 3a + 15
$$
Since $a+b+c=- fraca_2a_3 = 0$ (Vieta's formulas),
$$
I= 3a + 15 + 3b + 3c = 15 + 3(a+b+c)
= 15
$$
answered 9 hours ago
MonadologieMonadologie
3413 bronze badges
$endgroup$
add a comment |
$begingroup$
Let $a,b,c$ be the roots of
$$
x^3 -3x -15=0
$$
Let
$$
I=a^3 + 3b + 3c
$$
We have
$$
a^3 = 3a + 15
$$
Since $a+b+c=- fraca_2a_3 = 0$ (Vieta's formulas),
$$
I= 3a + 15 + 3b + 3c = 15 + 3(a+b+c)
= 15
$$
answered 9 hours ago
MonadologieMonadologie
3413 bronze badges
$endgroup$
Let $a,b,c$ be the roots of
$$
x^3 -3x -15=0
$$
Let
$$
I=a^3 + 3b + 3c
$$
We have
$$
a^3 = 3a + 15
$$
Since $a+b+c=- fraca_2a_3 = 0$ (Vieta's formulas),
$$
I= 3a + 15 + 3b + 3c = 15 + 3(a+b+c)
= 15
$$
answered 9 hours ago
MonadologieMonadologie
3413 bronze badges
answered 9 hours ago
MonadologieMonadologie
3413 bronze badges
answered 9 hours ago
MonadologieMonadologie
3413 bronze badges
answered 9 hours ago
MonadologieMonadologie
3413 bronze badges
3413 bronze badges
add a comment |
add a comment |
Aaron is a new contributor. Be nice, and check out our Code of Conduct.
Aaron is a new contributor. Be nice, and check out our Code of Conduct.
Aaron is a new contributor. Be nice, and check out our Code of Conduct.
Aaron is a new contributor. Be nice, and check out our Code of Conduct.
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%2f3279088%2fpolynomial-and-roots-problems%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
$begingroup$
@Monadologie's deleted answer is correct: $15$.
$endgroup$
– David G. Stork
9 hours ago