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Intuitive Explanation Of Descartes' Rule Of Signs
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Can someone please explain to me why, intuitively, does Descartes rule of signs work?
I realize there is a previous answer for this, that "Basically, at different values of $x$ different terms in the polynomial "dominate." So every the sign switches, there will be a change in the direction of the curve. Either
- This will result in crossing the $x$-axis and a root or
- There will have to be another change, meaning "losing roots" will always happen in pairs.
So the roots are equal to, or less than
by an even number, the number of sign changes." But perhaps because my
math understanding is not good enough, I still fail to see why this
ensures that the Descartes rule of signs works.
Why would different values of $x$ dominate in different areas?
And why would this result in crossing the $x$-axis or losing roots?
Can you please explain to me why the rule of signs can find the number of real zeroes for any polynomial? Please keep the explanation simple.
algebra-precalculus polynomials
edited Jul 17 at 14:48
Adrian Keister
3,61721533
asked Jul 17 at 13:59
Mathguy
735
 |Â
show 1 more comment
up vote
11
down vote
favorite
Can someone please explain to me why, intuitively, does Descartes rule of signs work?
I realize there is a previous answer for this, that "Basically, at different values of $x$ different terms in the polynomial "dominate." So every the sign switches, there will be a change in the direction of the curve. Either
- This will result in crossing the $x$-axis and a root or
- There will have to be another change, meaning "losing roots" will always happen in pairs.
So the roots are equal to, or less than
by an even number, the number of sign changes." But perhaps because my
math understanding is not good enough, I still fail to see why this
ensures that the Descartes rule of signs works.
Why would different values of $x$ dominate in different areas?
And why would this result in crossing the $x$-axis or losing roots?
Can you please explain to me why the rule of signs can find the number of real zeroes for any polynomial? Please keep the explanation simple.
algebra-precalculus polynomials
edited Jul 17 at 14:48
Adrian Keister
3,61721533
asked Jul 17 at 13:59
Mathguy
735
Note that implicitly the "number of roots" here means "with multiplicities", otherwise at the very moment when two roots disappear the curve is tangent to the $x$-axis and you would only have one root replacing two, breaking the parity.
– Arnaud Mortier
Jul 17 at 14:17
1
It is not true that "the rule of signs can find the number of real zeroes for any polynomial". You should be aware of the limitations of the rule, and various supplementary tactics to get more information when Descartes' Rule is ambiguous.
– hardmath
Jul 17 at 14:18
1
relevant earlier question can be found here
– Alexander Gruber♦
Jul 17 at 19:17
@hardmath Actually, in what situations will the descartes rule of signs be ambiguous, and in that situation, what can you do?
– Ethan Chan
Jul 18 at 14:01
@EthanChan: The OP hints at the limitation to knowing the number of (positive) real roots only up to some even number of possible overcounting. The OP omits to mention the counting of sign changes relates only to positive real roots, etc. A simple change of variable, replacing $x$ with $-x$, allows us to get information about negative real roots in similar fashion. More intricate changes of variable can be used to isolate real roots, but this Comment is not big enough to summarize the literature.
– hardmath
Jul 18 at 16:36
 |Â
show 1 more comment
up vote
11
down vote
favorite
up vote
11
down vote
favorite
Can someone please explain to me why, intuitively, does Descartes rule of signs work?
I realize there is a previous answer for this, that "Basically, at different values of $x$ different terms in the polynomial "dominate." So every the sign switches, there will be a change in the direction of the curve. Either
- This will result in crossing the $x$-axis and a root or
- There will have to be another change, meaning "losing roots" will always happen in pairs.
So the roots are equal to, or less than
by an even number, the number of sign changes." But perhaps because my
math understanding is not good enough, I still fail to see why this
ensures that the Descartes rule of signs works.
Why would different values of $x$ dominate in different areas?
And why would this result in crossing the $x$-axis or losing roots?
Can you please explain to me why the rule of signs can find the number of real zeroes for any polynomial? Please keep the explanation simple.
algebra-precalculus polynomials
edited Jul 17 at 14:48
Adrian Keister
3,61721533
asked Jul 17 at 13:59
Mathguy
735
Can someone please explain to me why, intuitively, does Descartes rule of signs work?
I realize there is a previous answer for this, that "Basically, at different values of $x$ different terms in the polynomial "dominate." So every the sign switches, there will be a change in the direction of the curve. Either
- This will result in crossing the $x$-axis and a root or
- There will have to be another change, meaning "losing roots" will always happen in pairs.
So the roots are equal to, or less than
by an even number, the number of sign changes." But perhaps because my
math understanding is not good enough, I still fail to see why this
ensures that the Descartes rule of signs works.
Why would different values of $x$ dominate in different areas?
And why would this result in crossing the $x$-axis or losing roots?
Can you please explain to me why the rule of signs can find the number of real zeroes for any polynomial? Please keep the explanation simple.
algebra-precalculus polynomials
edited Jul 17 at 14:48
Adrian Keister
3,61721533
asked Jul 17 at 13:59
Mathguy
735
edited Jul 17 at 14:48
Adrian Keister
3,61721533
edited Jul 17 at 14:48
Adrian Keister
3,61721533
edited Jul 17 at 14:48
Adrian Keister
3,61721533
3,61721533
asked Jul 17 at 13:59
Mathguy
735
asked Jul 17 at 13:59
Mathguy
735
asked Jul 17 at 13:59
Mathguy
735
735
Note that implicitly the "number of roots" here means "with multiplicities", otherwise at the very moment when two roots disappear the curve is tangent to the $x$-axis and you would only have one root replacing two, breaking the parity.
– Arnaud Mortier
Jul 17 at 14:17
1
It is not true that "the rule of signs can find the number of real zeroes for any polynomial". You should be aware of the limitations of the rule, and various supplementary tactics to get more information when Descartes' Rule is ambiguous.
– hardmath
Jul 17 at 14:18
1
relevant earlier question can be found here
– Alexander Gruber♦
Jul 17 at 19:17
@hardmath Actually, in what situations will the descartes rule of signs be ambiguous, and in that situation, what can you do?
– Ethan Chan
Jul 18 at 14:01
@EthanChan: The OP hints at the limitation to knowing the number of (positive) real roots only up to some even number of possible overcounting. The OP omits to mention the counting of sign changes relates only to positive real roots, etc. A simple change of variable, replacing $x$ with $-x$, allows us to get information about negative real roots in similar fashion. More intricate changes of variable can be used to isolate real roots, but this Comment is not big enough to summarize the literature.
– hardmath
Jul 18 at 16:36
 |Â
show 1 more comment
Note that implicitly the "number of roots" here means "with multiplicities", otherwise at the very moment when two roots disappear the curve is tangent to the $x$-axis and you would only have one root replacing two, breaking the parity.
– Arnaud Mortier
Jul 17 at 14:17
1
It is not true that "the rule of signs can find the number of real zeroes for any polynomial". You should be aware of the limitations of the rule, and various supplementary tactics to get more information when Descartes' Rule is ambiguous.
– hardmath
Jul 17 at 14:18
1
relevant earlier question can be found here
– Alexander Gruber♦
Jul 17 at 19:17
@hardmath Actually, in what situations will the descartes rule of signs be ambiguous, and in that situation, what can you do?
– Ethan Chan
Jul 18 at 14:01
@EthanChan: The OP hints at the limitation to knowing the number of (positive) real roots only up to some even number of possible overcounting. The OP omits to mention the counting of sign changes relates only to positive real roots, etc. A simple change of variable, replacing $x$ with $-x$, allows us to get information about negative real roots in similar fashion. More intricate changes of variable can be used to isolate real roots, but this Comment is not big enough to summarize the literature.
– hardmath
Jul 18 at 16:36
Note that implicitly the "number of roots" here means "with multiplicities", otherwise at the very moment when two roots disappear the curve is tangent to the $x$-axis and you would only have one root replacing two, breaking the parity.
– Arnaud Mortier
Jul 17 at 14:17
Note that implicitly the "number of roots" here means "with multiplicities", otherwise at the very moment when two roots disappear the curve is tangent to the $x$-axis and you would only have one root replacing two, breaking the parity.
– Arnaud Mortier
Jul 17 at 14:17
1
1
It is not true that "the rule of signs can find the number of real zeroes for any polynomial". You should be aware of the limitations of the rule, and various supplementary tactics to get more information when Descartes' Rule is ambiguous.
– hardmath
Jul 17 at 14:18
It is not true that "the rule of signs can find the number of real zeroes for any polynomial". You should be aware of the limitations of the rule, and various supplementary tactics to get more information when Descartes' Rule is ambiguous.
– hardmath
Jul 17 at 14:18
1
1
relevant earlier question can be found here
– Alexander Gruber♦
Jul 17 at 19:17
relevant earlier question can be found here
– Alexander Gruber♦
Jul 17 at 19:17
@hardmath Actually, in what situations will the descartes rule of signs be ambiguous, and in that situation, what can you do?
– Ethan Chan
Jul 18 at 14:01
@hardmath Actually, in what situations will the descartes rule of signs be ambiguous, and in that situation, what can you do?
– Ethan Chan
Jul 18 at 14:01
@EthanChan: The OP hints at the limitation to knowing the number of (positive) real roots only up to some even number of possible overcounting. The OP omits to mention the counting of sign changes relates only to positive real roots, etc. A simple change of variable, replacing $x$ with $-x$, allows us to get information about negative real roots in similar fashion. More intricate changes of variable can be used to isolate real roots, but this Comment is not big enough to summarize the literature.
– hardmath
Jul 18 at 16:36
@EthanChan: The OP hints at the limitation to knowing the number of (positive) real roots only up to some even number of possible overcounting. The OP omits to mention the counting of sign changes relates only to positive real roots, etc. A simple change of variable, replacing $x$ with $-x$, allows us to get information about negative real roots in similar fashion. More intricate changes of variable can be used to isolate real roots, but this Comment is not big enough to summarize the literature.
– hardmath
Jul 18 at 16:36
 |Â
show 1 more comment
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Note that implicitly the "number of roots" here means "with multiplicities", otherwise at the very moment when two roots disappear the curve is tangent to the $x$-axis and you would only have one root replacing two, breaking the parity.
– Arnaud Mortier
Jul 17 at 14:17
1
It is not true that "the rule of signs can find the number of real zeroes for any polynomial". You should be aware of the limitations of the rule, and various supplementary tactics to get more information when Descartes' Rule is ambiguous.
– hardmath
Jul 17 at 14:18
1
relevant earlier question can be found here
– Alexander Gruber♦
Jul 17 at 19:17
@hardmath Actually, in what situations will the descartes rule of signs be ambiguous, and in that situation, what can you do?
– Ethan Chan
Jul 18 at 14:01
@EthanChan: The OP hints at the limitation to knowing the number of (positive) real roots only up to some even number of possible overcounting. The OP omits to mention the counting of sign changes relates only to positive real roots, etc. A simple change of variable, replacing $x$ with $-x$, allows us to get information about negative real roots in similar fashion. More intricate changes of variable can be used to isolate real roots, but this Comment is not big enough to summarize the literature.
– hardmath
Jul 18 at 16:36