Can some cubic polynomial have two real roots?
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In $p(x) = x^3-x^2$, both $0$ and $1$ are possible roots of the polynomial; both are real. I had read that a cubic polynomial has either all real roots or just one real root. It can't have two. What is the problem in this case?
polynomials complex-numbers roots cubic-equations
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add a comment |
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In $p(x) = x^3-x^2$, both $0$ and $1$ are possible roots of the polynomial; both are real. I had read that a cubic polynomial has either all real roots or just one real root. It can't have two. What is the problem in this case?
polynomials complex-numbers roots cubic-equations
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1
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I think the statement is that if a cubic polynomial has three distinct roots, then either all 3 are real or only one of them is real and the other two are complex conjugates of each other.
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– harshit54
Jan 2 at 12:45
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The title is not good. A trivial answer would be: Yes, take $f(x)=x^3$. It has $3$ real roots, $0$, $0$ and again $0$.
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– Dietrich Burde
Jan 2 at 13:46
add a comment |
$begingroup$
In $p(x) = x^3-x^2$, both $0$ and $1$ are possible roots of the polynomial; both are real. I had read that a cubic polynomial has either all real roots or just one real root. It can't have two. What is the problem in this case?
polynomials complex-numbers roots cubic-equations
$endgroup$
In $p(x) = x^3-x^2$, both $0$ and $1$ are possible roots of the polynomial; both are real. I had read that a cubic polynomial has either all real roots or just one real root. It can't have two. What is the problem in this case?
polynomials complex-numbers roots cubic-equations
polynomials complex-numbers roots cubic-equations
edited Jan 2 at 14:01
Shailesh
3,95392134
3,95392134
asked Jan 2 at 12:40
RaenRaen
354
354
1
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I think the statement is that if a cubic polynomial has three distinct roots, then either all 3 are real or only one of them is real and the other two are complex conjugates of each other.
$endgroup$
– harshit54
Jan 2 at 12:45
$begingroup$
The title is not good. A trivial answer would be: Yes, take $f(x)=x^3$. It has $3$ real roots, $0$, $0$ and again $0$.
$endgroup$
– Dietrich Burde
Jan 2 at 13:46
add a comment |
1
$begingroup$
I think the statement is that if a cubic polynomial has three distinct roots, then either all 3 are real or only one of them is real and the other two are complex conjugates of each other.
$endgroup$
– harshit54
Jan 2 at 12:45
$begingroup$
The title is not good. A trivial answer would be: Yes, take $f(x)=x^3$. It has $3$ real roots, $0$, $0$ and again $0$.
$endgroup$
– Dietrich Burde
Jan 2 at 13:46
1
1
$begingroup$
I think the statement is that if a cubic polynomial has three distinct roots, then either all 3 are real or only one of them is real and the other two are complex conjugates of each other.
$endgroup$
– harshit54
Jan 2 at 12:45
$begingroup$
I think the statement is that if a cubic polynomial has three distinct roots, then either all 3 are real or only one of them is real and the other two are complex conjugates of each other.
$endgroup$
– harshit54
Jan 2 at 12:45
$begingroup$
The title is not good. A trivial answer would be: Yes, take $f(x)=x^3$. It has $3$ real roots, $0$, $0$ and again $0$.
$endgroup$
– Dietrich Burde
Jan 2 at 13:46
$begingroup$
The title is not good. A trivial answer would be: Yes, take $f(x)=x^3$. It has $3$ real roots, $0$, $0$ and again $0$.
$endgroup$
– Dietrich Burde
Jan 2 at 13:46
add a comment |
3 Answers
3
active
oldest
votes
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In your case, $0$ is a double root: you should count it as two roots. In other words, the following statement holds:
If the roots are counted with their multiplicities, then every cubic polynomial in one variable with real coefficients either has exactly one real root or it has three real roots.
$endgroup$
add a comment |
$begingroup$
Note: if you have complex coefficients instead, the statement isn’t true.
As an example, consider $f(z) = (z-1)(z+1)(z-i)$. $f$ has two real roots, but one complex root.
$endgroup$
add a comment |
$begingroup$
The point is that complex conjugation ${Bbb C}rightarrow {Bbb C}:zmapsto bar z$, where $bar z = a-ib$ if $z=a+ib$, is a (ring) automorphism. It follows that if you have a polynomial (with real coefficients in your case) $f(x)$ with $f(z)=0$ for some $zin{Bbb C}$, then $f(bar z) = overline {f(z)} = bar 0 = 0$. Thus complex roots always occur in pairs: $(z,bar z)$.
So actually a cubic polynomial with real coefficients can only have 1 or 3 real roots, but not 2. If if would have 2 real roots and 1 complex root $z$, then $bar z$ would also be a root and so (as argumented above) $z=bar z$ would be real.
$endgroup$
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
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votes
active
oldest
votes
$begingroup$
In your case, $0$ is a double root: you should count it as two roots. In other words, the following statement holds:
If the roots are counted with their multiplicities, then every cubic polynomial in one variable with real coefficients either has exactly one real root or it has three real roots.
$endgroup$
add a comment |
$begingroup$
In your case, $0$ is a double root: you should count it as two roots. In other words, the following statement holds:
If the roots are counted with their multiplicities, then every cubic polynomial in one variable with real coefficients either has exactly one real root or it has three real roots.
$endgroup$
add a comment |
$begingroup$
In your case, $0$ is a double root: you should count it as two roots. In other words, the following statement holds:
If the roots are counted with their multiplicities, then every cubic polynomial in one variable with real coefficients either has exactly one real root or it has three real roots.
$endgroup$
In your case, $0$ is a double root: you should count it as two roots. In other words, the following statement holds:
If the roots are counted with their multiplicities, then every cubic polynomial in one variable with real coefficients either has exactly one real root or it has three real roots.
answered Jan 2 at 12:43
José Carlos SantosJosé Carlos Santos
164k22131235
164k22131235
add a comment |
add a comment |
$begingroup$
Note: if you have complex coefficients instead, the statement isn’t true.
As an example, consider $f(z) = (z-1)(z+1)(z-i)$. $f$ has two real roots, but one complex root.
$endgroup$
add a comment |
$begingroup$
Note: if you have complex coefficients instead, the statement isn’t true.
As an example, consider $f(z) = (z-1)(z+1)(z-i)$. $f$ has two real roots, but one complex root.
$endgroup$
add a comment |
$begingroup$
Note: if you have complex coefficients instead, the statement isn’t true.
As an example, consider $f(z) = (z-1)(z+1)(z-i)$. $f$ has two real roots, but one complex root.
$endgroup$
Note: if you have complex coefficients instead, the statement isn’t true.
As an example, consider $f(z) = (z-1)(z+1)(z-i)$. $f$ has two real roots, but one complex root.
answered Jan 2 at 15:37
user458276user458276
674211
674211
add a comment |
add a comment |
$begingroup$
The point is that complex conjugation ${Bbb C}rightarrow {Bbb C}:zmapsto bar z$, where $bar z = a-ib$ if $z=a+ib$, is a (ring) automorphism. It follows that if you have a polynomial (with real coefficients in your case) $f(x)$ with $f(z)=0$ for some $zin{Bbb C}$, then $f(bar z) = overline {f(z)} = bar 0 = 0$. Thus complex roots always occur in pairs: $(z,bar z)$.
So actually a cubic polynomial with real coefficients can only have 1 or 3 real roots, but not 2. If if would have 2 real roots and 1 complex root $z$, then $bar z$ would also be a root and so (as argumented above) $z=bar z$ would be real.
$endgroup$
add a comment |
$begingroup$
The point is that complex conjugation ${Bbb C}rightarrow {Bbb C}:zmapsto bar z$, where $bar z = a-ib$ if $z=a+ib$, is a (ring) automorphism. It follows that if you have a polynomial (with real coefficients in your case) $f(x)$ with $f(z)=0$ for some $zin{Bbb C}$, then $f(bar z) = overline {f(z)} = bar 0 = 0$. Thus complex roots always occur in pairs: $(z,bar z)$.
So actually a cubic polynomial with real coefficients can only have 1 or 3 real roots, but not 2. If if would have 2 real roots and 1 complex root $z$, then $bar z$ would also be a root and so (as argumented above) $z=bar z$ would be real.
$endgroup$
add a comment |
$begingroup$
The point is that complex conjugation ${Bbb C}rightarrow {Bbb C}:zmapsto bar z$, where $bar z = a-ib$ if $z=a+ib$, is a (ring) automorphism. It follows that if you have a polynomial (with real coefficients in your case) $f(x)$ with $f(z)=0$ for some $zin{Bbb C}$, then $f(bar z) = overline {f(z)} = bar 0 = 0$. Thus complex roots always occur in pairs: $(z,bar z)$.
So actually a cubic polynomial with real coefficients can only have 1 or 3 real roots, but not 2. If if would have 2 real roots and 1 complex root $z$, then $bar z$ would also be a root and so (as argumented above) $z=bar z$ would be real.
$endgroup$
The point is that complex conjugation ${Bbb C}rightarrow {Bbb C}:zmapsto bar z$, where $bar z = a-ib$ if $z=a+ib$, is a (ring) automorphism. It follows that if you have a polynomial (with real coefficients in your case) $f(x)$ with $f(z)=0$ for some $zin{Bbb C}$, then $f(bar z) = overline {f(z)} = bar 0 = 0$. Thus complex roots always occur in pairs: $(z,bar z)$.
So actually a cubic polynomial with real coefficients can only have 1 or 3 real roots, but not 2. If if would have 2 real roots and 1 complex root $z$, then $bar z$ would also be a root and so (as argumented above) $z=bar z$ would be real.
edited Jan 3 at 8:48
answered Jan 2 at 12:50
WuestenfuxWuestenfux
4,7941513
4,7941513
add a comment |
add a comment |
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I think the statement is that if a cubic polynomial has three distinct roots, then either all 3 are real or only one of them is real and the other two are complex conjugates of each other.
$endgroup$
– harshit54
Jan 2 at 12:45
$begingroup$
The title is not good. A trivial answer would be: Yes, take $f(x)=x^3$. It has $3$ real roots, $0$, $0$ and again $0$.
$endgroup$
– Dietrich Burde
Jan 2 at 13:46