Mixing
zeros of polynomial on short interval and some reasoning.
$begingroup$
I read that polynomial of degree n can have up to n zeros and no more. My question is, which seem very obvious that you might bring number of zeros down by choosing smaller interval. For example -2 to 0 instead of -2 to 2.
Is that correct reasoning ?
algebra-precalculus
asked Jan 18 at 4:35
tt ztt z
214
$endgroup$
add a comment |
$begingroup$
I read that polynomial of degree n can have up to n zeros and no more. My question is, which seem very obvious that you might bring number of zeros down by choosing smaller interval. For example -2 to 0 instead of -2 to 2.
Is that correct reasoning ?
algebra-precalculus
asked Jan 18 at 4:35
tt ztt z
214
$endgroup$
$begingroup$
Your reasoning is fine : you are just restricting the domain of the polynomial. This also restricts the preimage of every point in the range of the polynomial, not just zero.
$endgroup$
– астон вілла олоф мэллбэрг
Jan 18 at 5:21
add a comment |
$begingroup$
I read that polynomial of degree n can have up to n zeros and no more. My question is, which seem very obvious that you might bring number of zeros down by choosing smaller interval. For example -2 to 0 instead of -2 to 2.
Is that correct reasoning ?
algebra-precalculus
asked Jan 18 at 4:35
tt ztt z
214
$endgroup$
I read that polynomial of degree n can have up to n zeros and no more. My question is, which seem very obvious that you might bring number of zeros down by choosing smaller interval. For example -2 to 0 instead of -2 to 2.
Is that correct reasoning ?
algebra-precalculus
algebra-precalculus
asked Jan 18 at 4:35
tt ztt z
214
asked Jan 18 at 4:35
tt ztt z
214
asked Jan 18 at 4:35
tt ztt z
214
asked Jan 18 at 4:35
tt ztt z
214
asked Jan 18 at 4:35
tt ztt z
214
214
$begingroup$
Your reasoning is fine : you are just restricting the domain of the polynomial. This also restricts the preimage of every point in the range of the polynomial, not just zero.
$endgroup$
– астон вілла олоф мэллбэрг
Jan 18 at 5:21
add a comment |
$begingroup$
Your reasoning is fine : you are just restricting the domain of the polynomial. This also restricts the preimage of every point in the range of the polynomial, not just zero.
$endgroup$
– астон вілла олоф мэллбэрг
Jan 18 at 5:21
$begingroup$
Your reasoning is fine : you are just restricting the domain of the polynomial. This also restricts the preimage of every point in the range of the polynomial, not just zero.
$endgroup$
– астон вілла олоф мэллбэрг
Jan 18 at 5:21
$begingroup$
Your reasoning is fine : you are just restricting the domain of the polynomial. This also restricts the preimage of every point in the range of the polynomial, not just zero.
$endgroup$
– астон вілла олоф мэллбэрг
Jan 18 at 5:21
add a comment |
1 Answer
1
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oldest
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$begingroup$
If what you're asking is if the number of roots in that interval decreases, then yes, sure, decrease the interval all you want and the amount of roots in that interval will decrease.
This is not always true though:
Consider the polynomial $x^2 + 3x + 2$. The roots of this polynomial are at $x = -1$ and $ x = -2$, so changing the interval from all real numbers to $(-infty, -2]$ will change the amount of roots $text{in that interval}$.
However, for the same polynomial, if the interval is changed from $(-infty, infty)$ to $(-10000, 10000)$, although the interval changes, the amount of roots in that interval doesn't.
TLDR, your statement is only true for some cases, not all.
Hope this answers your question.
answered Jan 18 at 4:46
Aniruddh VenkatesanAniruddh Venkatesan
156113
$endgroup$
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
If what you're asking is if the number of roots in that interval decreases, then yes, sure, decrease the interval all you want and the amount of roots in that interval will decrease.
This is not always true though:
Consider the polynomial $x^2 + 3x + 2$. The roots of this polynomial are at $x = -1$ and $ x = -2$, so changing the interval from all real numbers to $(-infty, -2]$ will change the amount of roots $text{in that interval}$.
However, for the same polynomial, if the interval is changed from $(-infty, infty)$ to $(-10000, 10000)$, although the interval changes, the amount of roots in that interval doesn't.
TLDR, your statement is only true for some cases, not all.
Hope this answers your question.
answered Jan 18 at 4:46
Aniruddh VenkatesanAniruddh Venkatesan
156113
$endgroup$
add a comment |
$begingroup$
If what you're asking is if the number of roots in that interval decreases, then yes, sure, decrease the interval all you want and the amount of roots in that interval will decrease.
This is not always true though:
Consider the polynomial $x^2 + 3x + 2$. The roots of this polynomial are at $x = -1$ and $ x = -2$, so changing the interval from all real numbers to $(-infty, -2]$ will change the amount of roots $text{in that interval}$.
However, for the same polynomial, if the interval is changed from $(-infty, infty)$ to $(-10000, 10000)$, although the interval changes, the amount of roots in that interval doesn't.
TLDR, your statement is only true for some cases, not all.
Hope this answers your question.
answered Jan 18 at 4:46
Aniruddh VenkatesanAniruddh Venkatesan
156113
$endgroup$
add a comment |
$begingroup$
If what you're asking is if the number of roots in that interval decreases, then yes, sure, decrease the interval all you want and the amount of roots in that interval will decrease.
This is not always true though:
Consider the polynomial $x^2 + 3x + 2$. The roots of this polynomial are at $x = -1$ and $ x = -2$, so changing the interval from all real numbers to $(-infty, -2]$ will change the amount of roots $text{in that interval}$.
However, for the same polynomial, if the interval is changed from $(-infty, infty)$ to $(-10000, 10000)$, although the interval changes, the amount of roots in that interval doesn't.
TLDR, your statement is only true for some cases, not all.
Hope this answers your question.
answered Jan 18 at 4:46
Aniruddh VenkatesanAniruddh Venkatesan
156113
$endgroup$
If what you're asking is if the number of roots in that interval decreases, then yes, sure, decrease the interval all you want and the amount of roots in that interval will decrease.
This is not always true though:
Consider the polynomial $x^2 + 3x + 2$. The roots of this polynomial are at $x = -1$ and $ x = -2$, so changing the interval from all real numbers to $(-infty, -2]$ will change the amount of roots $text{in that interval}$.
However, for the same polynomial, if the interval is changed from $(-infty, infty)$ to $(-10000, 10000)$, although the interval changes, the amount of roots in that interval doesn't.
TLDR, your statement is only true for some cases, not all.
Hope this answers your question.
answered Jan 18 at 4:46
Aniruddh VenkatesanAniruddh Venkatesan
156113
answered Jan 18 at 4:46
Aniruddh VenkatesanAniruddh Venkatesan
156113
answered Jan 18 at 4:46
Aniruddh VenkatesanAniruddh Venkatesan
156113
answered Jan 18 at 4:46
Aniruddh VenkatesanAniruddh Venkatesan
156113
156113
add a comment |
add a comment |
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$begingroup$
Your reasoning is fine : you are just restricting the domain of the polynomial. This also restricts the preimage of every point in the range of the polynomial, not just zero.
$endgroup$
– астон вілла олоф мэллбэрг
Jan 18 at 5:21