how to memorize the sum and product of roots for an $n^th$ degree equation The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Invertible and primitive polynomialsSum of squares of roots of a polynomial $P(x)$sufficient condition for a polynomial to have roots in $[0,1]$How often do polynomials in $bar mathbb F_p[x]$ have multiple roots?On factorization of polynomialsDo the integer roots of a polynomial $P(x) in Bbb Z[x]$ have to divide the constant coefficient?Can a polynomial $p(x)$ generate only primes and 2-almost primes $forall x ge 0 in Bbb N$ or there is also a restriction for this to happen?Deriving $x^n-1=(x-1)(x^n-1+x^n-2+…+x+1)$Prove the value of the $k$th symmetric sum via inductionGeneral $S_k$ case of the Vieta's Formulas
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how to memorize the sum and product of roots for an $n^th$ degree equation
The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Invertible and primitive polynomialsSum of squares of roots of a polynomial $P(x)$sufficient condition for a polynomial to have roots in $[0,1]$How often do polynomials in $bar mathbb F_p[x]$ have multiple roots?On factorization of polynomialsDo the integer roots of a polynomial $P(x) in Bbb Z[x]$ have to divide the constant coefficient?Can a polynomial $p(x)$ generate only primes and 2-almost primes $forall x ge 0 in Bbb N$ or there is also a restriction for this to happen?Deriving $x^n-1=(x-1)(x^n-1+x^n-2+…+x+1)$Prove the value of the $k$th symmetric sum via inductionGeneral $S_k$ case of the Vieta's Formulas
$begingroup$
For my exams I need to know the following equations by heart:
for a polynomial equation: $a_nx+a_n-1x^n-1+...+a_1x+a_0,$ the sum and product of the roots are given by
$$textrmSum=-fraca_n-1a_n$$
$$textrmProduct=(-1)^nfraca_0a_n.$$
I have never been able to memorize these, and for some reason, they are not on the formula booklet. If anyone has any mnemonic or trick of some sort for memorizing them, it would be very useful to me.
Thank you very much in advance!
polynomials quadratics
edited Mar 24 at 19:11
J. W. Tanner
4,7721420
asked Mar 24 at 19:03
Jorge RomeuJorge Romeu
142
$endgroup$
add a comment |
$begingroup$
For my exams I need to know the following equations by heart:
for a polynomial equation: $a_nx+a_n-1x^n-1+...+a_1x+a_0,$ the sum and product of the roots are given by
$$textrmSum=-fraca_n-1a_n$$
$$textrmProduct=(-1)^nfraca_0a_n.$$
I have never been able to memorize these, and for some reason, they are not on the formula booklet. If anyone has any mnemonic or trick of some sort for memorizing them, it would be very useful to me.
Thank you very much in advance!
polynomials quadratics
edited Mar 24 at 19:11
J. W. Tanner
4,7721420
asked Mar 24 at 19:03
Jorge RomeuJorge Romeu
142
$endgroup$
3
$begingroup$
Learn how to derive the results by yourself. Come the day of the exam, you many not remember the derivations but you will remember the results.
$endgroup$
– John Douma
Mar 24 at 19:07
$begingroup$
Do you mean $a_n x^pmb n + ...$ ?
$endgroup$
– J. W. Tanner
Mar 24 at 19:13
2
$begingroup$
You might want to read the wiki page, Vieta's formulas hope this helps you.
$endgroup$
– Max
Mar 24 at 19:17
add a comment |
$begingroup$
For my exams I need to know the following equations by heart:
for a polynomial equation: $a_nx+a_n-1x^n-1+...+a_1x+a_0,$ the sum and product of the roots are given by
$$textrmSum=-fraca_n-1a_n$$
$$textrmProduct=(-1)^nfraca_0a_n.$$
I have never been able to memorize these, and for some reason, they are not on the formula booklet. If anyone has any mnemonic or trick of some sort for memorizing them, it would be very useful to me.
Thank you very much in advance!
polynomials quadratics
edited Mar 24 at 19:11
J. W. Tanner
4,7721420
asked Mar 24 at 19:03
Jorge RomeuJorge Romeu
142
$endgroup$
For my exams I need to know the following equations by heart:
for a polynomial equation: $a_nx+a_n-1x^n-1+...+a_1x+a_0,$ the sum and product of the roots are given by
$$textrmSum=-fraca_n-1a_n$$
$$textrmProduct=(-1)^nfraca_0a_n.$$
I have never been able to memorize these, and for some reason, they are not on the formula booklet. If anyone has any mnemonic or trick of some sort for memorizing them, it would be very useful to me.
Thank you very much in advance!
polynomials quadratics
polynomials quadratics
edited Mar 24 at 19:11
J. W. Tanner
4,7721420
asked Mar 24 at 19:03
Jorge RomeuJorge Romeu
142
edited Mar 24 at 19:11
J. W. Tanner
4,7721420
asked Mar 24 at 19:03
Jorge RomeuJorge Romeu
142
edited Mar 24 at 19:11
J. W. Tanner
4,7721420
edited Mar 24 at 19:11
J. W. Tanner
4,7721420
edited Mar 24 at 19:11
J. W. Tanner
4,7721420
4,7721420
asked Mar 24 at 19:03
Jorge RomeuJorge Romeu
142
asked Mar 24 at 19:03
Jorge RomeuJorge Romeu
142
asked Mar 24 at 19:03
Jorge RomeuJorge Romeu
142
142
3
$begingroup$
Learn how to derive the results by yourself. Come the day of the exam, you many not remember the derivations but you will remember the results.
$endgroup$
– John Douma
Mar 24 at 19:07
$begingroup$
Do you mean $a_n x^pmb n + ...$ ?
$endgroup$
– J. W. Tanner
Mar 24 at 19:13
2
$begingroup$
You might want to read the wiki page, Vieta's formulas hope this helps you.
$endgroup$
– Max
Mar 24 at 19:17
add a comment |
3
$begingroup$
Learn how to derive the results by yourself. Come the day of the exam, you many not remember the derivations but you will remember the results.
$endgroup$
– John Douma
Mar 24 at 19:07
$begingroup$
Do you mean $a_n x^pmb n + ...$ ?
$endgroup$
– J. W. Tanner
Mar 24 at 19:13
2
$begingroup$
You might want to read the wiki page, Vieta's formulas hope this helps you.
$endgroup$
– Max
Mar 24 at 19:17
3
3
$begingroup$
Learn how to derive the results by yourself. Come the day of the exam, you many not remember the derivations but you will remember the results.
$endgroup$
– John Douma
Mar 24 at 19:07
$begingroup$
Learn how to derive the results by yourself. Come the day of the exam, you many not remember the derivations but you will remember the results.
$endgroup$
– John Douma
Mar 24 at 19:07
$begingroup$
Do you mean $a_n x^pmb n + ...$ ?
$endgroup$
– J. W. Tanner
Mar 24 at 19:13
$begingroup$
Do you mean $a_n x^pmb n + ...$ ?
$endgroup$
– J. W. Tanner
Mar 24 at 19:13
2
2
$begingroup$
You might want to read the wiki page, Vieta's formulas hope this helps you.
$endgroup$
– Max
Mar 24 at 19:17
$begingroup$
You might want to read the wiki page, Vieta's formulas hope this helps you.
$endgroup$
– Max
Mar 24 at 19:17
add a comment |
4 Answers
4
active
oldest
votes
$begingroup$
Write
$$a_n x^n + a_n-1x^n-1 + ... + a_1 x + a_0 = a_n(x-r_1)...(x-r_n)$$
and develop. You see immediately what is the constant term and the term of degree $n-1$.
answered Mar 24 at 19:06
TheSilverDoeTheSilverDoe
5,541216
$endgroup$
1
$begingroup$
You are missing an $a_n$ on the RHS.
$endgroup$
– Lord Shark the Unknown
Mar 24 at 19:08
add a comment |
$begingroup$
Consider $a_n(x-r_1)(x-r_2)...(x-r_n)=a_n x^n-a_n(r_1+r_2+r_3...)x^n-1+...+(-1)^na_nr_1r_2...r_n$.
And so product of roots are $frac(-1)^na_0a_n$ and sum of roots is $frac-a_n-1a_n$.
answered Mar 24 at 19:27
mathpadawanmathpadawan
2,021422
$endgroup$
add a comment |
$begingroup$
Since the OP is looking for an artificial device to recall the correct equations under duress rather than a way to derive the formulas quickly, consider the following suggestion.
Rewrite the equations as follows
begineqnarray
a_nS&=&-a_n-1\
a_nP&=&(-1)^na_0
endeqnarray
Next, make some observations about these equations:
$a_n$ occurs only on the left side of each equation.- The variable $n$ occurs only once on the right side of each equation.
- The Sum formula has a minus sign in front and the $n$ occurs in the Subscript as $n-1$.
- The Product formula has a factor in front, $(-1)^n$, containing the $n$.
answered Mar 24 at 19:47
John Wayland BalesJohn Wayland Bales
15.1k21238
$endgroup$
add a comment |
$begingroup$
Hint:
Write the polynomial as $$(x-r_1)(x-r_2)...(x-r_n),$$ where the roots are $r_1, r_2, ... r_n.$
answered Mar 24 at 19:06
J. W. TannerJ. W. Tanner
4,7721420
$endgroup$
add a comment |
Your Answer
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Write
$$a_n x^n + a_n-1x^n-1 + ... + a_1 x + a_0 = a_n(x-r_1)...(x-r_n)$$
and develop. You see immediately what is the constant term and the term of degree $n-1$.
answered Mar 24 at 19:06
TheSilverDoeTheSilverDoe
5,541216
$endgroup$
1
$begingroup$
You are missing an $a_n$ on the RHS.
$endgroup$
– Lord Shark the Unknown
Mar 24 at 19:08
add a comment |
$begingroup$
Write
$$a_n x^n + a_n-1x^n-1 + ... + a_1 x + a_0 = a_n(x-r_1)...(x-r_n)$$
and develop. You see immediately what is the constant term and the term of degree $n-1$.
answered Mar 24 at 19:06
TheSilverDoeTheSilverDoe
5,541216
$endgroup$
1
$begingroup$
You are missing an $a_n$ on the RHS.
$endgroup$
– Lord Shark the Unknown
Mar 24 at 19:08
add a comment |
$begingroup$
Write
$$a_n x^n + a_n-1x^n-1 + ... + a_1 x + a_0 = a_n(x-r_1)...(x-r_n)$$
and develop. You see immediately what is the constant term and the term of degree $n-1$.
answered Mar 24 at 19:06
TheSilverDoeTheSilverDoe
5,541216
$endgroup$
Write
$$a_n x^n + a_n-1x^n-1 + ... + a_1 x + a_0 = a_n(x-r_1)...(x-r_n)$$
and develop. You see immediately what is the constant term and the term of degree $n-1$.
answered Mar 24 at 19:06
TheSilverDoeTheSilverDoe
5,541216
edited Mar 24 at 19:11
answered Mar 24 at 19:06
TheSilverDoeTheSilverDoe
5,541216
answered Mar 24 at 19:06
TheSilverDoeTheSilverDoe
5,541216
answered Mar 24 at 19:06
TheSilverDoeTheSilverDoe
5,541216
5,541216
1
$begingroup$
You are missing an $a_n$ on the RHS.
$endgroup$
– Lord Shark the Unknown
Mar 24 at 19:08
add a comment |
1
$begingroup$
You are missing an $a_n$ on the RHS.
$endgroup$
– Lord Shark the Unknown
Mar 24 at 19:08
1
1
$begingroup$
You are missing an $a_n$ on the RHS.
$endgroup$
– Lord Shark the Unknown
Mar 24 at 19:08
$begingroup$
You are missing an $a_n$ on the RHS.
$endgroup$
– Lord Shark the Unknown
Mar 24 at 19:08
add a comment |
$begingroup$
Consider $a_n(x-r_1)(x-r_2)...(x-r_n)=a_n x^n-a_n(r_1+r_2+r_3...)x^n-1+...+(-1)^na_nr_1r_2...r_n$.
And so product of roots are $frac(-1)^na_0a_n$ and sum of roots is $frac-a_n-1a_n$.
answered Mar 24 at 19:27
mathpadawanmathpadawan
2,021422
$endgroup$
add a comment |
$begingroup$
Consider $a_n(x-r_1)(x-r_2)...(x-r_n)=a_n x^n-a_n(r_1+r_2+r_3...)x^n-1+...+(-1)^na_nr_1r_2...r_n$.
And so product of roots are $frac(-1)^na_0a_n$ and sum of roots is $frac-a_n-1a_n$.
answered Mar 24 at 19:27
mathpadawanmathpadawan
2,021422
$endgroup$
add a comment |
$begingroup$
Consider $a_n(x-r_1)(x-r_2)...(x-r_n)=a_n x^n-a_n(r_1+r_2+r_3...)x^n-1+...+(-1)^na_nr_1r_2...r_n$.
And so product of roots are $frac(-1)^na_0a_n$ and sum of roots is $frac-a_n-1a_n$.
answered Mar 24 at 19:27
mathpadawanmathpadawan
2,021422
$endgroup$
Consider $a_n(x-r_1)(x-r_2)...(x-r_n)=a_n x^n-a_n(r_1+r_2+r_3...)x^n-1+...+(-1)^na_nr_1r_2...r_n$.
And so product of roots are $frac(-1)^na_0a_n$ and sum of roots is $frac-a_n-1a_n$.
answered Mar 24 at 19:27
mathpadawanmathpadawan
2,021422
answered Mar 24 at 19:27
mathpadawanmathpadawan
2,021422
answered Mar 24 at 19:27
mathpadawanmathpadawan
2,021422
answered Mar 24 at 19:27
mathpadawanmathpadawan
2,021422
2,021422
add a comment |
add a comment |
$begingroup$
Since the OP is looking for an artificial device to recall the correct equations under duress rather than a way to derive the formulas quickly, consider the following suggestion.
Rewrite the equations as follows
begineqnarray
a_nS&=&-a_n-1\
a_nP&=&(-1)^na_0
endeqnarray
Next, make some observations about these equations:
$a_n$ occurs only on the left side of each equation.- The variable $n$ occurs only once on the right side of each equation.
- The Sum formula has a minus sign in front and the $n$ occurs in the Subscript as $n-1$.
- The Product formula has a factor in front, $(-1)^n$, containing the $n$.
answered Mar 24 at 19:47
John Wayland BalesJohn Wayland Bales
15.1k21238
$endgroup$
add a comment |
$begingroup$
Since the OP is looking for an artificial device to recall the correct equations under duress rather than a way to derive the formulas quickly, consider the following suggestion.
Rewrite the equations as follows
begineqnarray
a_nS&=&-a_n-1\
a_nP&=&(-1)^na_0
endeqnarray
Next, make some observations about these equations:
$a_n$ occurs only on the left side of each equation.- The variable $n$ occurs only once on the right side of each equation.
- The Sum formula has a minus sign in front and the $n$ occurs in the Subscript as $n-1$.
- The Product formula has a factor in front, $(-1)^n$, containing the $n$.
answered Mar 24 at 19:47
John Wayland BalesJohn Wayland Bales
15.1k21238
$endgroup$
add a comment |
$begingroup$
Since the OP is looking for an artificial device to recall the correct equations under duress rather than a way to derive the formulas quickly, consider the following suggestion.
Rewrite the equations as follows
begineqnarray
a_nS&=&-a_n-1\
a_nP&=&(-1)^na_0
endeqnarray
Next, make some observations about these equations:
$a_n$ occurs only on the left side of each equation.- The variable $n$ occurs only once on the right side of each equation.
- The Sum formula has a minus sign in front and the $n$ occurs in the Subscript as $n-1$.
- The Product formula has a factor in front, $(-1)^n$, containing the $n$.
answered Mar 24 at 19:47
John Wayland BalesJohn Wayland Bales
15.1k21238
$endgroup$
Since the OP is looking for an artificial device to recall the correct equations under duress rather than a way to derive the formulas quickly, consider the following suggestion.
Rewrite the equations as follows
begineqnarray
a_nS&=&-a_n-1\
a_nP&=&(-1)^na_0
endeqnarray
Next, make some observations about these equations:
$a_n$ occurs only on the left side of each equation.- The variable $n$ occurs only once on the right side of each equation.
- The Sum formula has a minus sign in front and the $n$ occurs in the Subscript as $n-1$.
- The Product formula has a factor in front, $(-1)^n$, containing the $n$.
answered Mar 24 at 19:47
John Wayland BalesJohn Wayland Bales
15.1k21238
answered Mar 24 at 19:47
John Wayland BalesJohn Wayland Bales
15.1k21238
answered Mar 24 at 19:47
John Wayland BalesJohn Wayland Bales
15.1k21238
answered Mar 24 at 19:47
John Wayland BalesJohn Wayland Bales
15.1k21238
15.1k21238
add a comment |
add a comment |
$begingroup$
Hint:
Write the polynomial as $$(x-r_1)(x-r_2)...(x-r_n),$$ where the roots are $r_1, r_2, ... r_n.$
answered Mar 24 at 19:06
J. W. TannerJ. W. Tanner
4,7721420
$endgroup$
add a comment |
$begingroup$
Hint:
Write the polynomial as $$(x-r_1)(x-r_2)...(x-r_n),$$ where the roots are $r_1, r_2, ... r_n.$
answered Mar 24 at 19:06
J. W. TannerJ. W. Tanner
4,7721420
$endgroup$
add a comment |
$begingroup$
Hint:
Write the polynomial as $$(x-r_1)(x-r_2)...(x-r_n),$$ where the roots are $r_1, r_2, ... r_n.$
answered Mar 24 at 19:06
J. W. TannerJ. W. Tanner
4,7721420
$endgroup$
Hint:
Write the polynomial as $$(x-r_1)(x-r_2)...(x-r_n),$$ where the roots are $r_1, r_2, ... r_n.$
answered Mar 24 at 19:06
J. W. TannerJ. W. Tanner
4,7721420
answered Mar 24 at 19:06
J. W. TannerJ. W. Tanner
4,7721420
answered Mar 24 at 19:06
J. W. TannerJ. W. Tanner
4,7721420
answered Mar 24 at 19:06
J. W. TannerJ. W. Tanner
4,7721420
4,7721420
add a comment |
add a comment |
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$begingroup$
Learn how to derive the results by yourself. Come the day of the exam, you many not remember the derivations but you will remember the results.
$endgroup$
– John Douma
Mar 24 at 19:07
$begingroup$
Do you mean $a_n x^pmb n + ...$ ?
$endgroup$
– J. W. Tanner
Mar 24 at 19:13
2
$begingroup$
You might want to read the wiki page, Vieta's formulas hope this helps you.
$endgroup$
– Max
Mar 24 at 19:17