I need help with this polynomial question.

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uaahelp

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I know the answer is -9 but I couldn't figure out how to solve it. Thanks for help.
 

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The basic clue is that, BY DEFINITION, every exponent in a polynomial must be a non-negative integer.

Why do you not work from that and see if that helps.

EDIT: By the way, I am getting minus 8.
 
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uaahelp said:
I know the answer is -9 but I couldn't figure out how to solve it. Thanks for help.
Click to expand...

Clever problem!! Did you work with the definition Jeff had provided?

Did you investigate the case 3a - 5 = 0?

Please share your work.
 
uaahelp said:
I know the answer is -9 but I couldn't figure out how to solve it. Thanks for help.
Click to expand...

It should state that the variable a must be an (allowable) integer.

If you do long division, (3a - 5)/(a + 1) = \(\displaystyle \ 3 \ - \ \dfrac{8}{a + 1}\).

\(\displaystyle \ 3 \ - \ \dfrac{8}{a + 1} \ \ge \ 0, \ \ \) and (a + 1) must divide 8.


(I got the answer of -9 using this.)
 
Why not include

a = 5/3

Since it "a" was not restricted!

then the answer would be:

Sum = -9 + 5/3 = -22/3
 
This is what I was originally going to post:

We need:

[MATH]\frac{3a-5}{a+1}=n[/MATH] where \(n\in\mathbb{N_0}\) (this means \(n\) must be a non-negative integer).

Solve for \(a\)

[MATH]a=\frac{n+5}{3-n}[/MATH] where \(n\ne3\)

Then:

[MATH]S_a=\frac{35}{3}+\sum_{k=4}^{\infty}\left(\frac{k+5}{3-k}\right)[/MATH]
This diverges. So, I initially decided not to post, because clearly I did something wrong. :(
 
Subhotosh Khan said:
Why not include a = 5/3 …
Click to expand...
If we allow Rational numbers for symbol a, then we need to consider all possibilities.

So far, I get the sum -70/3 by including the following.

-11/3, -7/,3, -5/3, -4/3, 5/3​
​
-13/5, -9/5, -7/5, -6/5​
​
I'm still ponderin'.

?
 
Why can't a be -11/3?

After all [3(-11/3) -5]/[-11/3 +1] = [-11-5]/[-8/3] = -16/(-8/3) = 6 which is an allowable exponent for a polynomial. There are many other values for a.
 
I go back to the definition of a polynomial. The exponent must be a non-negative integer.
 
JeffM said:
I go back to the definition of a polynomial. The exponent must be a non-negative integer.
Click to expand...
Jeff, I am sorry but I am missing your point. Are you saying that a=-11/3 or a=5/3 are not allowed in this problem?
 
Jomo said:
Jeff, I am sorry but I am missing your point. Are you saying that a=-11/3 or a=5/3 are not allowed in this problem?
Click to expand...
No. I am missing my own point. What is important is that the exponent when evaluated must be a non-negative integer. It goes back to lookagain's post. What is allowable for a?
 
Here is a list of values for \(a\) that give an exponent of 4-500:

{-9, -5, -11/3, -3, -13/5, -7/3, -15/7, -2, -17/9, -9/5, -19/11, -5/3, -21/13, -11/7, -23/15, -3/2, -25/17, -13/9, -27/19, -7/5, -29/21, -15/11, -31/23, -4/3, -33/25, -17/13, -35/27, -9/7, -37/29, -19/15, -39/31, -5/4, -41/33, -21/17, -43/35, -11/9, -45/37, -23/19, -47/39, -6/5, -49/41, -25/21, -51/43, -13/11, -53/45, -27/23, -55/47, -7/6, -57/49, -29/25, -59/51, -15/13, -61/53, -31/27, -63/55, -8/7, -65/57, -33/29, -67/59, -17/15, -69/61, -35/31, -71/63, -9/8, -73/65, -37/33, -75/67, -19/17, -77/69, -39/35, -79/71, -10/9, -81/73, -41/37, -83/75, -21/19, -85/77, -43/39, -87/79, -11/10, -89/81, -45/41, -91/83, -23/21, -93/85, -47/43, -95/87, -12/11, -97/89, -49/45, -99/91, -25/23, -101/93, -51/47, -103/95, -13/12, -105/97, -53/49, -107/99, -27/25, -109/101, -55/51, -111/103, -14/13, -113/105, -57/53, -115/107, -29/27, -117/109, -59/55, -119/111, -15/14, -121/113, -61/57, -123/115, -31/29, -125/117, -63/59, -127/119, -16/15, -129/121, -65/61, -131/123, -33/31, -133/125, -67/63, -135/127, -17/16, -137/129, -69/65, -139/131, -35/33, -141/133, -71/67, -143/135, -18/17, -145/137, -73/69, -147/139, -37/35, -149/141, -75/71, -151/143, -19/18, -153/145, -77/73, -155/147, -39/37, -157/149, -79/75, -159/151, -20/19, -161/153, -81/77, -163/155, -41/39, -165/157, -83/79, -167/159, -21/20, -169/161, -85/81, -171/163, -43/41, -173/165, -87/83, -175/167, -22/21, -177/169, -89/85, -179/171, -45/43, -181/173, -91/87, -183/175, -23/22, -185/177, -93/89, -187/179, -47/45, -189/181, -95/91, -191/183, -24/23, -193/185, -97/93, -195/187, -49/47, -197/189, -99/95, -199/191, -25/24, -201/193, -101/97, -203/195, -51/49, -205/197, -103/99, -207/199, -26/25, -209/201, -105/101, -211/203, -53/51, -213/205, -107/103, -215/207, -27/26, -217/209, -109/105, -219/211, -55/53, -221/213, -111/107, -223/215, -28/27, -225/217, -113/109, -227/219, -57/55, -229/221, -115/111, -231/223, -29/28, -233/225, -117/113, -235/227, -59/57, -237/229, -119/115, -239/231, -30/29, -241/233, -121/117, -243/235, -61/59, -245/237, -123/119, -247/239, -31/30, -249/241, -125/121, -251/243, -63/61, -253/245, -127/123, -255/247, -32/31, -257/249, -129/125, -259/251, -65/63, -261/253, -131/127, -263/255, -33/32, -265/257, -133/129, -267/259, -67/65, -269/261, -135/131, -271/263, -34/33, -273/265, -137/133, -275/267, -69/67, -277/269, -139/135, -279/271, -35/34, -281/273, -141/137, -283/275, -71/69, -285/277, -143/139, -287/279, -36/35, -289/281, -145/141, -291/283, -73/71, -293/285, -147/143, -295/287, -37/36, -297/289, -149/145, -299/291, -75/73, -301/293, -151/147, -303/295, -38/37, -305/297, -153/149, -307/299, -77/75, -309/301, -155/151, -311/303, -39/38, -313/305, -157/153, -315/307, -79/77, -317/309, -159/155, -319/311, -40/39, -321/313, -161/157, -323/315, -81/79, -325/317, -163/159, -327/319, -41/40, -329/321, -165/161, -331/323, -83/81, -333/325, -167/163, -335/327, -42/41, -337/329, -169/165, -339/331, -85/83, -341/333, -171/167, -343/335, -43/42, -345/337, -173/169, -347/339, -87/85, -349/341, -175/171, -351/343, -44/43, -353/345, -177/173, -355/347, -89/87, -357/349, -179/175, -359/351, -45/44, -361/353, -181/177, -363/355, -91/89, -365/357, -183/179, -367/359, -46/45, -369/361, -185/181, -371/363, -93/91, -373/365, -187/183, -375/367, -47/46, -377/369, -189/185, -379/371, -95/93, -381/373, -191/187, -383/375, -48/47, -385/377, -193/189, -387/379, -97/95, -389/381, -195/191, -391/383, -49/48, -393/385, -197/193, -395/387, -99/97, -397/389, -199/195, -399/391, -50/49, -401/393, -201/197, -403/395, -101/99, -405/397, -203/199, -407/399, -51/50, -409/401, -205/201, -411/403, -103/101, -413/405, -207/203, -415/407, -52/51, -417/409, -209/205, -419/411, -105/103, -421/413, -211/207, -423/415, -53/52, -425/417, -213/209, -427/419, -107/105, -429/421, -215/211, -431/423, -54/53, -433/425, -217/213, -435/427, -109/107, -437/429, -219/215, -439/431, -55/54, -441/433, -221/217, -443/435, -111/109, -445/437, -223/219, -447/439, -56/55, -449/441, -225/221, -451/443, -113/111, -453/445, -227/223, -455/447, -57/56, -457/449, -229/225, -459/451, -115/113, -461/453, -231/227, -463/455, -58/57, -465/457, -233/229, -467/459, -117/115, -469/461, -235/231, -471/463, -59/58, -473/465, -237/233, -475/467, -119/117, -477/469, -239/235, -479/471, -60/59, -481/473, -241/237, -483/475, -121/119, -485/477, -243/239, -487/479, -61/60, -489/481, -245/241, -491/483, -123/121, -493/485, -247/243, -495/487, -62/61, -497/489, -249/245, -499/491, -125/123, -501/493, -251/247, -503/495, -63/62, -505/497}

There are infinitely more, and their sum diverges.
 
MarkFL said:
This is what I was originally going to post:

We need:

[MATH]\frac{3a-5}{a+1}=n[/MATH] where \(n\in\mathbb{N_0}\) (this means \(n\) must be a non-negative integer).

Solve for \(a\)

[MATH]a=\frac{n+5}{3-n}[/MATH] where \(n\ne3\)

Then:

[MATH]S_a=\frac{35}{3}+\sum_{k=4}^{\infty}\left(\frac{k+5}{3-k}\right)[/MATH]
This diverges. So, I initially decided not to post, because clearly I did something wrong. :(
Click to expand...
Actually it looked fine to me the instant I saw it.
To others, just because the OP says the answer is -9 you assume that a itself must be positive? Corner to many tonight! Mark and I are going to go out, have a drink and think of you all in the corner.
 
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MarkFL said:
Here is a list of values for \(a\) that give an exponent of 4-500:

{-9, -5, -11/3, -3, -13/5, -7/3, -15/7, -2, -17/9, -9/5, -19/11, -5/3, -21/13, -11/7, -23/15, -3/2, -25/17, -13/9, -27/19, -7/5, -29/21, -15/11, -31/23, -4/3, -33/25, -17/13, -35/27, -9/7, -37/29, -19/15, -39/31, -5/4, -41/33, -21/17, -43/35, -11/9, -45/37, -23/19, -47/39, -6/5, -49/41, -25/21, -51/43, -13/11, -53/45, -27/23, -55/47, -7/6, -57/49, -29/25, -59/51, -15/13, -61/53, -31/27, -63/55, -8/7, -65/57, -33/29, -67/59, -17/15, -69/61, -35/31, -71/63, -9/8, -73/65, -37/33, -75/67, -19/17, -77/69, -39/35, -79/71, -10/9, -81/73, -41/37, -83/75, -21/19, -85/77, -43/39, -87/79, -11/10, -89/81, -45/41, -91/83, -23/21, -93/85, -47/43, -95/87, -12/11, -97/89, -49/45, -99/91, -25/23, -101/93, -51/47, -103/95, -13/12, -105/97, -53/49, -107/99, -27/25, -109/101, -55/51, -111/103, -14/13, -113/105, -57/53, -115/107, -29/27, -117/109, -59/55, -119/111, -15/14, -121/113, -61/57, -123/115, -31/29, -125/117, -63/59, -127/119, -16/15, -129/121, -65/61, -131/123, -33/31, -133/125, -67/63, -135/127, -17/16, -137/129, -69/65, -139/131, -35/33, -141/133, -71/67, -143/135, -18/17, -145/137, -73/69, -147/139, -37/35, -149/141, -75/71, -151/143, -19/18, -153/145, -77/73, -155/147, -39/37, -157/149, -79/75, -159/151, -20/19, -161/153, -81/77, -163/155, -41/39, -165/157, -83/79, -167/159, -21/20, -169/161, -85/81, -171/163, -43/41, -173/165, -87/83, -175/167, -22/21, -177/169, -89/85, -179/171, -45/43, -181/173, -91/87, -183/175, -23/22, -185/177, -93/89, -187/179, -47/45, -189/181, -95/91, -191/183, -24/23, -193/185, -97/93, -195/187, -49/47, -197/189, -99/95, -199/191, -25/24, -201/193, -101/97, -203/195, -51/49, -205/197, -103/99, -207/199, -26/25, -209/201, -105/101, -211/203, -53/51, -213/205, -107/103, -215/207, -27/26, -217/209, -109/105, -219/211, -55/53, -221/213, -111/107, -223/215, -28/27, -225/217, -113/109, -227/219, -57/55, -229/221, -115/111, -231/223, -29/28, -233/225, -117/113, -235/227, -59/57, -237/229, -119/115, -239/231, -30/29, -241/233, -121/117, -243/235, -61/59, -245/237, -123/119, -247/239, -31/30, -249/241, -125/121, -251/243, -63/61, -253/245, -127/123, -255/247, -32/31, -257/249, -129/125, -259/251, -65/63, -261/253, -131/127, -263/255, -33/32, -265/257, -133/129, -267/259, -67/65, -269/261, -135/131, -271/263, -34/33, -273/265, -137/133, -275/267, -69/67, -277/269, -139/135, -279/271, -35/34, -281/273, -141/137, -283/275, -71/69, -285/277, -143/139, -287/279, -36/35, -289/281, -145/141, -291/283, -73/71, -293/285, -147/143, -295/287, -37/36, -297/289, -149/145, -299/291, -75/73, -301/293, -151/147, -303/295, -38/37, -305/297, -153/149, -307/299, -77/75, -309/301, -155/151, -311/303, -39/38, -313/305, -157/153, -315/307, -79/77, -317/309, -159/155, -319/311, -40/39, -321/313, -161/157, -323/315, -81/79, -325/317, -163/159, -327/319, -41/40, -329/321, -165/161, -331/323, -83/81, -333/325, -167/163, -335/327, -42/41, -337/329, -169/165, -339/331, -85/83, -341/333, -171/167, -343/335, -43/42, -345/337, -173/169, -347/339, -87/85, -349/341, -175/171, -351/343, -44/43, -353/345, -177/173, -355/347, -89/87, -357/349, -179/175, -359/351, -45/44, -361/353, -181/177, -363/355, -91/89, -365/357, -183/179, -367/359, -46/45, -369/361, -185/181, -371/363, -93/91, -373/365, -187/183, -375/367, -47/46, -377/369, -189/185, -379/371, -95/93, -381/373, -191/187, -383/375, -48/47, -385/377, -193/189, -387/379, -97/95, -389/381, -195/191, -391/383, -49/48, -393/385, -197/193, -395/387, -99/97, -397/389, -199/195, -399/391, -50/49, -401/393, -201/197, -403/395, -101/99, -405/397, -203/199, -407/399, -51/50, -409/401, -205/201, -411/403, -103/101, -413/405, -207/203, -415/407, -52/51, -417/409, -209/205, -419/411, -105/103, -421/413, -211/207, -423/415, -53/52, -425/417, -213/209, -427/419, -107/105, -429/421, -215/211, -431/423, -54/53, -433/425, -217/213, -435/427, -109/107, -437/429, -219/215, -439/431, -55/54, -441/433, -221/217, -443/435, -111/109, -445/437, -223/219, -447/439, -56/55, -449/441, -225/221, -451/443, -113/111, -453/445, -227/223, -455/447, -57/56, -457/449, -229/225, -459/451, -115/113, -461/453, -231/227, -463/455, -58/57, -465/457, -233/229, -467/459, -117/115, -469/461, -235/231, -471/463, -59/58, -473/465, -237/233, -475/467, -119/117, -477/469, -239/235, -479/471, -60/59, -481/473, -241/237, -483/475, -121/119, -485/477, -243/239, -487/479, -61/60, -489/481, -245/241, -491/483, -123/121, -493/485, -247/243, -495/487, -62/61, -497/489, -249/245, -499/491, -125/123, -501/493, -251/247, -503/495, -63/62, -505/497}

There are infinitely more, and their sum diverges.
Click to expand...
And Subhotosh thought that the only non-integer was a=5/3. It may be night time here in the US but that doesn't mean you were born last night. Wake up folks.
 
JeffM said:
No. I am missing my own point. What is important is that the exponent when evaluated must be a non-negative integer. It goes back to lookagain's post. What is allowable for a?
Click to expand...
What is allowable for a you ask? Any number that makes (3a-5)/(a+1) a non negative integer
 
I hinted that this is a faulty problem in my earlier post. For the answer
to come out to -9, the variable a has to be restricted to integers. Also, the
expression for the exponent has to be allowed to be considered to be between
0 and 3, inclusive, where possible, not just greater than or equal to 4.

I saw that for including rational variable a values it is diverging.
 
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Jomo said:
What is allowable for a you ask? Any number that makes (3a-5)/(a+1) a non negative integer
Click to expand...
I do not see that. If a can be any real number that makes the exponent a non-negative integer, then, as Mark has already pointed out, there are an infinite number of possible solutions, and the sum of those solutions diverges. So, for the question to make any sense, lookagain's original post was correct.
 
MarkFL said:
… There are infinitely more [Rational values of a] …
Click to expand...
That's what I was ponderin'. You'd already posted that your series diverged (post #7), and I was considering that infinite values of Rational 'a' could be the reason why.

But, I'd run out of computation/motivation time, heh.

?
 
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