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anemone
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MHB
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For some integer $k$, the polynomial $x^3-2011x+k$ has three integer roots $p, q, r$. Evaluate $ |p|+|q|+|r|$.
anemone said:For some integer $k$, the polynomial $x^3-2011x+k$ has three integer roots $p, q, r$. Evaluate $ |p|+|q|+|r|$.
mente oscura said:Hello.
I do not know if I have interpreted the question well.
[tex]p+q+r=0[/tex]
Regards.
(Doh) Yes, I got a sign wrong (you can probably see where).mathbalarka said:@Opalg, $(p, q, r) = (49, 39, -88)$ does not seem to yield a $k$ such that the cubic is separable over $\mathbb{Z}[x]$, let alone being each of $p, q$ and $r$ a root.
Sorry you're not well. I hope you recover in time for Christmas.anemone said:Hey MHB,
I will reply to this thread after I gained much more energy, because now I am very sick and ill with symptoms including stomach pain, vomiting, diarrhea and nausea.
Opalg said:Sorry you're not well. I hope you recover in time for Christmas.
MarkFL said:A little birdie told me ...
sorry to hear itanemone said:Hey MHB,
I will reply to this thread after I gained much more energy, because now I am very sick and ill with symptoms including stomach pain, vomiting, diarrhea and nausea.
Sorry guys!
Opalg said:(Doh) Yes, I got a sign wrong (you can probably see where).
[sp]It should have been $(2p+q)^2 + 3q^2 = 8044$, and the result of that is $p=10$, $q=39$, and therefore $r=-49$, so that $|p| + |q| + |r| = 98$. The value of $k$ is then $-pqr = 19\,110$. Better?[/sp]
mathbalarka said:I am very sorry to hear that, hope you get better soon. :(
Opalg said:Sorry you're not well. I hope you recover in time for Christmas.
Albert said:sorry to hear it
take care of yourself please ,and hope you will get better soon
The purpose of evaluating |p|+|q|+|r| for $x^3-2011x+k$ is to determine the absolute value of the coefficients p, q, and r in the polynomial equation $x^3-2011x+k$. This can help us understand the behavior and characteristics of the polynomial, such as its roots, turning points, and overall shape.
To evaluate |p|+|q|+|r| for $x^3-2011x+k$, we first need to identify the values of p, q, and r. These values can be found by looking at the coefficients of x in the polynomial equation. Once we have identified the values, we simply take the absolute value of each one and add them together to get the final result.
The value of |p|+|q|+|r| represents the sum of the absolute values of the coefficients in the polynomial equation $x^3-2011x+k$. This value can give us insights into the behavior of the polynomial, such as the number of real roots, the number of turning points, and the overall shape of the graph.
No, the value of |p|+|q|+|r| cannot be negative. The absolute value of a number is always positive, so the sum of the absolute values of any numbers will also be positive.
The value of k does not affect the evaluation of |p|+|q|+|r| for $x^3-2011x+k$. This is because k is not part of the coefficients of the polynomial and therefore does not impact the sum of the absolute values of p, q, and r. However, k does have an effect on the overall behavior of the polynomial, such as shifting the graph up or down.