$p, q$ are the roots of equation $x^2 - 5x + 1 = 0$ and we need to calculate the value of expression $E = \frac{p^{15} + p^{11} + q^{15} + q^{11}}{p^{13} + q^{13}}$. I will list 2 methods that produce different values, please tell me which one is correct mathematically. (Method 1 is mine and method 2 is from the textbook where I have seen this question).
METHOD 1:
From equation, we get $x + \frac{1}{x} = 5$ which translates into $x^2 + \frac{1}{x^2} = 23 = p^2 + \frac{1}{p^2} = q^2 + \frac{1}{q^2}$
Now, expression $$E = \frac{p^{13}q^{13}[\frac{1}{q^{13}}(p^2 + \frac{1}{p^2}) + \frac{1}{p^{13}}(q^2 + \frac{1}{q^2})]}{p^{13}q^{13}[\frac{1}{p^{13}} + \frac{1}{q^{13}}]}$$
$$E = \frac{(p^2 + \frac{1}{p^2})(\frac{1}{p^{13}} + \frac{1}{q^{13}})}{\frac{1}{p^{13}} + \frac{1}{q^{13}}} \, since \, \, p^2 + \frac{1}{p^2} = q^2 + \frac{1}{q^2}$$
So, $E = p^2 + \frac{1}{p^2} = 23$
METHOD 2:
This method is shown in the textbook.
$$E = \frac{p^{15} + q^{15} + p^2q^2(p^{11} + q^{11})}{p^{13} + q^{13}} \, since\,\, pq = -1$$ Simplifying it more, we get -: $$E = \frac{(p^{13} + q^{13})(p^2 + q^2)}{p^{13} + q^{13}} = p^2 + q^2 = (p+q)^2 - 2pq = 25 - 2(-1) = 27$$
Explain which method is correct and which is not because both seem to be logical. Thanks in advance!
This is a simple sign error: in fact, in the second method, $pq=1$ and not $-1$ as claimed. So the correct answer should be $25-2(1)=23$, which agrees with the first method.
To check that $pq=1$, just expand $(x-p)(x-q)=x^2-(p+q)x+pq$ and compare coefficients, of course.