"Spoooooooky" Russian rational expressions 7

Algebra Level 3

E = ( p 2 q − p ) ( 4 p − 8 p 3 − 2 p 2 − q + 2 q 3 + 2 q 2 ) \mathscr{E} = \left(\dfrac{p}{2q-p} \right) \left( \dfrac{4p-8}{p^3 - 2p^2} - \dfrac{q+2}{q^3 + 2q^2}\right)

Let p = 2016 p = 2016 and q = 2017 q = 2017 . If E \mathscr{E} can be expressed in the form a b \dfrac{a}{b} , where a a and b b are coprime positive integers . Find a + b a+b .

You may use a calculator for the last step.

Credit: My former Trig teacher's worksheet of Russian rational expressions


The answer is 4100838337.

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Hobart Pao by Hobart Pao , 23, USA

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1 solution

Hobart Pao
Jul 5, 2016

= ( p 2 q − p ) ( 4 ( p − 2 ) p 2 ( p − 2 ) − q + 2 q 2 ( q + 2 ) ) = \left( \dfrac{p}{2q-p} \right) \left( \dfrac{4(p-2)}{p^2 (p-2) } - \dfrac{q+2}{q^2 (q+2)}\right)

= ( p 2 q − p ) ( 4 q 2 − p 2 p 2 q 2 ) = \left( \dfrac{p}{2q-p} \right) \left( \dfrac{4q^2 - p^2 }{p^2 q^2 } \right)

= ( p 2 q − p ) ( ( 2 q + p ) ( 2 q − p ) p 2 q 2 ) = \left( \dfrac{p}{2q-p} \right) \left( \dfrac{(2q+p)(2q-p) }{p^2 q^2 } \right)

= 2 q + p p q 2 = \dfrac{2q+p}{pq^2}

Letting p = 2016 , q = 2017 p =2016, q=2017 , we get 3025 4100835312 \boxed{\dfrac{3025}{4100835312}} , and a + b = 4100838337 a+b = \boxed{4100838337}

(Now there should not be an issue; I made a mistake earlier...)

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