"Spoooooooky" Russian rational expressions 8

Algebra Level 2

E = 1 − 2 x 2 x + 1 + x 2 + 3 x 4 x 2 − 1 ÷ 3 + x 4 x + 2 \mathscr{E} = \dfrac{1-2x}{2x+1} + \dfrac{x^2 + 3x}{4x^2 -1} \div \dfrac{3+x}{4x+2}

Let x = 2016 x = 2016 . If E \mathscr{E} can be represented 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 16269118.

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Hobart Pao by Hobart Pao , 23, USA

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1 solution

⇒ 1 − 2 x 2 x + 1 + x 2 + 3 x 4 x 2 − 1 ÷ 3 + x 4 x + 2 \Rightarrow \dfrac{1-2x}{2x+1} + \dfrac{x^2 + 3x}{4x^2 -1} \div \dfrac{3+x}{4x+2}

= 1 − 2 x 2 x + 1 + x ( x + 3 ) ( 2 x − 1 ) ( 2 x + 1 ) × 2 ( 2 x + 1 ) ( 3 + x ) =\dfrac{1-2x}{2x+1}+\dfrac{x(x+3)}{(2x-1)(2x+1)}×\dfrac{2(2x+1)}{(3+x)}

= 1 − 2 x 2 x + 1 + 2 x 2 x − 1 =\dfrac{1-2x}{2x+1}+\dfrac{2x}{2x-1}

= ( 1 − 2 x ) ( 2 x − 1 ) + 2 x ( 2 x + 1 ) ( 2 x + 1 ) ( 2 x − 1 ) =\dfrac{(1-2x)(2x-1)+2x(2x+1)}{(2x+1)(2x-1)}

= 2 x − 1 − 4 x 2 + 2 x + 4 x 2 + 2 x 4 x 2 − 1 =\dfrac{2x-1-4x^2+2x+4x^2+2x}{4x^2-1}

= 6 x − 1 4 x 2 − 1 =\dfrac{6x-1}{4x^2-1}

Pluging x = 2016 x=2016 .

= 12095 16257023 =\dfrac{12095}{16257023}

∴ a + b = 16269118 \therefore a+b=\boxed{16269118}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

You're missing a 1 in line 4 but great solution!

Hobart Pao - 4 years, 11 months ago

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Done...Thanks! :)

A Former Brilliant Member - 4 years, 11 months ago

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