What remains?

Algebra Level 4

If x^2017+x^2013 is divided by x^2-1. What is the first derivative of its remainder?


The answer is 2.

Jun Arro Estrella by Jun Arro Estrella , 23, Philippines

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2 solutions

Anatoliy Razin
Nov 17, 2014

x = 1 x = 1 and x = − 1 x = -1 are roots of x 2017 + x 2013 − 2 x x^{2017} + x^{2013} - 2x , therefore x 2017 + x 2013 − 2 x = ( x 2 − 1 ) P ( x ) x^{2017} + x^{2013} - 2x = (x^2 - 1)P(x) for some P P or x 2017 + x 2013 = ( x 2 − 1 ) P ( x ) + 2 x x^{2017} + x^{2013} = (x^2 - 1)P(x) + 2x

2 x 2x is the reminder, the answer is 2 \boxed{2}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Since P(x)=x^2013+x^2017, it follows that x(x^2016)+x(x^2012) since it can also be expressed as an even power, and by the relation x^2=1, it follows that the remainder is 2x and d(2x)/dx is 2

Jun Arro Estrella - 6 years, 6 months ago
Sai Ram
Aug 6, 2015

We know that x 2 = 1. x^2=1.

Therefore the given polynomial is ( x 2 ) 2017 2 + ( x 2 ) 2013 2 (x^2)^\dfrac{2017}{2} + (x^2)^\dfrac{2013}{2} ,Which is 1 + 1 = 2. 1+1 = 2.

Therefore the required answer is 2. 2.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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