Remainders Are Fun 3

Algebra Level 4

Let f ( x ) f(x) be the remainder when the expression 3 x 2017 − x 2015 + 4 x 1998 + x 100 + 5 x 2 + 9 3x^{2017} - x^{2015} + 4x^{1998} + x^{100} + 5x^2 + 9 is divided by x 4 + 1 x^4 + 1 .

Evaluate f ( 1 ) f(1) .

For more problems like this, try answering this set .


The answer is 13.

Christian Daang by Christian Daang , 20, Philippines

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1 solution

Christian Daang
Apr 3, 2017

By remainder theorem, x 4 = − 1 x^4 = -1

then,

3 x 2017 = 3 ( x 4 ) 504 ⋅ x ≡ 3 ( 1 ) x = 3 x − x 2015 = − ( x 4 ) 503 ⋅ x 3 ≡ − ( − 1 ) x 3 = x 3 4 x 1998 = 4 ( x 4 ) 499 ⋅ x 2 ≡ 4 ( − 1 ) x 2 = − 4 x 2 x 100 = ( x 4 ) 25 ≡ ( − 1 ) = − 1 5 x 2 = 5 x 2 ≡ 5 x 2 = 5 x 2 9 = 9 ≡ 9 = 9 \begin{aligned} 3x^{2017} & = 3(x^4)^{504} \cdot x & \equiv 3(1)x & = 3x \\ -x^{2015} & = -(x^4)^{503} \cdot x^3 & \equiv -(-1)x^3 & = x^3 \\ 4x^{1998} & = 4(x^4)^{499} \cdot x^2 & \equiv 4(-1)x^2 & = -4x^2 \\ x^{100} & = (x^4)^{25} & \equiv (-1) & = -1 \\ 5x^{2} & = 5x^2 & \equiv 5x^2 & = 5x^2 \\ 9 & = 9 & \equiv 9 & = 9 \end{aligned}

∴ f ( x ) = 3 x + x 3 − 4 x 2 − 1 + 5 x 2 + 9 = x 3 + x 2 + 3 x + 8 ⟹ f ( 1 ) = 1 + 1 + 3 + 8 = 13 \therefore f(x) = 3x + x^3 - 4x^2 - 1 + 5x^2 + 9 = x^3 + x^2 + 3x + 8 \implies \boxed{f(1) = 1 + 1 + 3 + 8 = 13}

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