Can you calculate this without an calculator?

Algebra Level 2

Find A = x 5 − 100 x 4 + 100 x 3 − 100 x 2 + 100 x − 9 A=x^{5}-100x^{4}+100x^{3}-100x^{2}+100x-9 when x = 99 x=99

Hint: You don't need to convert all x x into 99. Maybe you should do it in reverse?


The answer is 90.

Tin Le by Tin Le , 15, Vietnam

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1 solution

Chew-Seong Cheong
Jun 19, 2019

A = x 5 − 100 x 4 + 100 x 3 − 100 x 2 + 100 x − 9 For x = 99 ⟹ 100 = x + 1 = x 5 − ( x + 1 ) x 4 + ( x + 1 ) x 3 − ( x + 1 ) x 2 + ( x + 1 ) x − 9 = x 5 − x 5 − x 4 + x 4 + x 3 − x 3 − x 2 + x 2 + x − 9 = x − 9 = 99 − 9 = 90 \begin{aligned} A & = x^5 - 100x^4 + 100x^3 - 100x^2+100x - 9 & \small \color{#3D99F6} \text{For }x=99 \implies 100 = x+1 \\ & = x^5 - (x+1)x^4 + (x+1)x^3 - (x+1)x^2+(x+1)x - 9 \\ & = x^5 - x^5-x^4 + x^4+x^3 - x^3 - x^2+ x^2+x - 9 \\ & = x - 9 = 99 - 9 = \boxed{90} \end{aligned}

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