An algebra problem by Satwik Murarka

Algebra Level 2

f ( x ) = x 9 − 6 x 8 − 2 x 7 + 12 x 6 + x 4 − 7 x 3 + 6 x 2 + x − 3 f(x)=x^{9}-6x^{8}-2x^{7}+12x^{6}+x^{4}-7x^{3}+6x^{2}+x-3

Find f ( 6 ) f(6) .

Source: IIT-JEE 1979


The answer is 3.

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Satwik Murarka by Satwik Murarka , 20, India

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2 solutions

Satwik Murarka
Apr 6, 2017

Solution:

f ( x ) = x 9 − 6 x 8 − 2 x 7 + 12 x 6 + x 4 − 7 x 3 + 6 x 2 + x − 3 f ( 6 ) = [ 6 9 − 6 ( 6 8 ) ] + [ − 2 ( 6 7 ) + 12 ( 6 6 ) ] + [ 6 4 − 7 ( 6 3 ) + 6 ( 6 2 ) ] + 6 − 3 f ( 6 ) = ( 6 9 − 6 9 ) + ( − 12 ( 6 6 ) + 12 ( 6 6 ) ) + ( 6 4 + 6 3 − 7 ( 6 3 ) ) + 6 − 3 f ( 6 ) = 6 − 3 = 3 \begin{aligned}f(x)& =x^{9}-6x^{8}-2x^{7}+12x^{6}+x^{4}-7x^{3}+6x^{2}+x-3\\ f(6)& =[6^{9}-6(6^{8})]+[-2(6^{7})+12(6^{6})]+[6^4-7(6^{3})+6(6^{2})]+6-3\\ f(6) & = \cancel{\color{#3D99F6}(6^{9}-6^{9})}+ \cancel{\color{#D61F06}(-12(6^{6})+12(6^{6}))}+\cancel{\color{#20A900}(6^{4}+6^{3}-7(6^{3}))}+6-3 \\f(6)&=6-3=\boxed{3}\end{aligned}

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It is the standard in Brilliant.org that we don't enter text in LaTex. I have edited the problem and solution for you. When using align enter an "&" before "=" so that each line is align at "&" as "\begin{align} f(x) & = ... \end{align}". It is better to leave spaces for ease of editing. You need to enter \require{cancel} before you can use \cancel.

Chew-Seong Cheong - 4 years, 2 months ago

@Chew-Seong Cheong - Thank you Sir for your assistance.I am still learning and will keep in mind the points you mentioned.

Satwik Murarka - 4 years, 2 months ago
Roger Erisman
Apr 7, 2017

Use synthetic division:

6| 1 -6 -2 12 0 1 -7 6 1 -3 

  _6_0_-12_0_0_6_-6_0_6_

 1 0-2 0 0 1 -1 0 1 3

Remainder = 3 so f(6) = 3

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