Union of set of different methods

Algebra Level 5

If, x 9 + x 8 + x 5 + 7 x 2 2 x = 7 x^{9}+x^{8}+x^{5}+7x^{2}-2x=7

Let α , β , γ \alpha ,\beta ,\gamma be the real roots of this equation,where α < β < γ \alpha < \beta < \gamma

Let there be a triangle with sides as α , β , γ |\alpha|,|\beta|,|\gamma|

Find the Least Common Multiple of all 3 interior angles when rounded off to the nearest degree.


The answer is 1932.

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1 solution

G r a p h i n g i n T I 83 P l u s , ( α , β , γ ) = ( 1.35378 , 0.9068825 , 0.96833362 ) U s i n g C o s R u l e , w e g e t a n g l e s a s 42 , 46 , 92. L C M = 42 46 = 1932 Graphing ~in ~TI-83 ~Plus, ~~\\ (\alpha,~~\beta,~\gamma)=(-1.35378,~-0.9068825,~0.96833362) \\ Using~Cos~Rule, ~we~get~angles~as\\ 42,~46,~92.~~~~LCM=42*46=~~~\Large \color{#D61F06}{1932} B y g r a p h i n g By~~ graphing x 9 + x 8 + x 5 + 7 x 2 2 x 7 = 0 x^{9}+x^{8}+x^{5}+7x^{2}-2x-7=0 and noting down the three intersection with x-axis. To get accurate values, I went on narrowing range of x around each intersection. As we come to more accuracy, range of y in the window was made - 0.000001 to 0.000001. I use TI-83 Plus. You may use any graphing calculator. Any more questions ? You are welcome . Google too has many graphing calculator. O R OR Use zoom Box a few times up to required accuracy.

Perfect but how did you get the roots?

Vraj Mistry - 5 years, 9 months ago

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By graphing x 9 + x 8 + x 5 + 7 x 2 2 x 7 = 0 x^{9}+x^{8}+x^{5}+7x^{2}-2x-7=0 and noting down the three intersection with x-axis. To get accurate values, I went on narrowing range of x around each intersection. As we come to more accuracy, range of y in the window was made - 0.000001 to 0.000001. I use TI-83 Plus. You may use any graphing calculator. Any more questions ? You are welcome . Google too has many graphing calculator. O R OR If you have zoom box, use it up to desired accuracy.

Niranjan Khanderia - 5 years, 9 months ago

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Can you post this explanation in your answer by editing. It will explain others how to derive this answer.This will be the perfect answer to my Problem.

Vraj Mistry - 5 years, 9 months ago

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@Vraj Mistry Thanks. I have included in my solution .

Niranjan Khanderia - 5 years, 9 months ago

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