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Algebra Level 4

How many real solutions are there to

x 8 40 x 7 + 700 x 6 7000 x 5 + 43750 x 4 175000 x 3 + 437500 x 2 625000 x + 390625 x 5 = 0 ? \dfrac{x^8-40x^7+700x^6-7000x^5+43750x^4-175000x^3+437500x^2-625000x+390625}{x-5} = 0?


The answer is 0.

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1 solution

Siddharth Bhatt
Jan 17, 2015

As

( x 5 ) 8 = t h e n u m e r a t o r (x-5)^8 = the\ numerator

So it becomes

( x 5 ) 8 x 5 = 0 \dfrac{(x-5)^8}{x-5} = 0

This gives us the solution that x = 5 x=5

But this solution is not just as the denominator becomes 0 0 and

a n y t h i n g 0 \dfrac{anything}{0} is not defined

So there are 0 0 solutions

. 0 × 1.6 = 0 0\times1.6=0

@parthlohomi which friends??

siddharth bhatt - 6 years, 4 months ago

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SS+SG............

Parth Lohomi - 6 years, 4 months ago

How did you analysed the numerator?

U Z - 6 years, 4 months ago

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Haha!!

I saw (x-5) and then divided the numerator and got (x-5)^7

Parth Lohomi - 6 years, 4 months ago

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S you knew the expansion of (x-5)^7?

a tedious task I think , caught in a mistake while dividing the polynomial - what you will do?

U Z - 6 years, 4 months ago

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@U Z Yes it is in our books!!

Jai Gupta - 6 years, 4 months ago

@U Z I analysed the product of roots and noticed that 390625 = 5^8, so I tried (x-5)^8 and it happened XD

Devin Ky - 5 years, 11 months ago

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