Rationality

Algebra Level 2

How many rational roots does x 1000 − x 500 + x 100 + x + 1 = 0 { x }^{ 1000 }-{ x }^{ 500 }+{ x }^{ 100 }+x+1=0 have?


The answer is 0.

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Shivamani Patil by shivamani patil , 21, India

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2 solutions

Let the given polynomial be denoted by f ( x ) f(x) By the rational root theorem,the only possible rational roots are ± 1 \pm1 .It is easily seen that f ( 1 ) ≠ 0 , f ( − 1 ) ≠ 0 f(1)\neq0,f(-1)\neq0 .So the polynomial has no rational roots.

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What is the rational root theorem?

Mukul Sharma - 5 years, 8 months ago

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This .

Pi Han Goh - 5 years, 8 months ago

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I Think rational roots theorem is correct but when your graph this equation with a graphing calculator or online graph the solution x = − 1 x = -1 seems to be the answer

Vishal Yadav - 5 years, 8 months ago

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@Vishal Yadav – Nope. Check your working again.

Pi Han Goh - 5 years, 8 months ago

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@Pi Han Goh – Maybe an Error in looking at the co-ordinates of the point but algebraically x = − 1 x=-1 is not a solution. Thanks.

Vishal Yadav - 5 years, 8 months ago

This whole article is about this only......

Rishabh Kumar - 1 year, 1 month ago
Mr. Krabs
Mar 30, 2021

If u dont know rational roots consider Desmos 😏(I swear i didnt use it tho)

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