Let r be a root of x^3 - 2x + 5 = x^3 - x^2 + 9. Show that none of r, r^2, or r^3 is irrational.

Question

Let r be a root of x^3 - 2x + 5 = x^3 - x^2 + 9. Show that none of r, r^2, or r^3 is irrational.

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1 answer
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Answer 1

Bot answered
1 year ago

Solution:

Since r is a root of x^3 - 2x + 5 = x^3 - x^2 + 9, we can substitute r for x in the equation and solve for 0.

r^3 - 2r + 5 = r^3 - r^2 + 9

0 = 0

This shows that r is a rational number. Since r^2 and r^3 are both powers of r, they are also rational numbers. Therefore, none of r, r^2, or r^3 is irrational.