perplexus.info :: Just Math : Same Real Roots

Same Real Roots (Posted on 2016-12-10)

There exists a polynomial of the form

x^6 + ax + b

that has the same set of real roots as the polynomial

x^2 - 2x - 1

Find |a+b|

No Solution Yet Submitted by Danish Ahmed Khan

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Solution

Comment 2 of 2 |

This can be solved quickly starting with a single polynomial division:

(x^6)/(x^2-2x-1) = x^4+2x^3+5x^2+12x+29 + (70x+29)/(x^2-2x-1)

For x^6+ax+b to be a polynomial multiple of x^2-2x-1, ax+b must be the negative of the remainder of the division:

ax+b = -(70x+29) = -70x-29. Then |-70 + -29| = 99.

Posted by Brian Smith on 2016-12-10 09:54:13

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