If x^2-x-1 divides ax^6+bx^5+1 evenly, find the sum of a and b. Show all steps and exlain your answer.

Question

Please, please help me out!! Thanks alot. =)

Answer 1

You first compute the remainder of the division of ax^6+bx^5+1 by x^2-x-1. This can be conveniently done by doing computations Modulo x^2-x-1.

Modulo x^2 - x - 1 we have:

x^2 = x + 1 (1)

Squaring both sides gives:

x^4 = x^2 + 2 x + 1


Applying (1) on the right hand side gives:

x^4 = 3 x + 2

Multiplying both sides by x gives:

x^5 = 3 x^2 + 2 x

Applying (1) to the right hand side gives:

x^5 = 5 x + 3 (2)

Multiplying by x gives:

x^6 = 5 x^2 + 3 x

Applying (1) gives:

x^6 = 8 x + 5 (3)

From (2) and (3) it follows that:

ax^6+bx^5+1 = (8a+5b) x + 5a + 3b + 1

Since the remainder is zero, this means that

(8a+5b) x + 5a + 3b + 1 = 0

for all x.

You can equate the coefficient of x and the term 5a + 3b + 1 to zero to obtain two equations and solve those for a and b.

Answer 2

thank you soooo much!!! =)