Posted by **betty** on Tuesday, March 31, 2009 at 6:27pm.

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.

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

- algebra 2 -
**Count Iblis**, Tuesday, March 31, 2009 at 7:16pm
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.

- algebra 2 -
**betty**, Tuesday, March 31, 2009 at 7:21pm
thank you soooo much!!! =)

## Answer this Question

## Related Questions

- math - If x^2-x-1 divides ax^6+bx^5+1 evenly, find the sum of a and b. Show all ...
- math - If x^2-x-1 divides ax^6+bx^5+1 evenly, find the sum of a and b. Show all ...
- math ergent please please... - 10.a)Solve the inequality x^3-5x^2+2x+8<0 by i...
- Algebra - Help please-I'm doing something wrong- Solve 8x^2 + 6x + 5 = 0 I'm ...
- Math....Please help I have a deadline for tonight! - Use the geometric sequence ...
- math - Find the sum of all primes p, such that p divides up, where up is the p- ...
- Algebra 2 Urgent! please Help! - simplfy (10x^2-6x-x^3)+(9x+x^3+5x^2) please ...
- algebra - (38) please help me figure this out. The multiple choice answers did ...
- math ergent please... - 10.a)Solve the inequality x^3-5x^2+2x+8<0 by i) using...
- Algebra - Find all intervals on which the given expressions are positive. (x-1)(...