SOLUTION: Divide x^3 - 1 by x - 1, x^4 - 1 by x - 1, and x^5 - 1 by x - 1. What is the quotient when x^9 - 1 is divided by x - 1?

Algebra ->  Numeric Fractions Calculators, Lesson and Practice -> SOLUTION: Divide x^3 - 1 by x - 1, x^4 - 1 by x - 1, and x^5 - 1 by x - 1. What is the quotient when x^9 - 1 is divided by x - 1?      Log On


   

Question 16532: Divide x^3 - 1 by x - 1, x^4 - 1 by x - 1, and x^5 - 1 by x - 1. What is the quotient when x^9 - 1 is divided by x - 1?
Found 2 solutions by rahman, venugopalramana:
Answer by rahman(247) About Me  (Show Source):
You can put this solution on YOUR website!
%28x%5E3+-+1%29%2F%28x+-+1%29=%28x%5E2%2Bx%2B1%29%28x-1%29%2F%28x-1%29=x%5E2%2Bx%2B1
%28x%5E4+-+1%29%2F%28x+-+1%29=%28x%5E3%2Bx%5E2%2Bx%2B1%29%28x-1%29%2F%28x-1%29=x%5E3%2Bx%5E2%2Bx%2B1
%28x%5E5+-+1%29%2F%28x+-+1%29=%28x%5E4%2Bx%5E3%2Bx%5E2%2Bx%2B1%29%28x-1%29%2F%28x-1%29=x%5E4%2Bx%5E3%2Bx%5E2%2Bx%2B1
Can you see the pattern?
%28x%5E9+-+1%29%2F%28x+-+1%29=%28x%5E8%2Bx%5E7%2Bx%5E6%2Bx%5E5%2Bx%5E4%2Bx%5E3%2Bx%5E2%2Bx%2B1%29%28x-1%29%2F%28x-1%29=x%5E8%2Bx%5E7%2Bx%5E6%2Bx%5E5%2Bx%5E4%2Bx%5E3%2Bx%5E2%2Bx%2B1

Answer by venugopalramana(3286) About Me  (Show Source):
You can put this solution on YOUR website!
there is a procedure for short division called horner's method which can be used here. i shall show you 2 examples ..rest you can do in a similar manner..
first write the given expression in decreasing order of powers listing missing terms also as zeros...
For example x^3-1 = 1*x^3+0*x^2+0*x-1..........(A)
now check if we have the divisor in the form of x+a or x-a or not .we have here x-1 which is in the form x-a...then let x-1 =0 which gives us x=1...(B)
now do the division as follows
write 1 0 0 -1......these are coefficients of powers of x obtained above under (A)from the problem
put 1 as divisor as obtained under (B) above
do as follows
divisor...1] 1 0 0 -1......row 1
.................. 0 1 1 1.......row 2
--------------
.................. 1 1 1 0 ......row 3
explanation: row 1 has divisor on the left as 1 followed by a seperation bracket..then put coefficients of powers of x obtained above under (A)from the problem in different columns
row 2 ..start with a zero under the coefficient in row 1 in 1st.column..now add the row 1 and row 2 numbers in 1st. column (which are there vertically one below the other )and put the sum under the same column in row 3...here 1+0=1..now multiply this number in row 3 with divisor and put it in row 2 in second column...now add numbers in row 1 and row 2 in column 2 put the sum in row 3 under same column no.2....repeat the procedure till the end
row 3 represents the quotient and remainder...number in last column in row 3 is remainder here it is zero.the other numbers read from right to left give the coefficients of increasing powers of x ...that is the answer on division is 1x^2+1x+1..remainder =0
to explain i am giving another example below...x^2-5x+6 devide by x-3
put x-3 =0 so x=3 is the divisor...
3] 1 -5 6
..... 0 3 -6
--------------
..... 1 -2 0 answer is x-2 and remainder is 0