SOLUTION: f f(x) = x8 - 1 is divided by x -2, the remainder would be?

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Question 1134893: f f(x) = x8 - 1 is divided by x -2, the remainder would be?
Found 4 solutions by josgarithmetic, MathLover1, ikleyn, MathTherapy:
Answer by josgarithmetic(39618)   (Show Source):
You can put this solution on YOUR website!
Either try polynomial Division or if you know it, synthetic division.

2 | 1 0 0 0 0 0 0 0 -1 | | 2 4 8 16 32 64 128 256 |________________________________________ 1 2 4 8 16 32 64 128 255

Remainder is 255.

Answer by MathLover1(20850)   (Show Source):
You can put this solution on YOUR website!

you can use long division
Divide the leading coefficients of the numerator and the divisor

multiply by :
subtract from to get remainder

so far, you have

now repeat all with
..........multiply

subtract from

so far you have
then

and continue same process until you get

=> reminder is

Answer by ikleyn(52797)   (Show Source):
You can put this solution on YOUR website!
.
f f(x) = x8 - 1 is divided by x -2, the remainder would be?
~~~~~~~~~~~~~~~~~~

            Look how I edited your post to present it in the right form:

If f(x) = x^8 - 1 is divided by x -2, what the remainder is ?

Solution

A standard way to solve such problems is to apply the Remainder Theorem.

The Remainder Theorem says that for any polynomial f(x) the remainder of division by a binomial (x-a) is equal to the value
of the polynomial at x= a, i.e. f(a).

In the given case, the remainder of division of the polynomial by (x-2) is equal to = 256 - 1 = 255.

It is the shortest solution to the problem which requires MINIMUM calculations.

Solved.

----------------

   Theorem   (the remainder theorem)
   1. The remainder of division the polynomial    by the binomial    is equal to the value    of the polynomial.
   2. The binomial    divides the polynomial    if and only if the value of    is the root of the polynomial , i.e. .
   3. The binomial    factors the polynomial    if and only if the value of    is the root of the polynomial , i.e. .

See the lessons
    - Divisibility of polynomial f(x) by binomial x-a and the Remainder theorem
    - Solved problems on the Remainder thoerem
in this site.

Also, you have this free of charge online textbook in ALGEBRA-II in this site
    ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic
"Divisibility of polynomial f(x) by binomial (x-a). The Remainder theorem".

Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.

Answer by MathTherapy(10552)   (Show Source):
You can put this solution on YOUR website!

f f(x) = x8 - 1 is divided by x -2, the remainder would be?

REMAINDER THEOREM is the way to go......PERIOD!!

The REMAINDER THEOREM states that with a FACTOR being x - 2, the remainder is f(ROOT), or in this case, the remainder is: f(2).
When applied, we get:
THAT'S IT!!!