What is reminder of polynomial x4-x3-2x2-3x+1 if divided by x+1? division is not allow

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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The problem specifies that you must not perform long division to find the reinder, hence, you should remember that the value of polynomial at x=-1 is equal to the reminder that you obtain when the polynomial is divided by x+1 such that:

`f(-1) = r(-1)`

Hence, you should substitute -1 for x in polynomial such that:

`f(-1) = (-1)^4 - (-1)^3 - 2(-1)^2 - 3(-1) + 1`

`f(-1) = 1 + 1 - 2 + 3 + 1`

`f(-1) = 4 =gt r(-1) = 4`

Hence, evaluating the reminder of polynomial when you divide f(x) by x+1 yields `r(-1) = 4` .

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tutor4u | College Teacher | (Level 1) eNoter

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We have given the polynomial as x^4 - x^3 - 2x^2 - 3x + 1 need to divided by x + 1.

we can do this problem by two method one is by Remainder theorem and another by Synthetic division method. We will start with synthetic division method as it will be so easier and then we'll go for Remainder theorem.

In Synthetic division method, first we need to find the coefficient of the each term of the polynomial.i.e. the number multiplied with the variable with power.

we will get

x^4 - x^3 - 2x^2 - 3x + 1 

for x^4 = coefficient = 1

-x^3 = -1

-2x^2 = -2

 -3x = -3 and for 1 = 1

Since, we have given the divisor as x + 1. we can rewrite it as

x = -1

(as we know that divisor is simply one of the root of the equation so x+1 = 0, and subtracting both the sides by 1 will give x = -1).

Now,

       -1  ! 1  -1  -2  -3  1

            !     -1   2   0   3

            !_____________

              1   -2  0   -3  4

Here, the last term will be our remainder. which is 4.!!

we can also use the "Remainder theorem" to find the remainder of the given polynomial.

Remainder theorem states that " If any polynomial p(x) is divided by some linear factor (x - a) (where a is any number). than as a result of long polynomial division, we are left with polynomial solution q(x)(quotient polynomial) and some polynomial remainder remainder r(x)"

i.e. p(x)/(x - a) = q(x) + r(x)

or, p(x) = (x - a)*q(x) + r(x).

let's collect what values we have given.We have given

x - a as x + 1, which means we have given(by comparison) a = -1.

p(x) = x^4 - x^3 - 2x^2 - 3x + 1

Now, evaluating the polynomial at x = a i.e. x = -1 we will get the exact remainder.

just plug the value of x = -1 in the given polynomial and evaluate it to get the remainder..

we will get

p(x) = x^4 - x^3 - 2x^2 - 3x + 1

p(-1) = (-1)^4 - (-1)^3 - 2(-1)^2 - 3(-1) + 1

        = 1 + 1 - 2 + 3 + 1

        = 2 - 2 + 4

        = 4

We got the same result by using two different method here. that means both method will work fine while finding a remainder of a polynomial.

So, 4 will be our remainder.

Hope this will help you!!!

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