If the polynomial f=x^4+x^2+1 is divided by g=x^2+2x+3 what is the reminder of division ?

4 Answers | Add Yours

justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to  divide f = x^4 + x^2 + 1 by g=x^2 + 2x + 3.

 

x^2 + 2x + 3 | x^4 + x^2 + 1.................| x^2 -2x + 2

.....................  x^4 + 2x^3 + 3x^2

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

................................2x^3 - 2x^2 + 1

................................2x^3 - 4x^2 - 6x

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

.......................................2x^2 + 6x + 1

.......................................2x^2 + 4x + 6

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

....................................................2x - 5

Therefore we get the remainder as 2x - 5

hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

Given the polynomial f(x) = x^4 + x^2 + 1

Divided by g(x) = x^2 + 2x + 3.

We need to find the remainder.

First we will divide and determine the quotient.

==> f(x) = g(x)*P(x) + R where R is the remainder

==> f(x) = g(x) * (x^2 -2x +2) + (2x-5)

Then the quotient is ( x^2 - 2x +2)

And the remainder is ( 2x -5).

neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

To find the remainder if f(x) = x^4+x^2+1 is divided by g(x) = x^2+2x+1.

 We divide x^4+x^2+1 by x^2+2x+3.

x^2+2x+3) x^4+0*x^3 +x^2+0*x+1( x^2-2x+2

                   x^4  +2x^3   +3x^2

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

        x^2+2x+3)-2x^3   -2x^2 + 0*x+1(-2x

                          -2x^3 - 4x^2 -6x

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

                     x^2+2x+3)2x^2+6x+1 (2

                                       2x^2+4x+6

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

                                                2x -5 is the remainder.

Therefore f(x)/g(x) = (x^4+x^2+1)/ (x^2+2x+3) = x^2-2x+2 is the quotient and  2x-5 is the remainder.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll write the rule of division with reminder:

f = g*q + r

q is the quotient and r is the reminder.

By definition, the order of reminder has to be smaller than the order of g. Since g is of 2nd order, then the reminder is linear expression.

r = dx + e

x^4+x^2+1 = (x^2+2x+3)*(ax^2 + bx + c) + dx + e

We'll remove the brackets:

x^4+x^2+1 = ax^4 + bx^3 + cx^2 + 2ax^3 + 2bx^2 + 2cx + 3ax^2 + 3bx + 3c + dx + e

We'll combine like terms from the right side:

x^4+x^2+1 = ax^4 + x^3(b + 2a) + x^2(c + 2b + 3a) + x(2c + 3b + d) + 3c + e

We'll compare and we'll get:

a = 1

b + 2a = 0

2a = -b

b  =-2

c + 2b + 3a = 1

c - 4 + 3 = 1

c - 1 = 1

c = 2

2c + 3b + d = 0

d = -2c - 3b

d = -4 + 6

d = 2

3c + e = 1

e = 1 - 3c

e = 1 - 6

e = -5

The reminder r = dx + e is: r = 2x - 5.

We’ve answered 318,929 questions. We can answer yours, too.

Ask a question