What is the remainder when x^4-3x^2+7x+3x4−3x2+7x+3 is divided by x-2x−2?

2 Answers
Jim G.
Jul 11, 2017

"remainder "=21remainder =21

Explanation:

"one way of dividing is to use the divisor as a factor in "one way of dividing is to use the divisor as a factor in
"the numerator"the numerator

"consider the numerator"consider the numerator

color(red)(x^3)(x-2)color(magenta)(+2x^3)-3x^2+7x+3x3(x−2)+2x3−3x2+7x+3

=color(red)(x^3)(x-2)color(red)(+2x^2)(x-2)color(magenta)(+4x^2)-3x^2+7x+3=x3(x−2)+2x2(x−2)+4x2−3x2+7x+3

=color(red)(x^3)(x-2)color(red)(+2x^2)(x-2)color(red)(+x)(x-2)color(magenta)(+2x)+7x+3=x3(x−2)+2x2(x−2)+x(x−2)+2x+7x+3

=color(red)(x^3)(x-2)color(red)(+2x^2)(x-2)color(red)(+x)(x-2)color(red)(+9)(x-2)color(magenta)(+18)+3=x3(x−2)+2x2(x−2)+x(x−2)+9(x−2)+18+3

=color(red)(x^3)(x-2)color(red)(+2x^2)(x-2)color(red)(+x)(x-2)color(red)(+9)(x-2)+21=x3(x−2)+2x2(x−2)+x(x−2)+9(x−2)+21

"quotient "=color(red)(x^3+2x^2+x+9)," remainder "=21quotient =x3+2x2+x+9, remainder =21

sjc
Jul 12, 2017

2121

Explanation:

We can use the Remainder theorem , which states;

"For the Polynomial "P(x) " its remainder on division by "For the Polynomial P(x) its remainder on division by

(x-a) " is "P(a)(x−a) is P(a)

proof

(x-a)" a factor of "P(x)(x−a) a factor of P(x)

=>P(x)=(x-a)Q(x)+R---(1)⇒P(x)=(x−a)Q(x)+R−−−(1)

where

Q(x)" is the quotient polynomial"Q(x) is the quotient polynomial

R=" the remainder"R= the remainder

taking (1)(1)

P(a)=cancel((a-a)Q(a))+R

:.P(a)=R

so we have

P(x)=x^4-3x^2+7x+3

it is to be divided by

(x-2)

:. " we require "P(2)

P(2)=2^4-3xx2^2+7xx2+3

P(2)=16-12+14+3

P(2)=21

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