Question

Divide the polynomial p(X)=x4-3x square + 4 x + 5 by the polynomial g(X)=X square + 1 - X and find the quotient and remainder also verify division algorithm

Answer 1

the answer may be this

Answer 2

The quotient to the given polynomials is $x^2+x-3$  and remainder is 8

Hence Division algorithm is verified

Step-by-step explanation:

Given polynomials are $p(X)=x^4-3x^2+4x+5$ and

$g(X)=x^2+1-x$

Rewritting the g(X) as $x^2-x+1$

Given that divide the polynomial p(X) by the polynomial g(X)

To find the quotient and remainder also verify division algorithm :

                       $x^2+x-3$

                    _________________________

$x^2-x+1$     )  $x^4+0x^3-3x^2+4x+5$  

                                 $x^4-x^3+x^2$

                                __ _(-)__(+)__(-)______________

                                        $x^3-4x^2+4x$

                                        $x^3-x^2+x$

                                   _(-)__(+)__(-)____________

                                             $-3x^2+3x+5$

                                             $-3x^2+3x-3$

                                  ____(+)___(-)__(+)_________

                                                                 $8$

                                   ____________________________

Therefore the quotient is $x^2+x-3$  and remainder is 8

Now verify the Division algorithm :

$Dividend=quotient\times divisor+remainder$

$p(X)=q(X)\times g(X)+r(X)$

Substitute the values we get

• $x^4-3x^2+4x+5=(x^2-x+1)\times (x^2+x-3)+8$
• $=x^2(x^2)+x^2(x)+x^2(-3)-x(x^2)-x(x)-x(-3)+1(x^2)+1(x)+1(-3)+8$
• $=x^4+x^3-3x^2-x^3-x^2+3x+x^2+x-3+8$ ( by using the property$a^m.a^n=a^{m+n}$ )
• $=x^4-3x^2+4x+5=LHS$ ( by adding the like terms )

Therefore LHS=RHS

Therefore Division algorithm is verified