Question

divide the x4+2x3+3x2+2x+30 by x2-2x+2 and verify the division algori​

Answer 1

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Answer 2

Given that divide the polynomial $x^4+2x^3+3x^2+2x+30$ by $x^2-2x+2$

To verify :

The division algori​thm for the given polynomial.

Solution :

First divide the given polynomial

                                    $x^2+4x+9$

                         ____________________

      $x^2-2x+2$    ) $x^4+2x^3+3x^2+2x+30$

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

                          __(-)__(+)___(-)___________

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

                                       $4x^3-8x^2+8x$

                       _______(-)__(+)___(-)_________

                                                 $9x^2-6x+30$

                                                 $9x^2-18x+18$

                                        ____(-)___(+)__(-)_______

                                                         12x+12

                                         _____________________

From this we get the quotient is $x^2+4x+9$ and remainder is 12x+12

Now verify the Division Algorithm :

The formula for Division Algorithm is

$Dividend=divisor\times quotient+remainder$

Now substitute the values in the formula to verify we get

$x^4+2x^3+3x^2+2x+30=(x^2-2x+2)\times (x^2+4x+9)+(12x+12)$

By using the Distributive property a(x+y+z)=ax+ay+az :

$=x^2(x^2)+x^2(4x)+x^2(9)-2x(x^2)-2x(4x)-2x(9)+2(x^2)+2(4x)+2(9)+12x+12$

By using the formula $a^m.a^n=a^{m+n}$ :

$=x^{2+2}+4x^{2+1}+9x^2-2x^{1+2}-8x^{1+1}-18x+2x^2+8x+18+12x+12$  

$=x^4+4x^3+9x^2-2x^3-8x^2-18x+2x^2+8x+18+12x+12$

Now  adding the like terms.

$=x^4+2x^3+3x^2+2x+30$

Therefore $x^4+2x^3+3x^2+2x+30=x^4+2x^3+3x^2+2x+30$

Hence the Division Algorithm is verified .