Question

Find the quotient q(x) and remainder r(x) on dividing f(x) by g(x) and verify division algorithm

f(x)=15x3-20x2+13x-12, g(x)=2-2x+x2

Answer 1

Hope it helps .........

Answer 2

GIVEN :

The functions f is defined by $f(x)=15x^3-20x^2+13x-12$

and the function g is defined by, $g(x)=2-2x+x^2$

TO FIND :

The quotient q(x) and remainder r(x) on dividing f(x) by g(x) and verify division algorithm.

SOLUTION :

Now divide the polynomial f(x) by g(x)

$f(x)=15x^3-20x^2+13x-12$

$g(x)=2-2x+x^2$ can be written as $g(x)=x^2-2x+2$

                             $15x+10$

                   ______________________

$x^2-2x+2$ ) $15x^3-20x^2+13x-12$  

                    $15x^3-30x^2+30x$

             ___(-)___(+)_______________

                            $10x^2-17x$

                            $10x^2-20x$

                      ___(-)___(+)_______

                                      3x-32

                        ______________

The quotient is 15x+10 and remainder is 3x-32

Now verify the Division Algorithm :

$Dividend=quotient\times divisor+remainder$

Here Dividend is $f(x)=15x^3-20x^2+13x-12$ and divisor is $g(x)=2-2x+x^2$

Substitute the values in the formula we get

$15x^3-20x^2+13x-12=(15x+10)\times (2-2x+x^2)+3x-32$

Applying the Distributive property :

a(x+y)=ax+ay

$=15x(2)+15x(-2x)+15x(x^2)+10(2)+10(-2x)+10(x^2)+3x-32$

$=30x-30x^2+15x^3+20-20x+10x^2+3x-32$

Adding the like terms

$=15x^3-20x^2+13x-12$

∴ $15x^3-20x^2+13x-12=15x^3-20x^2+13x-12=f(x)$

∴ The Division algorithm is verified for the give polynomials.