Question

divide p(x) by q(x) and check your answer using division algorithm (dividend)=divisor×quotient +remainder p(x)=6x^5+4x^4-27x^3-7x^2-27x-6,q(x)=2x^2-3​

Answer 1

GIVEN :

The polynomials $p(x)=6x^5+4x^4-27x^3-7x^2-27x-6$ and ,$q(x)=2x^2-3$

TO DIVIDE :

The polynomials p(x) by q(x) and check your answer using division algorithm

SOLUTION :

From the given polynomials $p(x)=6x^5+4x^4-27x^3-7x^2-27x-6$ and ,$q(x)=2x^2-3$

Divide the p(x) by q(x) by using the Long Division Method

                          $3x^3+2x^2-9x-\frac{1}{2}$

                    ________________________________

$2x^2+0x-3$ ) $6x^5+4x^4-27x^3-7x^2-27x-6$

                      $6x^5+0x^4-9x^3$

                 __(-)___(-)___(+)________

                               $4x^4-18x^3-7x^2$

                               $4x^4+0x^3-6x^2$

                           __(-)___(-)__(+)__________

                                      $-18x^3-x^2-27x$

                                      $-18x^3+0x^2+27x$

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

                                                 $-x^2-54x-6$

                                                 $-x^2+0x+\frac{3}{2}$

                                              __(+)__(-)__(-)____

                                                      $-54x-6-\frac{3}{2}$

                                              _______________

∴ the quotient is $3x^3+2x^2-9x-\frac{1}{2}$ and remainder is $-54x-6-\frac{3}{2}$

Now verify the Division Algorithm

$dividend=divisor\times quotient +remainder$

Substitute the values we get

$6x^5+4x^4-27x^3-7x^2-27x-6=2x^2-3\times (3x^3+2x^2-9x-\frac{1}{2})+(-54x-6-\frac{3}{2})$ (by using the Distributive property)

$=6x^5+4x^4-18x^3-9x^3-x^2-9x^3-6x^2+27x+\frac{3}{2}-54x-6-\frac{3}{2}$ ( by adding the like terms)

$=6x^5+4x^4-27x^3-7x^2-27x-6$

∴ $6x^5+4x^4-27x^3-7x^2-27x-6=6x^5+4x^4-27x^3-7x^2-27x-6$

∴ Division Algorithm is verified.

Answer 2

Step-by-step explanation:

:

The polynomials p(x)=6x^5+4x^4-27x^3-7x^2-27x-6p(x)=6x

5

+4x

4

−27x

3

−7x

2

−27x−6 and ,q(x)=2x^2-3q(x)=2x

2

−3

:

The polynomials p(x) by q(x) and check your answer using division algorithm

:

From the given polynomials p(x)=6x^5+4x^4-27x^3-7x^2-27x-6p(x)=6x

5

+4x

4

−27x

3

−7x

2

−27x−6 and ,q(x)=2x^2-3q(x)=2x

2

−3

Divide the p(x) by q(x) by using the Long Division Method

3x^3+2x^2-9x-\frac{1}{2}3x

3

+2x

2

−9x−

2

1

________________________________

2x^2+0x-32x

2

+0x−3 ) 6x^5+4x^4-27x^3-7x^2-27x-66x

5

+4x

4

−27x

3

−7x

2

−27x−6

6x^5+0x^4-9x^36x

5

+0x

4

−9x

3

__(-)___(-)___(+)________

4x^4-18x^3-7x^24x

4

−18x

3

−7x

2

4x^4+0x^3-6x^24x

4

+0x

3

−6x

2

__(-)___(-)__(+)__________

-18x^3-x^2-27x−18x

3

−x

2

−27x

-18x^3+0x^2+27x−18x

3

+0x

2

+27x

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

-x^2-54x-6−x

2

−54x−6

-x^2+0x+\frac{3}{2}−x

2

+0x+

2

3

__(+)__(-)__(-)____

-54x-6-\frac{3}{2}−54x−6−

2

3

_______________

∴ the quotient is 3x^3+2x^2-9x-\frac{1}{2}3x

3

+2x

2

−9x−

2

1

and remainder is -54x-6-\frac{3}{2}−54x−6−

2

3

Now verify the Division Algorithm

dividend=divisor\times quotient +Remainder dividend=divisor×quotient+remainder

Substitute the values we get

6x^5+4x^4-27x^3-7x^2-27x-6=2x^2-3\times (3x^3+2x^2-9x-\frac{1}{2})+(-54x-6-\frac{3}{2})6x

5

+4x

4

−27x

3

−7x

2

−27x−6=2x

2

−3×(3x

3

+2x

2

−9x−

2

1

)+(−54x−6−

2

3

) by using distribution

=6x^5+4x^4-18x^3-9x^3-x^2-9x^3-6x^2+27x+\frac{3}{2}-54x-6-\frac{3}{2}=6x

5

+4x

4

−18x

3

−9x

3

−x

2

−9x

3

−6x

2

+27x+

2

3

−54x−6−

2

3

by adding the same terms

=6x^5+4x^4-27x^3-7x^2-27x-6=6x

5

+4x

4

−27x

3

−7x

2

−27x−6

∴ 6x^5+4x^4-27x^3-7x^2-27x-6=6x^5+4x^4-27x^3-7x^2-27x-66x

5

+4x

4

−27x

3

−7x

2

−27x−6=6x

5

+4x

4

−27x

3

−7x

2

−27x−6

verified.