## Division of Polynomials

1.     When dividing polynomials, the degree of the quotient is equal to the difference between the degrees of the
dividend and the divisor.

2.     Divisor) Dividend (quotient
------------
Remainder

Dividend = Divisor ∗ quotient + remainder

3.     If terms are missing in the dividend or divisor, leave spaces and treat them as terms with the coefficient zero.

We have observed that in the multiplication of polynomials, the degree of the product equals the sum of the
degrees of the multiplicand and the multiplier.

When dividing polynomials, the degree of the quotient is equal to the degree of the dividend minus the degree of
the divisor. The remainder may be zero or its degree is at least one less than that of the divisor.

#### Explaining Division of Polynomials:

1.     Divide the first term of the dividend by the first term of the divisor, i.e., x3 + x = x2. This is the first term of the
quotient.

x3(x + 2) = x2 + 2x2

Write this under the dividend as shown, and subtract. We get a new dividend.

2.     The first term of the new dividend is x2.

x2 + x = x

Write this as the second term of the quotient.

x(x + 2) = x2 + 2x. Write this under the dividend and subtract. You get a new dividend again.

3.     The first term of the new dividend is -4x.

-4x + x = -4. -4 is the third term of the quotient.

-4(x + 2) = -4x -8

Write this under the new dividend and subtract. 3 is left over.

We stop the process as the remainder's degree is less than that of the divisor's.

Quotient x2 + x - 4, remainder 3.

As in the case of natural numbers, the division algorithm, namely dividend = divisor quotient + remainder, can
be used to verify our computations.

Therefore,

(x + 2) (x2 + x - 4) + 3 = x3 + 3x2 - 2x - 5.

If terms are missing in the dividend or divisor, leave spaces and treat them as terms with the coefficient zero.

The example explains the method of division of polynomials involving more than one variable by first arranging
the dividend and divisor in descending powers using one of the variables, and then dividing as follows.

Example:

Divide ( x3 + x2 - 2x - 5 ) by ( x + 2 )

Here, both dividend and divisor are in descending powers of x. In the event that they are not so, write them in
descending powers of x.

Dividend

Divisor x + 2 ) x3 + 3x2 - 2x - 5 (x2 + x - 4 Quotient

x3 + 2x2

______________

x2 - 2x

x2+ 2x

______________

- 4x - 5

- 4x - 8

______________

3 Remainder.

#### Try these questions

1.     Divide ( x3- 5 x2+ 11x - 10) by ( x - 2 )

Divisor x - 2) x3 - 5x2+ 11x - 10 (x2 - 3x + 5 Quotient

- x32x2

__________________

-3x2 + 11x

3x2 ± 6x

__________________

5x - 10

-5x 10

__________________

0 Remainder

Verification:

Dividend = Divisor ∗Quotient + Remainder

= ( x - 2 ) ∗( x2 - 3x + 5 ) + 0

= x3 - 5 x2 + 11x - 10

= Dividend.

2.    Divide ( - 2 x3 - 7 x2+ 8x + 5 ) by ( 2x - 3 )

Answer: 2x - 3) 2x3 - 7x2 + 8x + 5 (x2 - 2x + 1

-2x3 3x2

_________________

- 4 x2 + 8x

4 x2 ± 6x

_________________

2x + 5

-2x 3

_________________

8

Verification:

( 2x -3 ) ( x2 -2x + 1 ) + 8 = 2x ( x2 -2x + 1 ) -3 ( x2 -2x + 1 ) + 8

= 2x3 - 4x2+ 2x - 3x2 + 3 + 6x - 3 + 8

= 2x3 - 7x2 + 8x + 5

= Dividend.

3.     Divide ( x4 + 0x3 - 4 x2 + 13x - 4 ) by ( x2- 2x + 3 )

Answer: x2 - 2x + 3) x4 + 0x3 - 4x2 + 13x - 4 (x2 + 2x - 3

- x4 2x3 ± 3x2

_____________________

2x3 - 7x2 + 13x

-2x3 4x2 ± 6x

____________________

- 3x2 + 7x - 4

3x2 ± 6x 9

____________________

x + 5

Verification:

( x2 - 2x + 3 ) ( x2 + 2x - 3 ) + ( x + 5 )

= x4+ 2 x3 - 3 x2 - 2 x3 - 4 x2 + 6x + 3 x2 + 6x - 9 + x + 5

= x4 - 4 x2 + 13x - 4

= Dividend.

4.     Divide ( 3x4 - 8x3 + 10x2- 8x - 2 ) by ( 3x2 - 2x + 5 )

Answer:3x2 – (2x + 5) 3x4 - 8x3 + 10x2 - 8x - 2 (x2 - 2x + 1/3

-3x4 2x3 ± 5x2

______________________

- 6x3 + 5x2 - 8x

6x3 ± 4x2 10x

______________________

x2 + 2x - 2

x2 (2/3)x ± 5/3

______________________

(8/3)x -11/3

Verification:

( 3x2- 2x + 5 ) ( x2- 2x + 1/3 ) + (8x)/3 - 11/3

= 3x4- 6x3+ x2 - 2x3 + 4x2- (2x)/3 + 5x2- 10x + 5/3+ (8x)/3 - 11/3

= 3x4- 8x3+ 10x2 - 8x - 2

= Dividend.

5.    Divide ( 4x4 - 8x3 + 9x2 + 3x - 7 ) by ( 2 x2 - x - 2 )

Answer: 2x2 - x - 2 ) 4x4 - 8x3 + 9x2 + 3x - 7 (2x2 - 3x + 5

-4x4 2x3 4x2

______________________

- 6x3 + 13 x2 + 3x

6x3 ± 3 x2 ± 6x

______________________

10 x2 - 3x - 7

-10 x2 5x 10

______________________

2x + 3

Verification:

( 2x2- x - 2 ) ( 2x2 - 3x + 5 ) + 2x + 3

= 4x4- 6x3 + 10x2 - 2x3 + 3x2- 5x - 4x2 + 6x -10 + 2x + 3

= 4x4 - 8x3+ 9x2 + 3x - 7

= Dividend.

6.    Divide ( 8x4 - 8x3 - 10x2 + 15x + 2 ) by ( 4x2 + 2x - 3 )

Answer: 4x2 + 2x - 3) 8x4 - 8x3 - 10x2 + 15x + 2 (2 x2 - 3x + 1/2

8x4 + 4x3 - 6x2

________________________

- 12x3 - 4x2+ 15x

-12x3 - 6x2 + 9x

________________________

2x2 + 6x + 2

2x2 + x - 3/2

________________________

5x + 3 1/2

Verification:

( 4x2 + 2x - 3 ) ( 2x2 - 3x + 1/2 ) + 5x + 7/2

= 8x4 - 12x3+ 2x2 + 4x3 - 6x2 + x - 6x2 + 9x - 3/2 + 5x + 7/2

= 8x4 -8x 3- 10x2 + 15x + 2

= Dividend.