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## 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:

 i. 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. x2(x + 2) = x3 + 2x2 Write this under the dividend as shown, and subtract. We get a new dividend. ii. 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. iii. 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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 - 8x3- 10x2 + 15x + 2
= Dividend.