# CLASS-6DIVISION OF ALGEBRAIC EXPRESSION OF POLYNOMIAL BY POLYNOMIAL

DIVISION OF ALGEBRAIC EXPRESSION OF POLYNOMIAL BY POLYNOMIAL -

Dividing algebraic expressions involves simplifying one expression by another. Here's how you can perform division of algebraic expressions, step by step:

Step 1:-  Write down the expression you want to divide, called the dividend, and the expression you want to divide by, called the divisor. For example, let's say we want to divide the polynomial P(x) by the polynomial Q(x):

Dividend:-  P(x) = 3x³− 5x²+ 2x − 7

Divisor:-    Q(x) = x²− 2x + 1

Step 2:-  Perform polynomial long division or synthetic division if the divisor is a monomial (a single term polynomial like ax). In this case, Q(x) is a polynomial, so we'll use polynomial long division.

Step 3:-  Divide the first term of the dividend (3x³) by the first term of the divisor (), which gives 3x. Write this as the first term of the quotient.

Step 4:- Multiply the divisor (Q(x)) by the term you found in the previous step (3x) and subtract this from the dividend (P(x)).

P(x) − (3x) (x²− 2x + 1)

Step 5:- Now, you should have a new expression to divide:

3x³− 5x²+ 2x − 7 − (3x) (x²− 2x + 1)

Step 6:-   Repeat steps 3 to 5 with the new expression. Divide the first term of this new expression by the first term of the divisor, which gives you the next term of the quotient.

Step 7:-  Continue this process until you've divided all the terms of the dividend. Your quotient will be the result of the division, and any remainder left over can be included as a fraction if you want to express it.

Here's a summary of the steps:

1. Divide the first term of the dividend by the first term of the divisor.
2. Multiply the divisor by the term you found in step 1 and subtract this from the dividend.
3. Repeat steps 1 and 2 until you've divided all the terms of the dividend.
4. The quotient obtained from the division is the result.

Keep in mind that division of algebraic expressions can sometimes result in fractional or polynomial expressions, and simplification may be required.

Other Way Of Understanding -

Division of algebraic expressions involves dividing one polynomial by another polynomial. To perform division, you can use long division or synthetic division, depending on the situation. I'll demonstrate both methods using an example.

Let's divide the polynomial P(x) by the polynomial D(x):

P(x) = 2x³− 5x²+ 3x − 9

D(x) = x − 2

Long Division Method:-

1. Set up the division as you would with numerical long division:

2x²
_______________________
x - 2 | 2x³- 5x²+ 3x - 9

2. Divide the highest-degree term of P(x) by the highest-degree term of D(x) and write the result above the line:

2x³ ​

-------- = 2x²

x

3. Multiply D(x) by the result from step 2 and write it below P(x), then subtract it:

2x²
_________________________
x - 2 | 2x³- 5x²+ 3x - 9

- (2x³- 4x²)

4. Bring down the next term of P(x) and repeat the process:

• Bring down 3x.

2x² + 3
__________________________
x - 2 | 2x³- 5x²+ 3x - 9

- (2x³- 4x²)
_________________
- x²

5. Continue this process until all terms are brought down and the degree of the remainder is less than the degree of D(x):

2x² + 3
__________________________
x - 2 | 2x³- 5x²+ 3x - 9

- (2x³- 4x²)
_________________
- x²

- (- x²)
_____________

0

The quotient is 2x²+ 3, and the remainder is 0.

So, the result of the division is 2x²+3.         (Ans.)