Division of a Polynomial by a Polynomial –
To divide one polynomial by another polynomial, we have to follow the rules which are given below –
Step.1) First we have to arrange the terms of the dividend and the divisor in descending powers of a variable.
Step.2) Then we have to Divide the first term of the dividend by the first term of the divisor to obtain the first term of the quotient.
Step.3) Then we need to multiply all the terms of the divisor by the first term of the quotient and subtract the product from the dividend.
Step.4) Treat this reminder as the new dividend. Divide the new dividend by the divisor using step.2 and step.3
Step.5) Continue until the reminder is either zero or an expression of lower degree than the degree of the divisor.
For your better understanding we are going to give you some example for your understanding –
Example.1) Divide (x² + 6x + 8) by (x + 2)
Ans.) as per the given condition -
x + 2 ) x² + 6x + 8 ( x + 4
x² + 2x [ x²÷ x = x and x (x+2) = x²+2x ]
- -
-----------------
4x + 8
4x + 8 [ 4x ÷ x = 4 and 4 (x + 2) = 4x+8 ]
- -
---------------
0 (Ans.)
Example.2) Divide (2xᶟ - x²- x - 10) by (x - 2)
Ans.) as per the given condition we have to divide (2xᶟ - x² - x - 10) by (x - 2)
x – 2 ) 2xᶟ - x² - x – 10 ( 2x² + 3x + 5
2xᶟ - 4x² [2xᶟ ÷ x = 2x² and 2x²(x-2) = 2xᶟ- 4x²]
- +
--------------
3x² - x
3x² - 6x [3x²÷ x = 3x and 3x (x-2) = 3x²-6x]
- +
--------------
5x – 10
5x – 10
- +
--------------
0 (Ans.)
Example.3) Divide (xᶟ + 5x²+ 9x + 12) by (x + 3)
Ans.) As per the given condition we have to divide (xᶟ+ 5x²+ 9x +12) by (x + 3)
x + 3 ) xᶟ + 5x² + 9x + 12 ( x²+ 2x + 3
xᶟ + 3x² [ xᶟ ÷ x = x² and x²(x + 3) = xᶟ + 3x² ]
- -
---------------
2x²+ 9x
2x²+ 6x [ 2x²÷ x = 2x and 2x (x+3) = 2x²+ 6x ]
- -
--------------
3x + 12
3x + 9 [ 3x÷x = 3 and 3 (x+3) = 3x+9 ]
- -
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3
So, here quotient = x² + 2x + 3 and reminder = 3 (Ans.)