Division of a Polynomial by a Binomial
Step.1) Write the divisor and the dividend in descending powers of a literal
Step.2) Divide the first term of the dividend by the first term of the divisor, place the result as a term in the quotient.
Step.3) Multiply each term of the divisor by the term of the quotient, and subtract the product from the dividend.
Step.4) The reminder is the new dividend, divide the new dividend by the divisor using step.2
Step.5) Continue the process until the reminder is either zero or a polynomial of lower degree than the degree of the divisor.
Example.1-
1) Divide x²- 7x + 12 by (x – 3) and verify the answer
Ans.) x – 3 ) x² - 7x + 12 ( x - 4
x² - 3x
- +
----------------
- 4x + 12
- 4x + 12
----------------
0
The quotient is x - 4
Step.1) The dividend and divisor are in descending powers of x
Step.2) x² ÷ x = x, we will write x as a term in the quotient
Step.3) (x-3) x = x² - 3x, we will subtract this from the dividend
Step.4) - 4x + 12 is the new dividend, 4x ÷ x = 4, so 4 becomes the second term in the quotient.
- 4 (x – 3) = - 4x + 12, now we will subtract it from newly obtained dividend.
If we would like to verify the result then –
as per the rules – Dividend = Quotient X Divisor + Reminder (Division Algorithm)
Divisor X Quotient = (x-3) X (x-4) = x² - 7x + 12
On the other way = Quotient X Divisor + Remainder
= (x – 3) X (x – 4) + 0
= (x – 3) x – (x – 3) 4
= x² - 3x – 4x + 12
= x² - 7x + 12
Example.2-
2) Divide 2x² - 13x + 15 by (x – 5) and verify the answer
Ans.) x – 5 ) 2x² - 13x + 15 ( 2x - 3
2x² - 10x
- +
-----------------
- 3x + 15
- 3x + 15
-------------
0
The quotient is 2x - 3
Step.1) The dividend and divisor are in descending powers of x
Step.2) 2x² ÷ x = 2x , we will write 2x as a term in the quotient
Step.3) (x - 5) 2x = 2x² - 10x , we will subtract this from the dividend
Step.4) - 3x + 15 is the new dividend, 3x ÷ x = 3, so 3 becomes the second term in the quotient.
- 3 (x – 5) = - 3x + 15, now we will subtract it from newly obtained dividend.
If We would like to verify the result then –
as per the rules => Dividend = Quotient X Divisor + Remainder (Division Algorithm)
Divisor X Quotient = (x - 5) X (2x - 3) = 2x (x - 5) - 3 (x - 5)
= 2x² -10x -3x + 15 = 2x² - 13 x + 15
On the other way = Quotient X Divisor + Reminder
= (2x – 3) X (x – 5) + 0
= (2x – 3) x – (2x – 3) 5
= 2x²- 3x – 10x + 15
= x²- 13x + 15