1. Division of Polynomials

2. Divide a polynomial by a monomial
We can carry out the division term by term.
dividend 2 ) 8 ( + x
divisor 2 8 + = x
Rough Work x 2 8 + 2 8 + = x x 1 4
 Divide the polynomial term by term. 4 + = x
The result (quotient) is x + 4 and the remainder is equal to 0. We say that 2x2 + 8x is divisible by 2x.

3. You may also use the method of long division.
Quotient x 2 x 2 8 x 4 +
Divisor 2 x
Dividend 2 x + 8 x 2 x 8 x 8 x
Remainder = 0

4. Long division (divisor is not a monomial)
Consider the long division of ). 1 2 ( ) 3 6 8 + x
Step 1 3 6 8 1 2 x + 4 x 2 8
Step 2 2 8 x + 1 3 6 4 8 2 x
(+ 1)(+ 4x)
Step 3 4 8 3 6 1 2 x + 4 x + 3 2 + x
 Subtract 8x2 + 4x from 8x2 + 6x + 3.

5. Step 4 4 8 3 6 1 2 x + + 1 x 2
Step 5 2 x 4 8 3 6 + 1 2 x
(+ 1)(+ 1)
Step 6 4 8 3 6 1 2 x + 1 +
Subtract 2x + 1 from 2x + 3. 2

6. 4 8 3 6 1 2 x +
divisor
∵ Degree of 2 = 0,
degree of 2x + 1 = 1
∴ Degree of 2 < degree of 2x + 1
∴ Stop the division process.
remainder
∴ Quotient = 4x + 1, remainder = 2
Since the remainder is not equal to 0, we say that 8x2 + 6x + 3 is not divisible by 2x + 1.

7. Follow-up question
Find the quotient and the remainder of (3x2 + 2x3 – 9)  (2x – 1).
1. Rearrange the terms in descending powers of x.
2. Insert the missing term ‘0x’. 2 x 2 + x 1 + 9 3 2 1 - + x 2 3 - x
 x2(2x – 1) = 2x3 – x2 4 2 + x
- 9 2 4 - x
 2x(2x – 1) = 4x2 – 2x 9 2 - x
∴ Quotient = x2 + 2x + 1
Remainder = -8 1 2 - x
 1(2x – 1) = 2x – 1 8 -

8. What is the relation among the dividend, divisor, quotient and remainder?4
divisor 29 6
dividend 24
29 =
remainder 5
In arithmetic,
dividend = divisor quotient + remainder

9. This is known as the division algorithm.
In fact, the mentioned relation is also true for polynomials.
Consider f(x)  p(x).
Let Q(x) and R(x) be the quotient and the remainder respectively. ) ( x Q R f p M
dividend = divisor quotient + remainder
f(x) = p(x)  Q(x) R(x)
degree of R(x) < degree of p(x)
This is known as the division algorithm.

10. From the above result, we have 8x2 + 6x + 3  (2x + 1)(4x + 1) + 2
e.g. (8x2 + 6x + 3)  (2x + 1) 4 8 3 6 1 2 x +
From the above result, we have
8x2 + 6x + 3  (2x + 1)(4x + 1) + 2
dividend
divisor
quotient
remainder

11. = (x + 3)(2x – 1) + 5 = (x + 3)(2x) – (x + 3)(1) + 5
When a polynomial is divided by x + 3 , the quotient and the remainder are 2x – 1 and 5 respectively. Find the polynomial.
By division algorithm, we have
the required polynomial
= (x + 3)(2x – 1) + 5
= (x + 3)(2x) – (x + 3)(1) + 5
= 2x2 + 6x – x – 3 + 5
= 2x2 + 5x + 2

12. Follow-up question
When 3x2 + 10x – 5 is divided by a polynomial, the quotient and the remainder are x + 4 and 3. Find the polynomial.
Let p(x) be the required polynomial.
By division algorithm, we have
dividend
divisor
quotient
remainder
∴ The required polynomial is 3x – 2.