Long Division of Polynomials

Long Division

To divide a polynomial , we make use of the long division process. Before bginning the long division process, write the dividend and the divisor in descending powers of x. In order to help align like terms, use 0 as the coefficient of any missing power.

Example 1: Divide

(2x3 + 3x2 - x + 16) / (x2 + 2x - 3)

Solution:

The dividend 2x
3 + 3x2 - x + 16 is a polynomial of degree 3. The divisor x2 + 2x - 3 is a polynomial of degree 2. By laws of the exponents, we expect that the quotient is a polynomial of degree 1.

STEP 1: We first divide the term with highest power in the dividend 2x3 by the term with the highest power in the divisor x2 to obtain a quotient equal to 2x and organize all three terms as follows.

2x
____________________
x2 + 2x - 3 | 2x3 + 3x2 - x + 16


STEP 2: We next multiply the divisor x2 + 2x - 3 by the quotient 2x and organize the result as follows
2x
____________________
x2+2x-3| 2x3 + 3x2 - x + 16
2x3 + 4x2 - 6x Multiply x2 + 2x - 3 by 2x


STEP 3: We next subtract the result of the multiplication from the dividend as follows

2x
____________________
x2+2x-3| 2x3 + 3x2 - x + 16
2x3 + 4x2 - 6x
_______________________
         -x2 + 5x + 16 Subtract

STEP 4: We now divide the term with the highest power in the subtraction result -x2 by the term with the highest power in the divisor x2 to obtain -1 and organize all terms as follows

2x - 1
____________________
x2+2x-3| 2x3 + 3x2 - x + 16
2x3 + 4x2 - 6x
_______________________
         -x2 + 5x + 16

STEP 5: We next multiply the divisor x2 + 2x - 3 by -1 and organize all terms as follows

2x - 1
____________________
x2+2x-3| 2x3 + 3x2 - x + 16
2x3 + 4x2 - 6x
_______________________
         -x2 + 5x + 16
         -x2 -2x + 3 Multiply 2 + 2x - 3 by -1

STEP 6: Subtract the result of the last multiplication from the term before it and organize the results as follows.

2x - 1
____________________
x2+2x-3| 2x3 + 3x2 - x + 16
2x3 + 4x2 - 6x
_______________________
         -x2 + 5x + 16
         -x2 -2x + 3
_______________________
7x + 13 Subtract

We now stop the process since the last term 7x + 13 has a degree smaller that that of the divisor 2 + 2x - 3.

The result of the long division may be written as follows

(2x3 + 3x2 - x + 16) / (x2 + 2x - 3) = (2x - 1) + (7x + 13) / (x2 + 2x - 3)

or also as follows

2x3 + 3x2 - x + 16 = (2x - 1)(x2 + 2x - 3) + (7x + 13)

Vocabulary associated with the long division process

2x3 + 3x2 - x + 16 is the dividend

x2 + 2x - 3 is the divisor

2x - 1 is the quotient

7x + 13 is the remainder


More on polynomial Functions.

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