Polynomial Division

Polynomial Division Definition

The division of polynomial \( P(x) \) by the polynomial \( S(x) \) may be written as follows [1]
\[ \dfrac{P(x)}{S(x)} = Q(x) + \dfrac{R(x)}{S(x)} \qquad (I) \]
where \( Q(x) \) is the quotient and \( R(x) \) is the remainder of the division.
If we multiply all terms in (I) above by \( S(x) \), we may write the division as
\( P(x) = Q(x) S(x) + R(x) \)

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.
An
online polynomial division calculator , that may be used to check results of division done by hand, is included



Example Polynomial Division

Example 1:
Divide
\[ \dfrac{2x^3 + 3x^2 - x + 16}{x^2 + 2x - 3} \]

Solution:
The dividend \( 2x^3 + 3x^2 - x + 16 \) is a polynomial of degree 3. The divisor \( \; x^2 + 2x - 3 \; \) is a polynomial of degree \( 2 \). By laws of the exponents, we expect that the quotient is a polynomial of degree \( 1 \) or \( 0 \).
STEP 1: We first divide the term with highest power in the dividend \( \; 2x^3 \; \) by the term with the highest power in the divisor \( \; x^2 \; \)to obtain a quotient equal to \( \; 2x \; \) and organize all three terms as follows.
polynomial long division step 1

STEP 2: We next multiply the divisor \( \; x^2 + 2x - 3 \; \) by the quotient \( \; 2x \; \)and organize the result as follows
polynomial long division step 2

STEP 3: We next subtract the result of the multiplication from the dividend as follows
polynomial long division step 3

STEP 4: We now divide the term with the highest power in the subtraction result \( \; -x^2 \; \) by the term with the highest power in the divisor \( \; x^2 \;\) to obtain \( \; -1 \; \) and organize all terms as follows
polynomial long division step 4

STEP 5: We next multiply the divisor \( \; x^2 + 2x - 3 \; \) by \( \; -1 \; \) and organize all terms as follows
polynomial long division step 5

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

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

The result of the long division may be written as follows
\( \dfrac{2x^3 + 3x^2 - x + 16}{x^2 + 2x - 3} = 2x - 1 + \dfrac{7x + 13}{x^2 + 2x - 3} \)
or also as follows
\( 2x^3 + 3x^2 - x + 16 = (2x - 1)(x^2 + 2x - 3) + 7x + 13 \)
Vocabulary associated with the long division process
\( 2x^3 + 3x^2 - x + 16 \) is the dividend
\( x^2 + 2x - 3 \) is the divisor
\( 2x - 1 \) is the quotient
\( 7x + 13 \) is the remainder


More References and Links to Polynomials

Algebra and Trigonometry - Swokowsky Cole - 1997 - ISBN: 0-534-95308-5
Algebra and Trigonometry with Analytic Geometry - R.E.Larson , R.P. Hostetler , B.H. Edwards, D.E. Heyd - 1997 - ISBN: 0-669-41723-8
polynomial Functions .