DIVISION OF POLYNOMIALS

Let us consider the numbers 13 and 5. When 13 is divided by 5 what is the quotient and remainder?

Yes, of course, the quotient is 2 and the remainder is 3. We write 13 = (5x2) + 3 Let us try.

divisionofpolynomials1.png

Dividend = (Divisor x Quotient) + Remainder

From the above examples, we observe that the remainder is less than the divisor.

Division Algorithm for Polynomials

Let p(x) and g(x) be two polynomials such that

degree of p(x)  degree of g(x)

and g(x)  0. Then there exists unique polynomials q(x) and r(x) such that

p(x) = g(x) ⋅ q(x) + r(x)

where r(x) = 0 or degree of r(x) < degree of g(x).

The polynomial p(x) is the Dividend, g(x) is the Divisor, q(x) is the Quotient and r(x) is the Remainder. Now (1) can be written as

Dividend = (Divisor x Quotient ) + Remainder

If r(x) is zero, then we say p(x) is a multiple of g(x). In other words, g(x) divides p(x).

If it looks complicated, don’t worry! it is important to know how to divide polynomials, and that comes easily with practice. The examples below will help you.

Example 1 :

Dividie (x3 - 4x2 + 6x) by x, where x ≠ 0.

Solution :

Example 2 :

Find the quotient and the remainder when

(5x2 - 7x + 2) ÷ (x - 1)

Solution :

We can use polynomial long division to get the quotient amd remainder when (5x2 - 7x + 2) is divided by (x - 1).

divisionofpolynomials2.png

Quotient = 5x - 2

Remainder  = 0

Video Lesson

Examples 3-4 : Find quotient and the remainder when f(x) is divided by g(x).

Example 3 :

f(x) = 8x3 - 6x2 + 15x - 7, g(x) = 2x + 1

Solution :

divisionofpolynomials3.png

Quotient = 4x2 - 5x + 10

Remainder  = -17

Example 4 :

f(x) = x4 - 3x3 + 5x2 - 7, g(x) = x2 + x + 1

Solution :

divisionofpolynomials4.png

Quotient = x2 - 4x + 8

Remainder  = -4x - 15

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. Finding the Slope of a Tangent Line Using Derivative

    Mar 01, 24 10:45 PM

    Finding the Slope of a Tangent Line Using Derivative

    Read More

  2. Implicit Differentiation

    Mar 01, 24 08:48 PM

    Implicit Differentiation - Concept - Examples

    Read More

  3. Logarithmic Differentiation

    Mar 01, 24 08:12 AM

    Logarithmic Differentiation

    Read More