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Division algorithm for polynomials

For polynomials p(x) and g(x), with g(x) not equal to zero, p(x)=g(x)q(x)+r(x), where the degree of r(x) is less than the degree of g(x).

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Student-friendly explanation

The division algorithm is the polynomial version of ordinary division. It tells us that dividend = divisor × quotient + remainder. This idea is useful for dividing one polynomial by another, checking factors, and finding remainders. Students should remember the rule about the degree of the remainder, because that is often asked in theory questions.

How to write this in exams

  1. 1

    Start with the exact idea

    For polynomials p(x) and g(x), with g(x) not equal to zero, p(x)=g(x)q(x)+r(x), where the degree of r(x) is less than the degree of g(x).

  2. 2

    Then show how to use it

    1. Arrange the dividend in descending powers. 2. Divide the leading term by the leading term of the divisor. 3. Multiply and subtract. 4. Repeat until the remainder has lower degree than the divisor.

  3. 3

    Add one concrete example

    If p(x)=x^2-1 and g(x)=x-1, then p(x)=(x-1)(x+1)+0.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to stop too early and leave a remainder with the same degree as the divisor. That breaks the division algorithm rule.

Definition

For polynomials p(x) and g(x), with g(x) not equal to zero, p(x)=g(x)q(x)+r(x), where the degree of r(x) is less than the degree of g(x).

Example

If p(x)=x^2-1 and g(x)=x-1, then p(x)=(x-1)(x+1)+0.

Rule to remember

p(x)=g(x)q(x)+r(x), with degree of r(x) < degree of g(x).

Memory hook

Dividend = divisor × quotient + remainder.

Examples and method

Worked example

Divide x^2+3x+2 by x+1. Since x^2+3x+2=(x+1)(x+2)+0, the quotient is x+2 and the remainder is 0.

Method to apply

1. Arrange the dividend in descending powers. 2. Divide the leading term by the leading term of the divisor. 3. Multiply and subtract. 4. Repeat until the remainder has lower degree than the divisor.

Diagram support

A long division layout can be drawn neatly in steps, with dividend, divisor, quotient, and remainder clearly separated.

How CBSE asks it

Perform polynomial division, state quotient and remainder, or write the division algorithm statement.

Avoid common mistakes

Common confusion

Students sometimes write a remainder whose degree is equal to or greater than the divisor. That is not allowed.

Common wrong answer

A common wrong answer is to stop too early and leave a remainder with the same degree as the divisor. That breaks the division algorithm rule.

Exam tip

Always check that the remainder has lower degree than the divisor before finalising the answer.

Quick check

What is the correct general form of the division algorithm for polynomials?

The correct form is p(x)=g(x)q(x)+r(x), where the degree of r(x) is less than the degree of g(x). This is the rule used in polynomial division.

Answer writing and exam use

1-mark answer

For polynomials p(x) and g(x), with g(x) not equal to zero, p(x)=g(x)q(x)+r(x), where the degree of r(x) is less than the degree of g(x).

2-mark answer

For polynomials p(x) and g(x), with g(x) not equal to zero, p(x)=g(x)q(x)+r(x), where the degree of r(x) is less than the degree of g(x). p(x)=g(x)q(x)+r(x), with degree of r(x) < degree of g(x). If p(x)=x^2-1 and g(x)=x-1, then p(x)=(x-1)(x+1)+0.

3-mark answer

The division algorithm is the polynomial version of ordinary division. It tells us that dividend = divisor × quotient + remainder. This idea is useful for dividing one polynomial by another, checking factors, and finding remainders. Students should remember the rule about the degree of the remainder, because that is often asked in theory questions. p(x)=g(x)q(x)+r(x), with degree of r(x) < degree of g(x). Divide x^2+3x+2 by x+1. Since x^2+3x+2=(x+1)(x+2)+0, the quotient is x+2 and the remainder is 0. Perform polynomial division, state quotient and remainder, or write the division algorithm statement. A common wrong answer is to stop too early and leave a remainder with the same degree as the divisor. That breaks the division algorithm rule.
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