medium difficulty5 min

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).

In one line: Dividend = divisor × quotient + remainder.

Practice This Concept

Understand it

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.

Example

If p(x)=x²-1 and g(x)=x-1, then p(x)=(x-1)(x+1)+0.

Formula and rule

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

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1Divide x²+3x+2 by x+1.
2Since x²+3x+2=(x+1)(x+2)+0, the quotient is x+2 and the remainder is 0.

Method you can reuse

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

Visual tip

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer

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.

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.

Don't mix this up

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

Exam tip

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

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

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

1
Enough for 1 mark
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
Add for 2 marks
Example: If p(x)=x²-1 and g(x)=x-1, then p(x)=(x-1)(x+1)+0.
3
Add for 3 marks
The division algorithm is the polynomial version of ordinary division.

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