Checking remainder through factor Class 10 Maths | Polynomials

Understand it

This is one of the fastest checks in the chapter.
Instead of doing full division every time, we can substitute a into the polynomial.
If the result is zero, then x-a is a factor.
If the result is not zero, that number is the remainder.
This idea links factor theorem and remainder theorem closely.

Example

For p(x)=x²-4x+3, p(1)=0, so x-1 is a factor.

Formula and rule

Remainder on division by x-a is p(a).

If p(a)=0, then x-a is a factor of p(x).

See it worked out

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

Worked example

1For p(x)=x²-5x+6 and divisor x-2, compute p(2)=4-10+6=0.
2So x-2 is a factor and the remainder is 0.

Method you can reuse

1Match the divisor x-a with the test value a.
2Substitute a into the polynomial.
3Calculate the value.
4If the value is 0, the binomial is a factor; otherwise it is the remainder.

Visual tip

A short division layout can be paired with a substitution check to show why the remainder is zero.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If a polynomial is divided by x-3, what gives the remainder?

Think first, then show answer

Substitute 3 into the polynomial. The value p(3) is the remainder, and if p(3)=0 then x-3 is a factor.

Common wrong answer

A common wrong answer is to test the negative of the number in the divisor. The sign should be read from x-a carefully.

Don't mix this up

Students often reverse the factor sign and test the wrong value. For x-a, the test value is a, not -a.

Exam tip

When the divisor is x-a, substitute a directly into p(x).

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

Check whether a given binomial is a factor, find remainder quickly, or verify a factor without long division.

1
Enough for 1 mark
If a polynomial p(x) is divided by x-a, then the remainder is p(a). If p(a)=0, then x-a is a factor of the polynomial.
2
Add for 2 marks
Example: For p(x)=x²-4x+3, p(1)=0, so x-1 is a factor.
3
Add for 3 marks
This is one of the fastest checks in the chapter.

Use the concept graph around this idea instead of revising it in isolation.

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).

Verification of given zeros

To verify whether a number is a zero of a polynomial, substitute that number into the polynomial and check whether the value becomes 0.

Geometrical meaning of zero of a polynomial

A zero of a polynomial is the value of x for which the polynomial becomes 0. On the graph, it is the x-coordinate where the curve touches or cuts the x-axis.

Linear polynomial and its zero

A linear polynomial has highest power 1, usually written as ax+b where a is not zero. Its zero is the value of x that makes ax+b equal to 0.

Previous: Division algorithm for polynomials Next: Sign pattern and graph interpretation Back to full mind map

Checking remainder through factor

Understand it

See it worked out

Check yourself

Write it in the exam

Practice this concept with focused MCQs

Help improve this page

Checking remainder through factor

Understand it

See it worked out

Check yourself

Write it in the exam

Related concepts

Practice this concept with focused MCQs

Help improve this page