C
CraftExam
medium importancemedium8 min

Checking remainder through factor

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.

Practice This Concept

Learn the concept

Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    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. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

    For p(x)=x^2-4x+3, p(1)=0, so x-1 is a factor.

  4. 4

    Avoid this incomplete 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.

Definition

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.

Example

For p(x)=x^2-4x+3, p(1)=0, so x-1 is a factor.

Rule to remember

Remainder on division by x-a is p(a). If p(a)=0, then x-a is a factor of p(x).

Memory hook

Factor x-a means test a.

Examples and method

Worked example

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

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

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.

Exam tip

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

Quick check

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

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

Answer writing and exam use

1-mark answer

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-mark answer

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. Remainder on division by x-a is p(a). If p(a)=0, then x-a is a factor of p(x). For p(x)=x^2-4x+3, p(1)=0, so x-1 is a factor.

3-mark answer

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. Remainder on division by x-a is p(a). If p(a)=0, then x-a is a factor of p(x). For p(x)=x^2-5x+6 and divisor x-2, compute p(2)=4-10+6=0. So x-2 is a factor and the remainder is 0. Check whether a given binomial is a factor, find remainder quickly, or verify a factor without long division. A common wrong answer is to test the negative of the number in the divisor. The sign should be read from x-a carefully.
MCQ Quiz

Practice this concept with focused MCQs

Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.

10 MCQs5 MinutesInstant Results
Practice This Concept

Help improve this page

Found something confusing, incorrect, or missing?

Practice This Concept