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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.

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Student-friendly explanation

Verification is a simple but important exam skill. If a number is claimed to be a zero, substitute it carefully into the polynomial. If the result is 0, the claim is correct. If the result is not 0, then the number is not a zero. This method also helps students avoid blind guessing.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

    1. Replace x by the given number. 2. Compute each term carefully. 3. Add or subtract all terms. 4. Conclude whether the result is 0 or not.

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to stop after getting a non-zero part of the expression. Verification must use the full polynomial value.

Definition

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

Example

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

Rule to remember

A number a is a zero of p(x) if and only if p(a)=0.

Memory hook

Substitute and test for zero.

Examples and method

Worked example

Check whether 2 is a zero of x^2-5x+6. Substituting x=2 gives 4-10+6=0, so 2 is a zero.

Method to apply

1. Replace x by the given number. 2. Compute each term carefully. 3. Add or subtract all terms. 4. Conclude whether the result is 0 or not.

Diagram support

On a graph, the same check appears as the point where the curve meets the x-axis at x=a.

How CBSE asks it

Verify a given value as a zero, or show that a number is not a zero by direct substitution.

Avoid common mistakes

Common confusion

Students often substitute only one part of the expression and forget to use the full polynomial. That gives a wrong check.

Common wrong answer

A common wrong answer is to stop after getting a non-zero part of the expression. Verification must use the full polynomial value.

Exam tip

Always write the substitution step clearly. One small sign mistake can change the final answer.

Quick check

How do you check whether a number is a zero of a polynomial?

Substitute the number into the polynomial and see whether the value becomes 0. If the result is 0, the number is a zero; otherwise it is not.

Answer writing and exam use

1-mark answer

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

2-mark answer

To verify whether a number is a zero of a polynomial, substitute that number into the polynomial and check whether the value becomes 0. A number a is a zero of p(x) if and only if p(a)=0. For p(x)=x^2-4x+3, p(1)=1-4+3=0, so 1 is a zero.

3-mark answer

Verification is a simple but important exam skill. If a number is claimed to be a zero, substitute it carefully into the polynomial. If the result is 0, the claim is correct. If the result is not 0, then the number is not a zero. This method also helps students avoid blind guessing. A number a is a zero of p(x) if and only if p(a)=0. Check whether 2 is a zero of x^2-5x+6. Substituting x=2 gives 4-10+6=0, so 2 is a zero. Verify a given value as a zero, or show that a number is not a zero by direct substitution. A common wrong answer is to stop after getting a non-zero part of the expression. Verification must use the full polynomial value.
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