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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.

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Student-friendly explanation

A linear polynomial has only one zero because its graph is a straight line. To find the zero, set the expression equal to zero and solve. For ax+b=0, the zero is x=-b/a. This chapter uses that simple idea many times in later topics.

How to write this in exams

  1. 1

    Start with the exact idea

    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.

  2. 2

    Then show how to use it

    1. Write the polynomial equal to zero. 2. Move the constant term to the other side. 3. Divide by the coefficient of x. 4. Check by substitution if time permits.

  3. 3

    Add one concrete example

    For 2x-6, the zero is x=3 because 2(3)-6=0.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is x=+b/a. That is incorrect because the constant changes sign when shifted to the other side.

Definition

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.

Example

For 2x-6, the zero is x=3 because 2(3)-6=0.

Rule to remember

For p(x)=ax+b, a zero is x=-b/a, with a≠0. A linear polynomial has one zero and its graph is a straight line.

Memory hook

Linear means one line, one zero.

Examples and method

Worked example

For p(x)=3x+9, set 3x+9=0. Then 3x=-9, so x=-3. Therefore the zero is -3.

Method to apply

1. Write the polynomial equal to zero. 2. Move the constant term to the other side. 3. Divide by the coefficient of x. 4. Check by substitution if time permits.

Diagram support

On a graph, a linear polynomial appears as a straight line cutting the x-axis at one point if the zero is real.

How CBSE asks it

Find the zero of a given linear polynomial, or check whether a stated value is a zero.

Avoid common mistakes

Common confusion

Students often change the sign wrongly while moving the constant term. They should solve carefully: ax=-b, not ax=b.

Common wrong answer

A common wrong answer is x=+b/a. That is incorrect because the constant changes sign when shifted to the other side.

Exam tip

For a linear polynomial ax+b, write the zero directly as -b/a after checking that a is not zero.

Quick check

How many zeros can a linear polynomial have, and how do you find them?

A linear polynomial has exactly one zero. You find it by setting ax+b=0 and solving, which gives x=-b/a when a is not zero.

Answer writing and exam use

1-mark answer

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.

2-mark answer

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. For p(x)=ax+b, a zero is x=-b/a, with a≠0. A linear polynomial has one zero and its graph is a straight line. For 2x-6, the zero is x=3 because 2(3)-6=0.

3-mark answer

A linear polynomial has only one zero because its graph is a straight line. To find the zero, set the expression equal to zero and solve. For ax+b=0, the zero is x=-b/a. This chapter uses that simple idea many times in later topics. For p(x)=ax+b, a zero is x=-b/a, with a≠0. A linear polynomial has one zero and its graph is a straight line. For p(x)=3x+9, set 3x+9=0. Then 3x=-9, so x=-3. Therefore the zero is -3. Find the zero of a given linear polynomial, or check whether a stated value is a zero. A common wrong answer is x=+b/a. That is incorrect because the constant changes sign when shifted to the other side.
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