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Linear Polynomial in One Variable

A linear polynomial in one variable is a polynomial of degree 1, usually written as ax + b, where a and b are constants and a is not equal to 0.

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Student-friendly explanation

In ax + b, a is the coefficient of x and b is the constant term. The zero of the polynomial is the value of x that makes ax + b equal to 0. This is found by solving ax + b = 0.

How to write this in exams

  1. 1

    Start with the exact idea

    A linear polynomial in one variable is a polynomial of degree 1, usually written as ax + b, where a and b are constants and a is not equal to 0.

  2. 2

    Then show how to use it

    Identify a and b, equate ax + b to 0, isolate x, divide by the coefficient, and substitute the answer back to check.

  3. 3

    Add one concrete example

    For p(x) = 2x - 6, the coefficient of x is 2, the constant term is -6, and the zero is x = 3 because 2(3) - 6 = 0.

  4. 4

    Avoid this incomplete answer

    For 2x - 8, students may write x = -4 by using the wrong sign, but 2x - 8 = 0 gives x = 4.

Definition

A linear polynomial in one variable is a polynomial of degree 1, usually written as ax + b, where a and b are constants and a is not equal to 0.

Example

For p(x) = 2x - 6, the coefficient of x is 2, the constant term is -6, and the zero is x = 3 because 2(3) - 6 = 0.

Rule to remember

General form: ax + b, a ≠ 0. Zero: ax + b = 0, so x = -b/a.

Memory hook

Zero means output becomes zero.

Examples and method

Worked example

Find the zero of 5x + 20. Put 5x + 20 = 0. Then 5x = -20, so x = -4. Check: 5(-4) + 20 = 0.

Method to apply

Identify a and b, equate ax + b to 0, isolate x, divide by the coefficient, and substitute the answer back to check.

Diagram support

A number-line mark can show the zero as the value where the polynomial becomes 0.

How CBSE asks it

Questions may ask for the coefficient, constant term, degree, zero, or verification of a given zero.

Avoid common mistakes

Common confusion

Students often ignore the sign of the constant term. In 4x - 12, the constant is -12, not 12.

Common wrong answer

For 2x - 8, students may write x = -4 by using the wrong sign, but 2x - 8 = 0 gives x = 4.

Exam tip

When finding the zero, write p(x) = 0 first and then solve step by step. Always verify by substitution.

Quick check

Find the zero of the linear polynomial p(x) = 3x - 15.

The zero is x = 5 because 3x - 15 = 0 gives 3x = 15, so x = 5.

Study the linear polynomial in one variable diagram carefully

Use the labelled diagram to keep linear polynomial in one variable clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of linear polynomial in one variable in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on linear polynomial in one variable.

Revision cue

Revise linear polynomial in one variable through the labels before writing the answer.

Answer writing and exam use

1-mark answer

A linear polynomial in one variable is a polynomial of degree 1, usually written as ax + b, where a and b are constants and a is not equal to 0.

2-mark answer

A linear polynomial in one variable is a polynomial of degree 1, usually written as ax + b, where a and b are constants and a is not equal to 0. General form: ax + b, a ≠ 0. Zero: ax + b = 0, so x = -b/a. For p(x) = 2x - 6, the coefficient of x is 2, the constant term is -6, and the zero is x = 3 because 2(3) - 6 = 0.

3-mark answer

In ax + b, a is the coefficient of x and b is the constant term. The zero of the polynomial is the value of x that makes ax + b equal to 0. This is found by solving ax + b = 0. General form: ax + b, a ≠ 0. Zero: ax + b = 0, so x = -b/a. Find the zero of 5x + 20. Put 5x + 20 = 0. Then 5x = -20, so x = -4. Check: 5(-4) + 20 = 0. Questions may ask for the coefficient, constant term, degree, zero, or verification of a given zero. For 2x - 8, students may write x = -4 by using the wrong sign, but 2x - 8 = 0 gives x = 4.
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