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Forming cubic polynomial from given zeros

If α, β, and γ are the zeros of a cubic polynomial, then the polynomial can be written as k(x-α)(x-β)(x-γ), where k is a non-zero constant.

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Student-friendly explanation

To form a cubic polynomial, write one linear factor for each zero and multiply them. If the problem gives a leading coefficient, include it before expanding. This method is useful for direct construction questions and for checking whether given zeros are correct.

How to write this in exams

  1. 1

    Start with the exact idea

    If α, β, and γ are the zeros of a cubic polynomial, then the polynomial can be written as k(x-α)(x-β)(x-γ), where k is a non-zero constant.

  2. 2

    Then show how to use it

    1. Write one factor for each zero. 2. Include the leading coefficient if given. 3. Multiply two factors first, then multiply by the third. 4. Expand and simplify fully.

  3. 3

    Add one concrete example

    If the zeros are 1, 2, and 3, a cubic polynomial is (x-1)(x-2)(x-3).

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to stop after writing only two factors. That gives a quadratic, not a cubic.

Definition

If α, β, and γ are the zeros of a cubic polynomial, then the polynomial can be written as k(x-α)(x-β)(x-γ), where k is a non-zero constant.

Example

If the zeros are 1, 2, and 3, a cubic polynomial is (x-1)(x-2)(x-3).

Rule to remember

Cubic with zeros α, β, γ: k(x-α)(x-β)(x-γ), k≠0.

Memory hook

Three zeros need three brackets.

Examples and method

Worked example

Zeros are -1, 2, and 3. So the polynomial can be written as (x+1)(x-2)(x-3). First multiply (x+1)(x-2)=x^2-x-2, then multiply by (x-3) to get x^3-4x^2+x+6.

Method to apply

1. Write one factor for each zero. 2. Include the leading coefficient if given. 3. Multiply two factors first, then multiply by the third. 4. Expand and simplify fully.

Diagram support

A cubic graph may cross the x-axis at up to three points, which correspond to the three zeros used in construction.

How CBSE asks it

Form a cubic polynomial from given zeros, sometimes with a specified leading coefficient.

Avoid common mistakes

Common confusion

Students sometimes forget one zero and write only a quadratic polynomial. A cubic needs three linear factors.

Common wrong answer

A common wrong answer is to stop after writing only two factors. That gives a quadratic, not a cubic.

Exam tip

For three zeros, always write three factors before expanding. Do not skip any root.

Quick check

How many factors should be written when forming a cubic polynomial from its zeros?

You should write three linear factors, one for each zero, in the form k(x-α)(x-β)(x-γ). That is because a cubic polynomial has three zeros counting repetition.

Answer writing and exam use

1-mark answer

If α, β, and γ are the zeros of a cubic polynomial, then the polynomial can be written as k(x-α)(x-β)(x-γ), where k is a non-zero constant.

2-mark answer

If α, β, and γ are the zeros of a cubic polynomial, then the polynomial can be written as k(x-α)(x-β)(x-γ), where k is a non-zero constant. Cubic with zeros α, β, γ: k(x-α)(x-β)(x-γ), k≠0. If the zeros are 1, 2, and 3, a cubic polynomial is (x-1)(x-2)(x-3).

3-mark answer

To form a cubic polynomial, write one linear factor for each zero and multiply them. If the problem gives a leading coefficient, include it before expanding. This method is useful for direct construction questions and for checking whether given zeros are correct. Cubic with zeros α, β, γ: k(x-α)(x-β)(x-γ), k≠0. Zeros are -1, 2, and 3. So the polynomial can be written as (x+1)(x-2)(x-3). First multiply (x+1)(x-2)=x^2-x-2, then multiply by (x-3) to get x^3-4x^2+x+6. Form a cubic polynomial from given zeros, sometimes with a specified leading coefficient. A common wrong answer is to stop after writing only two factors. That gives a quadratic, not a cubic.
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