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medium difficulty7 min

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.

In one line: Three zeros need three brackets.

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Understand it

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.

Example

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

Formula and rule

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

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Zeros are -1, 2, and 3.

  1. 1So the polynomial can be written as (x+1)(x-2)(x-3).
  2. 2First multiply (x+1)(x-2)=x2-x-2, then multiply by (x-3) to get x3-4x2+x+6.

Method you can reuse

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

Visual tip

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer
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.

Common wrong answer

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

Don't mix this up

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

Exam tip

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

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Form a cubic polynomial from given zeros, sometimes with a specified leading coefficient.
  1. 1

    Enough for 1 mark

    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

    Add for 2 marks

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

  3. 3

    Add for 3 marks

    To form a cubic polynomial, write one linear factor for each zero and multiply them.

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