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Forming quadratic polynomial from given zeros

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

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Student-friendly explanation

To form a quadratic polynomial from its zeros, first write the factors x-α and x-β. Then multiply them and, if needed, adjust the leading coefficient using the given information. This is a standard exam skill and often appears as a short construction question.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

    1. Write the factors x-α and x-β. 2. Multiply the factors. 3. Include the given leading coefficient if it is provided. 4. Expand and simplify.

  3. 3

    Add one concrete example

    If the zeros are 2 and 3, one quadratic polynomial is (x-2)(x-3)=x^2-5x+6.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to place the same sign as the zero inside the factor. That changes the roots completely.

Definition

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

Example

If the zeros are 2 and 3, one quadratic polynomial is (x-2)(x-3)=x^2-5x+6.

Rule to remember

Polynomial with zeros α and β: k(x-α)(x-β), k≠0. Expanding gives k[x^2-(α+β)x+αβ].

Memory hook

Zeros change sign inside factors.

Examples and method

Worked example

Zeros are -1 and 4. So the polynomial is (x+1)(x-4)=x^2-3x-4.

Method to apply

1. Write the factors x-α and x-β. 2. Multiply the factors. 3. Include the given leading coefficient if it is provided. 4. Expand and simplify.

Diagram support

A graph is not required for construction, but the zeros can be imagined as the x-intercepts of the parabola.

How CBSE asks it

Construct a quadratic polynomial when the zeros are given, sometimes with an extra condition on leading coefficient.

Avoid common mistakes

Common confusion

Students sometimes write (x+α)(x+β) even when the zeros are positive numbers. The sign inside each factor must be opposite to the zero.

Common wrong answer

A common wrong answer is to place the same sign as the zero inside the factor. That changes the roots completely.

Exam tip

From zeros α and β, use x-α and x-β, then expand carefully.

Quick check

How do you form a quadratic polynomial if its zeros are known?

Write the polynomial as k(x-α)(x-β), where α and β are the zeros and k is any non-zero constant. If no other condition is given, take k=1.

Answer writing and exam use

1-mark answer

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

2-mark answer

If α and β are the zeros of a quadratic polynomial, then the polynomial can be written as k(x-α)(x-β), where k is a non-zero constant. Polynomial with zeros α and β: k(x-α)(x-β), k≠0. Expanding gives k[x^2-(α+β)x+αβ]. If the zeros are 2 and 3, one quadratic polynomial is (x-2)(x-3)=x^2-5x+6.

3-mark answer

To form a quadratic polynomial from its zeros, first write the factors x-α and x-β. Then multiply them and, if needed, adjust the leading coefficient using the given information. This is a standard exam skill and often appears as a short construction question. Polynomial with zeros α and β: k(x-α)(x-β), k≠0. Expanding gives k[x^2-(α+β)x+αβ]. Zeros are -1 and 4. So the polynomial is (x+1)(x-4)=x^2-3x-4. Construct a quadratic polynomial when the zeros are given, sometimes with an extra condition on leading coefficient. A common wrong answer is to place the same sign as the zero inside the factor. That changes the roots completely.
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