Forming quadratic polynomial from given zeros Class 10 Maths | Polynomials

Understand it

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.

Example

If the zeros are 2 and 3, one quadratic polynomial is (x-2)(x-3)=x²-5x+6.

Formula and rule

Polynomial with zeros α and β: k(x-α)(x-β), k≠0.

Expanding gives k[x²-(α+β)x+αβ].

See it worked out

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

Worked example

1Zeros are -1 and 4.
2So the polynomial is (x+1)(x-4)=x²-3x-4.

Method you can reuse

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

Visual tip

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer

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.

Common wrong answer

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

Don't mix this up

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.

Exam tip

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

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

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

1
Enough for 1 mark
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
Add for 2 marks
Example: If the zeros are 2 and 3, one quadratic polynomial is (x-2)(x-3)=x²-5x+6.
3
Add for 3 marks
To form a quadratic polynomial from its zeros, first write the factors x-α and x-β.

Use the concept graph around this idea instead of revising it in isolation.

Relationship between zeros and coefficients of a quadratic polynomial

If α and β are the zeros of the quadratic polynomial ax²+bx+c, then α+β=-b/a and αβ=c/a, where a is not zero.

Verification of given zeros

To verify whether a number is a zero of a polynomial, substitute that number into the polynomial and check whether the value becomes 0.

Quadratic polynomial and its zeros

A quadratic polynomial is a polynomial of degree 2, usually written as ax²+bx+c where a is not zero. Its zeros are the values of x that make the polynomial equal to 0.

Geometrical meaning of zero of a polynomial

A zero of a polynomial is the value of x for which the polynomial becomes 0. On the graph, it is the x-coordinate where the curve touches or cuts the x-axis.

Previous: Relationship between zeros and coefficients of a cubic polynomial Next: Forming cubic polynomial from given zeros Back to full mind map

Forming quadratic polynomial from given zeros

Understand it

See it worked out

Check yourself

Write it in the exam

Practice this concept with focused MCQs

Help improve this page

Forming quadratic polynomial from given zeros

Understand it

See it worked out

Check yourself

Write it in the exam

Related concepts

Practice this concept with focused MCQs

Help improve this page