Relationship between zeros and coefficients of a quadratic polynomial Class 10 Maths | Polynomials

Understand it

This relationship is one of the most useful results in Class 10 polynomials.
It helps us find the sum and product of zeros without factorising every time.
It also helps in constructing polynomials and checking answers quickly.
Students should remember that the coefficient signs matter carefully.

Formula and rule

For a quadratic polynomial ax²+bx+c with zeros α and β: α+β=-b/a and αβ=c/a.

See it worked out

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

Worked example

1For x²-5x+6, the sum of zeros is 5 and the product is 6.
2So the zeros are 2 and 3 because 2+3=5 and 2×3=6.

Method you can reuse

1Compare the polynomial with ax²+bx+c.
2Take a, b, and c with signs carefully.
3Use sum = -b/a and product = c/a.
4Check by direct multiplication or factorisation if needed.

Visual tip

A parabola with two x-intercepts can be linked to its zeros, and those zeros can then be matched with the coefficients.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What are the sum and product of the zeros of ax²+bx+c?

Think first, then show answer

The sum of the zeros is -b/a and the product of the zeros is c/a, provided a is not zero.

Common wrong answer

A common wrong answer is to write sum as b/a. That fails because the standard formula has a negative sign in front of b/a.

Don't mix this up

Students often forget the minus sign in the sum formula or write product as -c/a. The product is c/a, not negative c/a.

Exam tip

For ax²+bx+c, always write sum as -b/a and product as c/a before substituting numbers.

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

Find sum/product of zeros, verify a relation, or form a polynomial from a given pair of zeros and coefficient.

1
Enough for 1 mark
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.
2
Add for 2 marks
Example: For 2x²-7x+3, the sum of zeros is 7/2 and the product of zeros is 3/2.
3
Add for 3 marks
It helps us find the sum and product of zeros without factorising every time.

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

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.

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.

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.

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.

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Relationship between zeros and coefficients of a quadratic polynomial

Understand it

See it worked out

Check yourself

Write it in the exam

Practice this concept with focused MCQs

Help improve this page

Relationship between zeros and coefficients of a quadratic polynomial

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