Exercise 3.3 — Zeroes and Coefficients

Relationship Between Zeroes and Coefficients of a Polynomial

Exercise 3.3 of Class 10 Polynomials builds on the idea of zeroes of a polynomial and connects them directly to the polynomial's coefficients. This relationship is one of the most frequently tested concepts in CBSE, Telangana, and Andhra Pradesh board exams, and it forms the foundation for solving quadratic equations, factorisation problems, and many algebra questions in Class 10.

For a quadratic polynomial written as ax² + bx + c, if α and β are its zeroes, then the sum and product of the zeroes follow fixed formulas based on the coefficients a, b, and c.

Extending the Idea to Cubic Polynomials

The same logic extends to a cubic polynomial of the form ax³ + bx² + cx + d, which has three zeroes — usually written as α, β, and γ. For these three zeroes, three separate relationships connect them to the coefficients of the polynomial.

• Sum of zeroes (α + β + γ) — equal to -b/a
• Sum of products taken two at a time (αβ + βγ + γα) — equal to c/a
• Product of all zeroes (αβγ) — equal to -d/a

These three identities allow students to find unknown zeroes, verify given zeroes, or even construct a polynomial when the zeroes are already known — all without solving the equation from scratch.

How to Find Zeroes and Verify the Relationship

A common type of question in Exercise 3.3 asks students to find the zeroes of a quadratic polynomial by factorisation, and then verify that the sum and product of those zeroes match the formulas above. The general method follows three clear steps.

• Step 1: Split the middle term of the quadratic so it can be factorised into two linear factors.
• Step 2: Set each factor equal to zero and solve for the variable to get the two zeroes.
• Step 3: Compare the polynomial with ax² + bx + c, identify a, b, and c, and check that α + β = -b/a and αβ = c/a.

For example, for the polynomial x² - 2x - 8, splitting the middle term gives factors (x - 4) and (x + 2), so the zeroes are 4 and -2. Their sum is 2, which matches -b/a, and their product is -8, which matches c/a — confirming the relationship holds true.

Constructing a Polynomial from Given Zeroes

Another important skill covered in this exercise is the reverse process — building a quadratic polynomial when the sum and product of its zeroes are already given. This uses a standard formula that works for any pair of values.

Here, k can be any nonzero real number, since multiplying a polynomial by a constant does not change its zeroes. Students typically choose a value of k that clears any fractions, giving a polynomial with whole-number coefficients. For instance, if the sum of zeroes is 1/4 and the product is -1, choosing k = 4 gives the polynomial 4x² - x - 4.

Verifying Zeroes of a Cubic Polynomial

The exercise also includes problems where students must verify whether specific numbers are zeroes of a given cubic polynomial, and then check the three relationships between the zeroes and coefficients. This involves substituting each value into the polynomial and confirming that the result is zero — a process called the remainder check. Once confirmed, students compare the sum, pairwise products, and overall product of the zeroes against -b/a, c/a, and -d/a respectively, demonstrating that the relationships hold for cubic polynomials just as they do for quadratics.

Common Mistakes to Avoid

• Forgetting the negative sign while comparing the sum of zeroes with -b/a.
• Mixing up the signs of b, c, and d when reading coefficients from the polynomial.
• Choosing a value of k that leaves fractions in the final polynomial instead of simplifying to integer coefficients.
• Not double-checking factorisation by multiplying the factors back to confirm they give the original polynomial.

What This Lesson Prepares You For

Mastering the relationship between zeroes and coefficients is essential before moving on to Exercise 3.4, which deals with the division algorithm for polynomials. It also strengthens the algebraic skills needed for the chapter on Quadratic Equations, where factorisation and root-finding techniques are used extensively. Students who have studied Polynomials in Class 9 will find this exercise a natural extension of those basics, now applied with deeper coefficient-based reasoning that is frequently tested in board exam papers across CBSE, Telangana, and Andhra Pradesh syllabi.