What is a zero of a polynomial?

A zero of a polynomial p(x) is a value k such that p(k) = 0. Geometrically, the zeroes of p(x) are exactly the x-coordinates of the points where the graph of y = p(x) crosses the x-axis.

What is the relationship between the zeroes and coefficients of a quadratic polynomial?

For a quadratic ax² + bx + c with zeroes α and β: the sum of the zeroes α + β = −b/a, and the product αβ = c/a.

What is the relationship between the zeroes and coefficients of a cubic polynomial?

For a cubic ax³ + bx² + cx + d with zeroes α, β and γ: the sum α+β+γ = −b/a, the sum of products taken two at a time αβ+βγ+γα = c/a, and the product αβγ = −d/a.

How do I find a quadratic polynomial given the sum and product of its zeroes?

Use x² − (sum of zeroes)x + (product of zeroes). Any nonzero constant multiple of this polynomial has the same zeroes.

Chapter 2 · CBSE Class 10 Maths

Polynomials

Q: How many zeroes can a polynomial have?

A polynomial of degree n has at most n zeroes — a linear polynomial has exactly 1, a quadratic has at most 2 (it could have 2, 1 repeated, or 0 real zeroes), and a cubic has at most 3.

Two ways to learn — a crisp formula reference, or a story that makes the sum/product trick feel obvious.

Zero of a polynomialGeometrical meaningSum & product (quadratic)Cubic relationships

Also useful:Chapter 1 — Real Numbers Chapter 3 — Pair of Linear Equations NCERT Solutions — Class 10 Maths 1-on-1 CBSE Class 10 Tuition

CBSEClass 10MathematicsChapter 2⏱ 8 min crisp · 12 min story

Four scenes that build the ideas from scratch, step by step.

How this works

Four scenes, one thread: a polynomial's zeroes are exactly where its graph crosses the x-axis — and that same picture explains why their sum and product can be read straight off the coefficients. Read the story, then tap "The Maths" for the formal result.

The Search for Zero

Take the polynomial p(x) = x² − 3x − 4. Plug in x = 4: p(4) = 16 − 12 − 4 = 0. Plug in x = −1: p(−1) = 1 + 3 − 4 = 0. Both give exactly zero — so 4 and −1 are called zeroes of this polynomial.

Student

Why do we care about the specific values that make a polynomial equal zero? What's special about zero?

Draw the graph of y = x² − 3x − 4. It's a parabola, and it happens to cross the x-axis at exactly x = −1 and x = 4 — the same two numbers. That's not a coincidence: the x-axis itself is the line y = 0, so anywhere the graph touches it, p(x) must equal 0 by definition.

Zeroes of p(x) = x-coordinates where the graph y = p(x) meets the x-axis

For x² − 3x − 4: crosses at x = −1 and x = 4 → zeroes are −1, 4

This graph picture also answers a harder question: how many zeroes CAN a polynomial have? A straight line (linear polynomial) crosses the x-axis exactly once — always. A parabola (quadratic) can cross it twice, touch it once, or miss it completely — so a quadratic has 2, 1, or 0 real zeroes. A cubic curve can wiggle across the x-axis up to three times.

Reading Without Solving

Factorise p(x) = 2x² − 8x + 6 by splitting the middle term: 2x² − 6x − 2x + 6 = 2x(x−3) − 2(x−3) = 2(x−1)(x−3). So the zeroes are 1 and 3.

Teacher

Now don't just find the zeroes — add them, and multiply them. Compare what you get to the original coefficients 2, −8, 6.

Sum of zeroes = 1 + 3 = 4 −(−8)/2 = 4 ✓ matches!

Product of zeroes = 1 × 3 = 3 6/2 = 3 ✓ matches!

That's not a coincidence either. If α and β are the zeroes of ax² + bx + c, then (x−α) and (x−β) must be its factors — so ax² + bx + c = k(x−α)(x−β) for some constant k.

k(x − α)(x − β) = k[x² − (α+β)x + αβ] = kx² − k(α+β)x + kαβ

Compare with ax² + bx + c: a = k, b = −k(α+β), c = kαβ

→ α + β = −b/a and αβ = c/a

Once you trust this derivation, you never need to factorise just to find the sum or product — read them straight off a, b, and c.

One Step Further

Does the same trick work for a cubic — three zeroes instead of two? Take p(x) = 2x³ − 5x² − 14x + 8, which has zeroes 4, −2, and 1/2 (you can check each one gives p(x) = 0).

Student

There are three zeroes now, not two. Is there still just a sum and a product, or does something else show up?

Sum: 4 + (−2) + 1/2 = 5/2 −b/a = −(−5)/2 = 5/2 ✓

Product: 4 × (−2) × (1/2) = −4 −d/a = −8/2 = −4 ✓

Sum and product both still work — but with three zeroes, there's a THIRD combination hiding in between: the sum of the products taken two zeroes at a time.

αβ + βγ + γα = (4)(−2) + (−2)(1/2) + (1/2)(4)

= −8 + (−1) + 2 = −7

c/a = −14/2 = −7 ✓ matches!

This middle relationship has no quadratic equivalent — with only two zeroes there's nothing "in between" sum and product. It only appears once you go to three or more zeroes.

Building a Polynomial Backwards

So far, every problem started with a polynomial and asked for its zeroes (or their sum/product). Now flip it: you're told the sum should be −3 and the product should be 2 — find a polynomial that has exactly those zeroes.

Teacher

You don't need to guess two numbers that add to −3 and multiply to 2. Just plug the sum and product directly into one formula.

x² − (sum)x + (product)

x² − (−3)x + 2 = x² + 3x + 2

Check it: factorise x² + 3x + 2 = (x+1)(x+2), giving zeroes −1 and −2. Sum: −1 + (−2) = −3 ✓. Product: (−1)(−2) = 2 ✓. Exactly what was asked for.

Is x² + 3x + 2 the ONLY correct answer? No — any constant multiple like 2x² + 6x + 4 or (1/2)x² + (3/2)x + 1 has the exact same two zeroes, since multiplying a whole polynomial by a nonzero constant never changes where it crosses the x-axis. Unless a question says "the" polynomial with a specific leading coefficient, any one valid multiple is a correct answer.

The Big Picture

Every idea in this chapter branches from one picture: a zero is where the graph meets the x-axis. That single picture explains the zero-count limit, and the algebra of matching a factored form to the coefficients explains every sum/product relationship — for quadratics and cubics alike.

What is a zero, geometrically?

x-coordinate where y = p(x) meets the x-axis

Max zeroes for degree n?

At most n — not guaranteed exactly n

Quadratic sum & product?

α+β = −b/a, αβ = c/a

Cubic's three relationships?

Σα = −b/a, Σαβ = c/a, αβγ = −d/a

Build a quadratic from sum & product?

x² − (sum)x + (product)

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