Polynomials | Maths Grade 10 CBSE

Play with it

Zeroes & coefficients, live

Drag a, b, c in p(x) = ax² + bx + c. The zeroes (orange dots) are where the curve meets the x-axis — and watch how sum of zeroes = −b/a and product = c/a hold every time.

p(x) = a x² + b x + cdrag a, b, c

a 1

b -1

c -6

Try:

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The four ideas in this chapter

A polynomial is an expression built from a variable (say x) using only whole-number powers, like 4x³ − 2x + 7. Each piece (4x³, −2x, 7) is a term, and the numbers multiplying the powers of x are the coefficients.

The degree is the highest power of x. It names the polynomial:

Degree 1 → linear (e.g. 2x + 3) — graph is a straight line.
Degree 2 → quadratic (e.g. x² − 5x + 6) — graph is a parabola.
Degree 3 → cubic (e.g. x³ − x) — graph is an S-shaped curve.

Watch out: expressions with a variable in the denominator (1/x) or under a root (√x) are not polynomials — powers must be whole numbers (0, 1, 2, …).

A zero of a polynomial p(x) is any value of x for which p(x) = 0. Finding zeroes usually means factorising and setting each factor to zero.

Worked example · zeroes of x² − 5x + 6

Factorise: x² − 5x + 6 = (x − 2)(x − 3).
Set each factor to 0: x − 2 = 0 or x − 3 = 0.
So the zeroes are x = 2 and x = 3. Check: p(2) = 4 − 10 + 6 = 0 ✓.

How many?

A polynomial of degree n has at most n zeroes. So a quadratic has at most 2, a cubic at most 3.

This is the big idea of the chapter: a zero is exactly where the graph crosses the x-axis (because there, y = p(x) = 0). So you can read the zeroes straight off a graph.

Parabola cuts the x-axis at 2 points → 2 distinct zeroes.
Parabola touches the x-axis at 1 point → 2 equal zeroes (a repeated zero).
Parabola misses the x-axis → no real zeroes.

Use the grapher near the top of the page — drag the sliders and watch the orange zero-dots appear, merge, and vanish.

Common mistake: confusing the y-intercept with a zero. A zero is where the graph meets the x-axis (y = 0), not the y-axis.

For a quadratic polynomial ax² + bx + c with zeroes α and β, the coefficients tell you their sum and product directly:

α + β = −b/a α × β = c/a

Worked example · verify for x² − 5x + 6

Here a = 1, b = −5, c = 6; the zeroes are 2 and 3.
Sum: α + β = 2 + 3 = 5, and −b/a = −(−5)/1 = 5 ✓.
Product: α × β = 2 × 3 = 6, and c/a = 6/1 = 6 ✓.

Worked example · build a polynomial from its zeroes

Find a quadratic polynomial whose zeroes are −3 and 4.

Sum = −3 + 4 = 1; Product = (−3)(4) = −12.
Use x² − (sum)x + (product): x² − x − 12.
Check: x² − x − 12 = (x + 3)(x − 4) → zeroes −3 and 4 ✓.

Common mistake: the sign. Sum of zeroes is −b/a (note the minus), not b/a.

Why this matters

Where you'll actually use Polynomials

Quadratics aren't abstract — every thrown ball, every satellite dish, every profit-maximising business runs on the curve you're studying.

Every throw is a parabola

Throw a ball, shoot a basketball, switch on a fountain — the path is the graph of a quadratic, h(x) = −ax² + bx + c. Its zeroes are exactly where the object leaves and hits the ground, and the peak is the vertex. Sports analysts and game engines compute trajectories with this every day.

Zeroes & vertex

Satellite dishes & headlights

A parabola has a magic property: every signal arriving parallel to its axis bounces to a single focus. That is why a satellite dish, a telescope mirror and a car headlight are all parabolic — weak signals get gathered to one point, or a small bulb gets spread into a strong beam.

Parabola property

💰 Profit optimisation

Revenue often follows a downward parabola; its vertex gives the price that maximises profit.

🌉 Suspension bridges

The main cable of a suspension bridge hangs in a near-perfect parabola — engineers model it as a polynomial.

🎮 Game & design curves

Polynomial (Bézier) curves draw the smooth shapes in fonts, logos and game graphics.

🔒 More real-world applications

Profit optimisation, bridge cables, design curves and more — each explained with a diagram. Free to unlock.

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Competency quiz

Modelled on CBSE's competency pattern — MCQ, assertion–reason and case-study items.

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