📐 Class 10 Mathematics — Polynomials
Interactive Visual Revision Notes
Aligned with the Class 10 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.
🌎 Real World Connection: A roller-coaster track curves up and down — its shape can be modelled by a polynomial equation. When you throw a ball, its height over time follows a quadratic (degree 2) curve: h = −5t² + 20t. Engineers use polynomials to predict profits, design bridges, and calculate the safe load a beam can carry. Every curve in nature has a polynomial hiding inside it.
A polynomial is an expression with variables.
It uses addition, subtraction, and multiplication.
The powers of the variable are whole numbers.
Examples: 3x + 5, x² + 2x − 1
NOT a polynomial: 1/x, √x, x⁻²
📐 Topic 2: Degree of a Polynomial
The degree = highest power of the variable.
5x³ + 2x has degree 3.
4x² − 7 has degree 2.
9x + 1 has degree 1.
Just the number 6 has degree 0.
📐 Topic 3: Types of Polynomials by Degree
Degree 1 = Linear (makes a straight line).
Form: ax + b, where a ≠ 0.
Degree 2 = Quadratic (makes a curve).
Form: ax² + bx + c, where a ≠ 0.
Degree 3 = Cubic (makes an S-curve).
Form: ax³ + bx² + cx + d, where a ≠ 0.
📐 Topic 4: Zeroes of a Polynomial
A zero is a value that makes the polynomial = 0.
If p(k) = 0, then k is a zero.
On a graph, zeroes are where the curve crosses the x-axis.
A linear polynomial has 1 zero.
A quadratic can have 0, 1, or 2 zeroes.
A cubic can have 1, 2, or 3 zeroes.
📐 Topic 5: Graph Shapes
Linear (ax + b): a straight line.
Quadratic (ax² + bx + c): a parabola.
If a > 0, the parabola opens upward (U shape).
If a < 0, the parabola opens downward (∩ shape).
Cubic: an S-shaped curve.
📐 Topic 6: Sum of Zeroes (Quadratic)
For ax² + bx + c, let zeroes be α and β.
Sum of zeroes: α + β = −b/a
That is: −(coefficient of x) ÷ (coefficient of x²).
Example: 3x² − 12x + 9
Sum = −(−12)/3 = 4.
α + β = −b / a
📐 Topic 7: Product of Zeroes (Quadratic)
For ax² + bx + c, let zeroes be α and β.
Product of zeroes: α × β = c/a
That is: constant term ÷ coefficient of x².
Example: 3x² − 12x + 9
Product = 9/3 = 3.
α × β = c / a
📐 Topic 8: Finding a Polynomial from Zeroes
If you know the sum (S) and product (P):
The polynomial is x² − Sx + P.
Example: S = 5, P = 6.
Polynomial = x² − 5x + 6.
Check: zeroes are 2 and 3. Sum = 5, Product = 6. ✓
p(x) = x² − (sum)x + (product)
📐 Topic 9: Cubic Polynomial Zeroes
For ax³ + bx² + cx + d, zeroes are α, β, γ.
Sum: α + β + γ = −b/a
Sum of products (pairs): αβ + βγ + αγ = c/a
Product of all three: αβγ = −d/a
These are useful for checking answers.
📐 Topic 10: Why Zeroes Matter
Zeroes tell us where a graph crosses the x-axis.
They help us factorise polynomials.
Engineers use them to find break-even points.
Scientists use them to find equilibrium.
Zeroes connect algebra with graphs.
🎨 Section 2 — Easy Illustrations
Picture 1: Types of Polynomials
Three types of polynomials and their graph shapes. Red dots = zeroes.
Picture 2: Parabola Opens Up vs Down
The sign of "a" decides which way the parabola faces.
Picture 3: Zeroes on a Graph
The parabola crosses the x-axis at x = 1 and x = 4. These are the zeroes.
Picture 4: Sum and Product of Zeroes
Two golden formulas connecting zeroes to the coefficients.
Picture 5: Building a Polynomial from Zeroes
Know the zeroes? Sum and product give you the polynomial.
✏️ Section 3 — Worked Examples
Example 1: Find Zeroes and Verify the Relationship
Question: Find zeroes of x² − 5x + 4 and verify the sum/product formulas.
1Factorise: find two numbers that multiply to 4 and add to −5.
2 Those numbers are −1 and −4.
3 x² − 5x + 4 = (x − 1)(x − 4).
4 Zeroes: x = 1 and x = 4.
5Verify sum: 1 + 4 = 5. Formula: −(−5)/1 = 5. ✓
6Verify product: 1 × 4 = 4. Formula: 4/1 = 4. ✓
Zeroes = 1 and 4. Sum = 5, Product = 4.
Example 2: Find a Quadratic Polynomial from Sum and Product
Question: Find a quadratic polynomial whose zeroes have sum = −2 and product = −8.
1 Use the formula: p(x) = x² − (sum)x + (product).