Polynomials

Why This Chapter Matters

Polynomials are the building blocks of algebra. Every equation you solve in Class 10 and beyond involves polynomials. Understanding zeroes and their relationship with coefficients is a direct board exam topic — questions appear every year worth 4-6 marks.

Prerequisites

Core Concepts

1. What is a Polynomial?

A polynomial in x is an expression of the form:

$$p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$$

where all powers of x are non-negative integers and coefficients are real numbers.

Degree = highest power of the variable.

|---|---|---|---|

What is NOT a polynomial: 1/x, √x, x⁻², 2^x

2. Zeroes of a Polynomial

A value k is called a zero of polynomial p(x) if p(k) = 0.

Example: For p(x) = x² − 3x − 4:

Number of zeroes: A polynomial of degree n has AT MOST n zeroes.

3. Geometrical Meaning of Zeroes

The zeroes of a polynomial p(x) are exactly the x-coordinates of the points where the graph of y = p(x) cuts the x-axis.

#### Graph of Linear Polynomial y = ax + b

#### Graph of Quadratic Polynomial y = ax² + bx + c

A parabola. Number of zeroes depends on where it intersects the x-axis:

Case 1: 2 distinct zeroes    Case 2: 1 repeated zero   Case 3: No real zeroes
     ∩                              ∩                         ∩
    / \                              |                        (above x-axis)
___/   \___                     ____⌣____                _______________
   x₁  x₂                          x₁
Graph cuts x-axis twice       Touches x-axis once        Doesn't touch x-axis

|---|---|---|

4. Relationship Between Zeroes and Coefficients

#### For Quadratic Polynomial p(x) = ax² + bx + c

If α (alpha) and β (beta) are the two zeroes:

$$\alpha + \beta = \frac{-b}{a} \quad \text{(Sum of zeroes)}$$

$$\alpha \cdot \beta = \frac{c}{a} \quad \text{(Product of zeroes)}$$

Memory trick: Sum = −(middle coefficient)/(leading coefficient) | Product = (constant term)/(leading coefficient)

Example: For p(x) = 2x² − 5x + 3:

Verification: Zeroes are x = 3/2 and x = 1 (by factoring: (2x−3)(x−1))

#### For Cubic Polynomial p(x) = ax³ + bx² + cx + d

If α, β, γ (gamma) are the three zeroes:

$$\alpha + \beta + \gamma = \frac{-b}{a}$$

$$\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$$

$$\alpha\beta\gamma = \frac{-d}{a}$$

5. Finding a Quadratic Polynomial Given Zeroes

If zeroes are α and β, the quadratic polynomial is:

$$p(x) = k[x^2 - (\alpha + \beta)x + \alpha\beta]$$

where k is any non-zero real number.

Standard answer (with k = 1): p(x) = x² − (Sum)x + (Product)

Example: Find a quadratic polynomial whose zeroes are 2 and −3.

Verify: p(2) = 4 + 2 − 6 = 0 ✓ | p(−3) = 9 − 3 − 6 = 0 ✓

6. Division Algorithm for Polynomials

Statement: If p(x) and g(x) are polynomials where degree of g(x) > 0, then:

$$p(x) = g(x) \times q(x) + r(x)$$

where r(x) = 0 OR degree of r(x) < degree of g(x).

This is exactly like dividing numbers: Dividend = Divisor × Quotient + Remainder

Example: Divide p(x) = 3x³ + x² + 2x + 5 by g(x) = 1 + 2x + x²

Arrange in standard form: g(x) = x² + 2x + 1

         3x − 5
       ___________
x²+2x+1 | 3x³ + x² + 2x + 5
          3x³ + 6x² + 3x        ← (multiply 3x by divisor)
          ─────────────────
               −5x² − x + 5
               −5x² − 10x − 5   ← (multiply −5 by divisor)
               ───────────────
                       9x + 10   ← remainder

So: p(x) = (x² + 2x + 1)(3x − 5) + (9x + 10)

Check: degree of remainder (9x + 10) = 1 < degree of divisor (x² + 2x + 1) = 2 ✓

Solved Examples

Example 1

Q: Find the zeroes of the polynomial p(x) = x² − 2x − 8 and verify the relationship.

Solution:

Factorise: x² − 2x − 8 = x² − 4x + 2x − 8 = x(x−4) + 2(x−4) = (x+2)(x−4)

Zeroes: x + 2 = 0 → x = −2 | x − 4 = 0 → x = 4

Verification:

Example 2

Q: Find a quadratic polynomial each with the given numbers as sum and product of zeroes: (i) 1/4, −1 (ii) √2, 1/3

Solution:

(i) Sum = 1/4, Product = −1

p(x) = k[x² − (1/4)x + (−1)] = k[x² − x/4 − 1]

Taking k = 4: p(x) = 4x² − x − 4

(ii) Sum = √2, Product = 1/3

p(x) = k[x² − √2x + 1/3]

Taking k = 3: p(x) = 3x² − 3√2x + 1

Example 3

Q: Verify that 3, −1, −1/3 are the zeroes of the cubic polynomial p(x) = 3x³ − 5x² − 11x − 3.

Solution:

Verification of relationships: a = 3, b = −5, c = −11, d = −3

α + β + γ = 3 + (−1) + (−1/3) = 3 − 1 − 1/3 = 5/3 = −(−5)/3 = −b/a ✓

αβ + βγ + γα = (3)(−1) + (−1)(−1/3) + (−1/3)(3) = −3 + 1/3 − 1 = −11/3 = c/a ✓

αβγ = 3 × (−1) × (−1/3) = 1 = −(−3)/3 = −d/a ✓

Previous Year Questions (PYQs)

2023

Q: If one zero of p(x) = 5x² + 13x + k is reciprocal of the other, find k.

Answer: If zeroes are α and 1/α, then product = α × (1/α) = 1 = k/5 → k = 5

2022

Q: Find zeroes of p(x) = 4s² − 4s + 1 and verify relationship.

Solution: 4s² − 4s + 1 = (2s − 1)² → zero: s = 1/2 (repeated)

Sum: 1/2 + 1/2 = 1 = 4/4 = −(−4)/4 ✓ | Product: 1/2 × 1/2 = 1/4 = 1/4 ✓

2021

Q: A quadratic polynomial whose product and sum of zeroes are −12 and 1 respectively is:

Answer: p(x) = x² − (sum)x + (product) = x² − x − 12

2020

Q: If α and β are zeroes of x² − 4x + 1, find the value of 1/α + 1/β.

Solution: 1/α + 1/β = (α + β)/αβ = 4/1 = 4

MCQ Practice

Easy

Q1. Number of zeroes of a cubic polynomial:

(A) 1 (B) 2 (C) At most 3 ✓ (D) Exactly 3

Q2. If zeroes of x² − kx + 6 are 2 and 3, then k =

(A) 5 ✓ (B) −5 (C) 6 (D) 1 [Sum = 2+3 = 5 = k/1 = k]

Medium

Q3. The graph of y = p(x) is given below. Number of zeroes is:

(Graph showing parabola touching x-axis once)

(A) 0 (B) 1 ✓ (C) 2 (D) 3

Q4. If one zero of 2x² + 3x + λ is 1/2, then λ =

(A) −2 ✓ (B) 2 (C) −1 (D) 1

[p(1/2) = 2(1/4) + 3(1/2) + λ = 0 → 1/2 + 3/2 + λ = 0 → λ = −2]

Hard

Q5. If α and β are zeroes of f(x) = x² − p(x+1) − c, then (α+1)(β+1) =

(A) c−1 ✓ (B) 1−c (C) c+1 (D) 1

[f(x) = x² − px − p − c. Product = −p−c = αβ. (α+1)(β+1) = αβ + α + β + 1 = (−p−c) + p + 1 = 1−c]

Revision Notes

Quadratic p(x) = ax² + bx + c with zeroes α, β:
  α + β = −b/a         (sum of zeroes)
  αβ   = c/a           (product of zeroes)

Form polynomial from zeroes:
  p(x) = x² − (sum)x + (product)

Division Algorithm:
  p(x) = g(x)·q(x) + r(x)
  degree(r) < degree(g) or r = 0

Common Mistakes:

❌ Confusing −b/a with b/a for sum — always the NEGATIVE of coefficient ratio

❌ Forgetting that zeroes of y = p(x) are where the GRAPH meets x-axis, not y-axis

❌ In cubic, forgetting the third relationship: αβγ = −d/a

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