Polynomials — Class 10 IMO

Zeroes of a Polynomial

What is a zero of a polynomial? A zero of a polynomial p(x) is a value of x for which p(x) = 0. In other words, it is the value that makes the entire polynomial equal to zero. Zeros are also called roots of the polynomial.

Example: For p(x) = x − 3, when x = 3, p(3) = 3 − 3 = 0. So 3 is a zero of p(x).

Real-life analogy: Think of a polynomial as a machine. You put a number (x) into the machine, and it gives an output. A zero is that special input number that makes the machine output exactly zero — like finding the exact temperature where water freezes (0°C) on a thermometer scale.

Key properties:

A linear polynomial (degree 1, like ax + b) has exactly 1 zero
A quadratic polynomial (degree 2, like \(ax^{2} + bx + c\)) has at most 2 zeros
A cubic polynomial (degree 3) has at most 3 zeros
Graphically, the zeros of a polynomial are the x‑intercepts (where the graph crosses the x‑axis)
Finding zeros means solving the equation p(x) = 0

┌─────────────────────────────────────────────────────────────┐
│              ZEROS OF A POLYNOMIAL - VISUAL MAP              │
└─────────────────────────────────────────────────────────────┘

GRAPHICAL REPRESENTATION ON XY-PLANE:

     LINEAR: p(x) = 2x - 4            QUADRATIC: p(x) = x² - 4
     
        y                                   y
        │                                   │
    10  +                                  10+
        │                                   │
     5  +                                  5 +
        │                                   │
     0  +══════╪══════► x                 0 ══════╪══════► x
        │     2                               -2   │   2
     -5 +                                   -5+
        │                                      │
     -10+                                   -10+
        │ ZERO at x = 2                         ZEROS at x = -2 and x = 2


FINDING ZEROS - STEP BY STEP:

     POLYNOMIAL p(x) = x² - 5x + 6
                │
                ▼
     Set p(x) = 0  →  x² - 5x + 6 = 0
                │
                ▼
     Factorise: (x - 2)(x - 3) = 0
                │
                ▼
     Set each factor = 0:
        x - 2 = 0  →  x = 2
        x - 3 = 0  →  x = 3
                │
                ▼
     ZEROS of p(x) are 2 and 3


RELATION BETWEEN DEGREE AND NUMBER OF ZEROS:

     ┌────────────────┬────────────────┬────────────────────┐
     │    DEGREE      │    POLYNOMIAL   │   MAXIMUM ZEROS    │
     ├────────────────┼────────────────┼────────────────────┤
     │       1        │     ax + b      │         1          │
     │       2        │   ax² + bx + c  │         2          │
     │       3        │  ax³ + bx² +..  │         3          │
     └────────────────┴────────────────┴────────────────────┘


REAL-LIFE APPLICATION (PROJECTILE MOTION):

     Height of ball thrown upward: h(t) = -5t² + 20t
     
     Find when ball hits ground (h=0):
     -5t(t - 4) = 0  →  t = 0 (start) or t = 4 seconds
     
     ZERO t=4 means ball returns to ground after 4 seconds!

Example 1: Check whether x = 2 is a zero of the polynomial p(x) = \(x^{2} - 4x + 4\).

Step 1: Substitute x = 2 into p(x): p(2) = (2)\(^{2} - 4\)(2) + 4
Step 2: Calculate: 4 − 8 + 4 = 0
Step 3: Since p(2) = 0, x = 2 is a zero

Answer: Yes, x = 2 is a zero

Example 2: Find the zero of the linear polynomial p(x) = 5x − 15.

Step 1: Set p(x) = 0 → 5x − 15 = 0
Step 2: Add 15 to both sides: 5x = 15
Step 3: Divide both sides by 5: x = 3
Step 4: Verify: 5(3) − 15 = 15 − 15 = 0 ✓

Answer: The zero is x = 3

Example 3: Find all zeros of p(x) = \(x^{3} - 4x^{2} + x + 6\), given that x = 2 is a zero.

Step 1: Since x = 2 is a zero, (x − 2) is a factor
Step 2: Divide p(x) by (x − 2): (\(x^{3} - 4x^{2} + x + 6\)) ÷ (x − 2) = \(x^{2} - 2x - 3
Step\) 3: Factor the quotient: \(x^{2} - 2x - 3\) = (x − 3)(x + 1)
Step 4: Set each factor to zero: x − 2 = 0 → x = 2; x − 3 = 0 → x = 3; x + 1 = 0 → x = −1

Answer: Zeros are x = 2, x = 3, and x = −1

Quick recap

A zero of a polynomial p(x) is a value c such that p(c) = 0
Zeros are also called roots of the polynomial
A polynomial of degree n has at most n zeros
Graphically, zeros are the x‑intercepts of the polynomial's graph
To find zeros, set the polynomial equal to zero and solve for x
Linear polynomials (ax + b) have exactly one zero: x = −b/a

✓ Quick check

Is x = 1 a zero of x² − 3x + 2?

Substituting: 1 − 3 + 2 = 0, so x = 1 is a zero.

A quadratic polynomial with zeroes 2 + √3 and 2 − √3 is:

Sum = 4 and product = (2)² − (√3)² = 1, so the polynomial is x² − 4x + 1.

Zeroes and Coefficients

What is the relationship between zeros and coefficients? For a quadratic polynomial p(x) = \(ax^{2} + bx + c\) (a ≠ 0), if α (alpha) and β (beta) are its two zeros, then:

Sum of zeros: α + β = −(coefficient of x) / (coefficient of \(x^{2}\)) = −b/a

Product of zeros: α × β = (constant term) / (coefficient of \(x^{2}\)) = c/a

Why does this matter? These formulas allow us to:

Find the sum and product of zeros without solving the equation
Construct a quadratic polynomial when we know its zeros
Verify if given values could be zeros of a polynomial

Real-life example: Imagine you have a rectangular garden. The length (α) and width (β) are zeros of the polynomial representing the area. If you know the sum and product of length and width, you can find the polynomial that describes the garden's dimensions!

How to form a quadratic from given zeros: If α and β are zeros, the polynomial is: k[\(x^{2}\) − (α+β)x + αβ], where k is any non‑zero constant.

┌─────────────────────────────────────────────────────────────┐
│     ZEROS AND COEFFICIENTS RELATIONSHIP - FORMULA MAP       │
└─────────────────────────────────────────────────────────────┘

GENERAL QUADRATIC: p(x) = ax² + bx + c  (a ≠ 0)

              ZEROS:  α  and  β
                 │
        ┌────────┴────────┐
        ▼                 ▼
   SUM (α+β)        PRODUCT (α×β)
        │                 │
        ▼                 ▼
      -b                  c
      ──                  ──
       a                  a


VERIFICATION WITH EXAMPLE:

     p(x) = 2x² - 5x + 3
     a=2, b=-5, c=3

     Formula gives:
     Sum = -b/a = -(-5)/2 = 5/2 = 2.5
     Product = c/a = 3/2 = 1.5

     Solve actually: 2x²-5x+3=0 → (2x-3)(x-1)=0
     Zeros: α = 3/2 = 1.5, β = 1
     
     Sum = 1.5 + 1 = 2.5 ✓
     Product = 1.5 × 1 = 1.5 ✓


CONSTRUCTING POLYNOMIAL FROM ZEROS:

     GIVEN: α = 3, β = -2
     
     Sum = 3 + (-2) = 1
     Product = 3 × (-2) = -6
     
     Polynomial = x² - (Sum)x + Product
                = x² - (1)x + (-6)
                = x² - x - 6
     
     VERIFY: p(3) = 9 - 3 - 6 = 0 ✓
             p(-2) = 4 + 2 - 6 = 0 ✓


TRICKY CASES TABLE:

┌─────────────────┬───────────────────┬──────────────────────┐
│   TYPE OF ZEROS │   SUM (α+β)       │   PRODUCT (αβ)       │
├─────────────────┼───────────────────┼──────────────────────┤
│ Both positive   │ Positive          │ Positive             │
│ Both negative   │ Negative          │ Positive             │
│ Opposite signs  │ Positive or Neg.  │ Negative             │
│ One zero = 0    │ = other zero      │ 0                    │
│ α = β (equal)   │ = 2α              │ = α²                 │
└─────────────────┴───────────────────┴──────────────────────┘

Example 1: Find the sum and product of the zeros of the polynomial p(x) = \(3x^{2} - 7x + 4\).

Step 1: Identify coefficients: a = 3, b = −7, c = 4
Step 2: Sum of zeros = −b/a = −(−7)/3 = 7/3
Step 3: Product of zeros = c/a = 4/3

Answer: Sum = 7/3, Product = 4/3

Example 2: Find a quadratic polynomial whose zeros are 4 and −5.

Step 1: Let α = 4, β = −5
Step 2: Sum = α + β = 4 + (−5) = −1
Step 3: Product = α × β = 4 × (−5) = −20
Step 4: Polynomial = \(x^{2}\) − (sum)x + product = \(x^{2}\) − (−1)x + (−20) = \(x^{2} + x - 20
Step\) 5: For standard form, we can take k=1

Answer: \(x^{2} + x - 20\) (or any multiple like \(2x^{2} + 2x - 40\))

Example 3: If α and β are zeros of \(x^{2} - 6x + k\), and α − β = 2, find the value of k.

Step 1: From polynomial: a = 1, b = −6, c = k
Step 2: Sum α + β = −b/a = 6
Step 3: We are given α − β = 2
Step 4: Solve for α and β: Adding the two equations: (α+β)+(α−β)=6+2 → 2α=8 → α=4
Then β = 6 − α = 2
Step 5: Product αβ = c/a = k/1 = k → k = 4 × 2 = 8

Answer: k = 8

Quick recap

For quadratic \(ax^{2} + bx + c\) with zeros α, β: α + β = −b/a, αβ = c/a
These relationships work without solving the polynomial
To form a polynomial from zeros: \(x^{2}\) − (sum)x + (product)
If sum and product are fractions, multiply to clear denominators
When one zero is zero, product = 0 and c = 0
If zeros are equal (α = β), then discriminant \(b^{2} - 4ac\) = 0

✓ Quick check

The polynomial whose zeroes are the reciprocals of the zeroes of 2x² + 3x + 1 is:

Reversing the coefficients gives x² + 3x + 2 (its zeroes are the reciprocals).

The degree of the product of a quadratic polynomial and a cubic polynomial is:

Degrees add on multiplication: 2 + 3 = 5.

Division Algorithm and Identities

For polynomials, p(x) = g(x)·q(x) + r(x), where the remainder r(x) is 0 or has degree less than the divisor g(x).

Factorisation by splitting the middle term and the standard algebraic identities speed up working with polynomials.

Example 1: Divide x² − 5x + 6 by (x − 2).

Quotient (x − 3), remainder 0, so (x − 2)(x − 3).

Example 2: Factorise x² + 7x + 12.

Split 7 = 3 + 4: (x + 3)(x + 4).

Quick recap

p(x) = g(x)q(x) + r(x), deg r < deg g.
Split the middle term to factorise quadratics.

✓ Quick check

The product of the zeroes of x² − 5x + 6 is:

For ax² + bx + c, product of zeroes = c/a = 6/1 = 6.

The sum of the zeroes of 2x² − 8x + 6 is:

Sum of zeroes = −b/a = −(−8)/2 = 4.