Examples on relation between zeroes and coefficient of Polynomial

Polynomial Questions and Solutions

i) Find the zeroes of the quadratic polynomial x² - x - 6 and verify the relationship between zeroes and coefficients of x.

Given polynomial: x² - x - 6

To find the zeroes, factor the polynomial:
x² - x - 6 = (x - 3)(x + 2)

The roots of the equation are x = 3 and x = -2.
Their sum is 3 + (–2) = 1, and their product is 3 × (–2) = –6

From the polynomial: ax² + bx + c → a = 1, b = -1, c = -6
Sum of zeroes = -b/a = -(-1)/1 = 1 ✔
Product of zeroes = c/a = -6/1 = -6 ✔

So, the connection between the zeroes and the polynomial's coefficients holds true.
  

ii) Find the quadratic polynomial where sum of zeroes = 3 and product = -√2 respectively.

Let α and β be the zeroes.
Given:
Sum (α + β) = 3
Product (α × β) = -√2

Quadratic polynomial = x² - (sum)x + product
= x² - 3x - √2
  

iii) Find the cubic polynomial where three zeroes are 3, -1, and 2.

Let the zeroes be 3, -1, and 2.

Cubic polynomial = (x - 3)(x + 1)(x - 2)

Start by multiplying (x - 3) and (x + 1):
(x - 3)(x + 1) = x² - 2x - 3
Next, multiply this result by (x - 2):
(x² - 2x - 3)(x - 2) 
= x³ - 2x² - 3x - 2x² + 4x + 6
= x³ - 4x² + x + 6

Final polynomial: x³ - 4x² + x + 6