Mathematics

Let zeros or roots of a quadratic quadrilateral be a and b

Example: Form the quadratic polynomial whose zeroes or roots are 4 and 6. 

Solution: Sum of the zeroes = 4 + 6 = 10
Product of the zeroes = 4 × 6 = 24

Hence the polynomial formed by the given equation
= x² – (sum of zeroes) x + Product of zeroes
= x² – 10x + 24

Example: Form the quadratic polynomial whose zeroes are –3, 5.

Solution: Here, zeroes are – 3 and 5.
Sum of the zeroes = – 3 + 5 = 2
Product of the zeroes = (–3) × 5 = – 15

Hence the polynomial formed by  
= x² – (sum of zeroes) x + Product of zeroes
= x² – 2x – 15

Example: Find a quadratic polynomial whose sum of zeroes and product of zeroes are respectively (i) , –1 (ii) √2, (iii) 0, √5 

Solution: Let the polynomial be ax² + bx + c and its zeroes be a and b. 

(i) Here, α + β =  and α . β = – 1, therefore, the polynomial equation is formed by the formula
= x² – (Sum of zeroes) x + Product of zeroes
= x² – x – 1 
The other polynomial are k.
If k = 4, then the polynomial is 4x² – x – 4.

(ii) Here, α + β = √2 , α . β = . Thus the polynomial formed by
= x² – (Sum of zeroes) x + Product of zeroes
= x² – (√2) x +  
Other polynomial are k.
If k = 3, then the polynomial is 3x² – 3√2x + 1

(iii) Here, α + β = 0 and α . β = √5, Thus the polynomial formed 
= x² – (Sum of zeroes) x + Product of zeroes 
= x² – (0) x + √5 = x² + √5

Example: Find a cubic polynomial with the sum of its zeroes, sum of the products of its zeroes taken two at a time, and product of its zeroes as 2, – 7 and –14, respectively.

Solution: Let the cubic polynomial be ax³ + bx² + cx + d, divide the whole polynomial by a, we get 
⇒ x³ + x² +  x +  ....(1) and its zeroes are a, b and g, then 
a + b + g = 2 = – 
ab + bg + ag = – 7 =
abg = – 14 = –

Putting the values of , and , in equtaion number 1, we get the cubic polynomial equation is
    x³ + (–2) x² + (–7)x + 14
⇒ x³ – 2x² – 7x + 14

Example: Find the cubic polynomial with the sum, sum of the product of its zeroes taken two at a time and product of its zeroes as 0, –7 and –6 respectively.

Solution: Let the cubic polynomial be ax³ + bx² + cx + d, then
⇒ x³ + x² +  x +  ....(1) and its zeroes are a, b, g. Then 
a + b + g = 0 = – 
ab + bg + ag = – 7 = 
abg = – 6 = –

Putting the values of , and ,, and in (1), we get 
x³ – (0) x² + (–7) x + (–6) 
or x³ – 7x + 6

Example: If a and b are the zeroes of the polynomials ax² + bx + c then form the polynomial whose zeroes are  and .

Solution: Since a and b are the zeroes of ax² + bx + c 
So a + b =  , ab =
Sum of the zeroes =  +  = 
= =  
Product of the zeroes 
= × == 
But required polynomial is given by the equation x² – (sum of zeroes) x + Product of zeroes
⇒ x² – x + 
⇒ c

Example: If a and b are the zeroes of the polynomial x² + 4x + 3, form the polynomial whose zeroes are 1 + and 1 + . 

Solution: Since a and b are the zeroes of the polynomial x² + 4x + 3. Then, a + b = – 4, ab = 3

Sum of the zeroes 
= 1 +  + 1 + = 
= = = =

Product of the zeroes 
= (1 + )(1 + )= 1 + + + 1 
= 2 + = 
= ==
But required polynomial is given by the quadratic formula
x² – (sum of zeroes) x + product of zeroes
or x² – x + or k
or 3(if k = 3)
⇒ 3x² – 16x + 16