Mathematics

Example: If a and b are the zeroes of ax² + bx + c, a ≠ 0 then verify the relation between the zeroes and its coefficients. 

Solution: Since a and b are the zeroes of polynomial ax² + bx + c. Therefore, (x – a), (x – b) are the factors of the quadratic polynomial ax² + bx + c.
⇒ ax² + bx + c = k (x – a) (x – b)
⇒ ax² + bx + c = k {x² – (a + b) x + ab}
⇒ ax² + bx + c = kx² – k(a + b) x + kab ...(1)

Comparing the coefficients of x², x and constant terms of (1) on both sides, we get a = k, b = – k(a + b) and c = kab
⇒ a + b = –and ab =
   a + b = and ab =  [k = a]

Sum of the zeroes = = 

Product of the zeroes = = 

Example: Prove relation between the zeroes and the coefficient of the quadratic polynomial ax²+ bx + c.

Solution: Let a and b be the zeroes of the polynomial ax² + bx + c
⇒ a =  ....(1)
⇒ b = ....(2)
By adding (1) and (2), we get
a + b =  += = 

Hence, sum of the zeroes of the polynomial ax² + bx + c is . 

By multiplying (1) and (2), we get
ab =  
     = =  
     =   = = 

Hence, product of the zeroes of the polynomial ax² + bx + c is  = .

Example: Find the zeroes of the quadratic polynomial x² – 2x – 8 and verify a relationship between zeroes and its coefficients.

Solution: x² – 2x – 8 =
= x² – 4x + 2x – 8
= x (x – 4) + 2 (x – 4)
= (x – 4) (x + 2)
So, the value of a quadratic equation x² – 2x – 8 is zero when
x – 4 = 0 or x + 2 = 0 i.e., when x = 4 or x = – 2.
So, the zeroes of x² – 2x – 8 are 4, – 2.

Sum of the zeroes : 4 + (– 2) = 2 = =

Product of the zeroes : = 4 × (–2) = –8 = =

Example: Verify that the numbers given along side of the cubic polynomials are their zeroes. Also verify the relationship between the zeroes and the coefficients. 2x³ + x² – 5x + 2, , 1, – 2 

Solution: Here, the polynomial p(x) is 2x³ + x² – 5x + 2

Value of the polynomial 2x³ + x² – 5x + 2 when x = 
= 2()³ + ()² – 5 () + 2 =1/4 + 1/4 + –5/2 + 2 = 0
So,  is a zero of p(x).

On putting x = 1 in the cubic polynomial 2x³ + x² – 5x + 2
= 2(1)³ + (1)² –­ 5(1) + 2 = 2 + 1 – 5 + 2 = 0

On putting x = –2 in the cubic polynomial 2x³ + x² – 5x + 2
= 2(–2)³ + (–2)² – 5(–2) + 2 = –16 + 4 + 10 + 2 = 0

Hence, , 1, – 2 are the zeroes of the given polynomial.

Sum of the zeroes of p(x) :   + 1 – 2 = – = 

Sum of the products of two zeroes taken at a time
= × 1 +× (–2) + 1 × (–2)
= – 1 – 2 = –  =

Product of all the three zeroes are  
= × (1) × (–2) = –1 =