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Relationship between Zeroes and Coefficients of a Polynomial

If a0, a1, a2, …, an are real numbers and 'n' is a non-negative integer, then a function p(x) = (a0 + a1x + a2x2 + a3 x3 +…+ an xn) is called a polynomial in x over reals. The value of 'n' in a given polynomial is called the degree of the polynomial. A polynomial of degree two is called a quadratic polynomial, e.g. 3x2 + 5x + 9, x2 + 9, 2x2 + 5x + 7, x2 - x +2 and .

The most general form of quadratic polynomials is ax2 + bx + c where a, b and c are real constants, a ≠ 0 and x is a real variable.
 
Zeros of a Quadratic Polynomial
If α and β are real numbers and by putting x = a and β in the quadratic polynomial ax2 + bx + c, it becomes zero, then α and β are  called the zeros of the (quadratic) polynomial, e.g. by putting x = 2 in the quadratic polynomial 3x2 - 2x - 8 it becomes zero, then '2' is the zero of the quadratic polynomial. At the most, any given quadratic polynomial can have two zeros, e.g. x2 + 6x + 8 have -4 and -2 as zeroes.
 

Sum and Product of the Zeroes of a Quadratic Polynomial

Let there be a quadratic polynomial ax2+bx+c, whose discriminant D = b2 - 4ac ≥ 0.

Then its two real zeroes α and β will be written as

  a= and β = 

Therefore, the sum of the zeroes of the polynomial,

α + β = 
        = 
and the product of the zeroes will be written as, 

α β = 
     =  

 

Therefore, the sum of the zeroes = and product of the zeroes = 

 

 
Example

If α and β are the zeroes of the polynomial 2x2 - 4x + 1, find the sum and product of the zeroes.

Solution

In the given polynomial 2x2 - 4x + 1

            a = 2, b = -4, c = 1
   ∴Sum of the zeroes = α + β = =2
 Product of the zeroes =

 

Example

If α, β are the zeroes of the polynomial x2 - 3x - 2, find the sum and product of the zeroes.

Solution

In the polynomial x2 - 3x - 2, a = 1, b = -3, c = -2
     Sum of the zeroes = α + β= 
 Product of the zeroes =αβ = 

 
Example

If α and β are the zeroes of the polynomial x2 - x - 2, find the sum and product of the zeroes.

Solution

In the polynomial x2 - x - 2
a = 1, b = -1, c = -2
    Sum of the zeroes = α + β = 
 Product of the zeroes = α β   =

 

Example

If α and β are the zeroes of the polynomial 5x2 - px + 1 and α - β =1, then find the value of 'p'.

Solution

The given polynomial is 5x2 - px + 1 and a = 5, b = -p, c = 1
Sum of the zeroes = α+β = 

Product of the zeroes = α β =
Now α-β=1 (given)
    (α-ß)2=1

or (α+β )2 - 4αß = 1


 

Example

If α and β are the zeroes of the polynomial ax2 + bx + c, find the sum and product of the zeroes.

Solution

In the given polynomial,

Sum of the zeroes = α+b,  Product of the zeroes = α β

 

 

Example

If α and β are the zeroes of x2 - 5x + 4, then, find the sum and product of the zeroes

Solution

Since α and β are the zeroes of x2 - 5x + 4 = 0
Sum of the zeroes =α + β = 5 and

Product of the zeroes = α β = 4





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