# Relationship between Zeroes and Coefficients of a Polynomial

If a_{0}, a

_{1}, a

_{2}, â€¦, a

_{n}are real numbers and 'n' is a non-negative integer, then a function p(x) = (a

_{0}+ a

_{1}x + a

_{2}x

^{2}+ a

_{3}x

^{3}+â€¦+ a

_{n}x

^{n}) 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. 3x

^{2 }+ 5x + 9, x

^{2 }+ 9, 2x

^{2 }+ 5x + 7, x

^{2 }- x +2 and .

The most general form of quadratic polynomials is ax

^{2 }+ bx + c where a, b and c are real constants, a â‰ 0 and x is a real variable.

**Zeros of a Quadratic Polynomial**

If Î± and Î² are real numbers and by putting x = a and Î² in the quadratic polynomial ax

^{2 }+ bx + c, it becomes zero, then Î± and Î² are called the zeros of the (quadratic) polynomial, e.g. by putting x = 2 in the quadratic polynomial 3x

^{2 }- 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. x

^{2 }+ 6x + 8 have -4 and -2 as zeroes.

# Sum and Product of the Zeroes of a Quadratic Polynomial

Let there be a quadratic polynomial ax^{2}+bx+c, whose discriminant D = b

^{2 }- 4ac â‰¥ 0.

Then its two real zeroes Î± and Î² will be written as

a= and Î² =

Therefore, the sum of the zeroes of the polynomial,

Î± + Î² =

=

and the product of the zeroes will be written as,

Î± Î² =

=

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

If Î± and Î² are the zeroes of the polynomial 2x^{2 }- 4x + 1, find the sum and product of the zeroes.

In the given polynomial 2x^{2 }- 4x + 1

a = 2, b = -4, c = 1

âˆ´Sum of the zeroes = Î± + Î² = =2

Product of the zeroes =

If Î±, Î² are the zeroes of the polynomial x^{2 }- 3x - 2, find the sum and product of the zeroes.

In the polynomial x^{2 }- 3x - 2, a = 1, b = -3, c = -2

Sum of the zeroes = Î± + Î²=

Product of the zeroes =Î±Î² =

If Î± and Î² are the zeroes of the polynomial x^{2 }- x - 2, find the sum and product of the zeroes.

In the polynomial x^{2 }- x - 2

a = 1, b = -1, c = -2

Sum of the zeroes = Î± + Î² =

Product of the zeroes = Î± Î² =

If Î± and Î² are the zeroes of the polynomial 5x^{2 }- px + 1 and Î± - Î² =1, then find the value of 'p'.

The given polynomial is 5x^{2 }- px + 1 and a = 5, b = -p, c = 1

Sum of the zeroes = Î±+Î² =

Product of the zeroes = Î± Î² =

Now Î±-Î²=1 (given)

(Î±-ÃŸ)^{2}=1

or (Î±+Î² )^{2 }- 4Î±ÃŸ = 1

If Î± and Î² are the zeroes of the polynomial ax^{2 }+ bx + c, find the sum and product of the zeroes.

In the given polynomial,

Sum of the zeroes = Î±+b, Product of the zeroes = Î± Î²

If Î± and Î² are the zeroes of x^{2 } - 5x + 4, then, find the sum and product of the zeroes

Since Î± and Î² are the zeroes of x^{2} - 5x + 4 = 0

Sum of the zeroes =Î± + Î² = 5 and

Product of the zeroes = Î± Î² = 4