Page 4 - Lesson Notes-Relationhip between Zeroes and coefficients Ch-2 (Polynomals)

In general, it can be proved that if α, β, γ are the zeroes of the cubic polynomial
2
3
ax + bx + cx + d, a ≠ 0 then
− b − coefficien tofx 2
 +  +  = = =
a coefficien tofx 3
c − coefficien tofx
 +  +  = = =
a coefficien tofx 3
− d
 = =
a
Formation of Quadratic and Cubic Polynomials:

(i) If α and β are the zeroes of a quadratic polynomial then quadratic polynomial
will be
2
x – (sum of zeroes)x + product of zeroes
2
i.e. x – (α + β ) x + α β

(ii) If α , β and γ are the zeroes of cubic polynomial then the cubic polynomial
will be
3
2
x – (sum of zeroes)x + sum of the product of zeroes taking two at a time)x –
product of zeroes
2
3
i.e.x – (α + β + γ)x +( αβ + βγ + γα)x - α β γ

Example : Find a quadratic polynomial, the sum and product of whose zeroes are

– 3 and 2, respectively.

Solution : Let the quadratic polynomial be ax + bx + c, and its zeroes be α and β.
2
We have

− b c
α + β = -3 = , αβ = 2 =
a a
If a = 1, then b = 3 and c = 2.

2
So, one quadratic polynomial which fits the given conditions is x + 3x + 2.

Example : Find the cubic polynomial whose three zeroes are 3, 1, –1/3

Solution: Let α , β , γ are the zeroes of cubic polynomial be
α = 3, β = 1, and γ = –1/3
Then α + β + γ)= 3+(–1)+( –1/3) = 5/3
αβ + βγ + γα =(3)( –1)+( –1)( –1/3)+( –1/3)(3)= –11/3
α β γ =3(–1)( –1/3) = 1
3
2
Now the required cubic polynomial= x – (α + β + γ)x +( αβ + βγ + γα)x – α β γ
2
3
3
2
= x – (5/3)x +(–11/3) x –1 =(3x – 5 x –11x –3) / 3
2
3
Also, the required cubic polynomial 3x – 5 x –11x –3.

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