Zeros of Polynomial

Zeros of Polynomial : It is a solution to the polynomial equation, P(x) = 0. It is that value of x that makes the polynomial equal to 0.

The value of a polynomial f(x) at x = a is obtained by substituting x = a in the given polynomial and is denoted by f(a) which is Zeros of Polynomial.

Examples Zeros of Polynomial :

Find the value of f(x) = x 3 - 2x2 + 2x – 4 at x = -1

Solution:
f (x) = x 3 - 2x2 + 2x – 4

f(-1) = (-1) 3 - 2(-1)2 + 2(-1) – 4

= -1 -2 – 2 – 4
= - 9

Note : If f(a) = 0 then x = a is the root of the polynomial f(x).

Relationship between the zeroes and coefficient of a Polynomial


Case 1 : Quadratic Polynomial

ax2 +bx + c

If α and β are the roots of the equation then

Sum of zeroes = α + β =
Product of zeroes = αβ =

From the above the equation of polynomial is given by
x 2 -( α + β) x + αβ

Examples :

1) Write the equation of the polynomial, if sum of zeroes = -8 and product of zeroes is 15

Solution : As Sum of zeroes = α + β = - 8
Product of zeroes = αβ = 15

Equation is x 2 -( α + β) x + αβ

x 2 - ( - 8) x + 15

x 2 + 8x + 15

Case 2 : Cubic Polynomial
ax3 + bx2 + cx + d , a ≠0
Sum of zeroes = α + β+γ = - b/a
Sum of the product of zeroes taken two at a time = αβ + βγ+αγ = c/a
Product of zeroes = αβγ = -d/a

The cubic polynomial can be written as
x3 - (α + β+γ)x2 + (αβ + βγ+αγ)x - αβγ

Example :

1) Find the cubic polynomial with the sum, sum of the product of zeroes taken two at a time, and product of its zeroes as 2,-7 ,-14 respectively.

Solution :
If α,β and γ are the zeroes of a cubic polynomial then

x3 - (α + β+γ)x2 + (αβ + βγ+αγ)x - αβγ

α + β+γ = 2

αβ + βγ+αγ = -7

αβγ = -14

x3 -(2 )x2 + (-7)x – (-14)

x3 -2 x2 -7x + 14
Polynomial

• Degree of the Polynomial
• Zeros of Polynomial
• Remainder Theorem
• Find remainder by Synthetic Division
• Rational root test in Polynomial
• Solved Examples on Polynomial

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