Zeros of Polynomial
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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 - 2x 2 + 2x – 4 at x = -1
f (x) = x 3 - 2x 2 + 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
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
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.
If α,β and γ are the zeroes of a cubic polynomial then
x 3 - (α + β+γ)x 2 + (αβ + βγ+αγ)x - αβγ
α + β+γ = 2
αβ + βγ+αγ = -7
αβγ = -14
x 3 -(2 )x 2 + (-7)x – (-14)
x 3 -2 x 2 -7x + 14
• 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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