Find the zeros of Cubic Polynomial and verify that the relationship between the zeros and the coefficients.
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Answers
= 3x³-5x²-11x-3
3x³-5x²-11x-3= 2x²-11x-3
3x³-5x²-11x-3= 2x²-11x-3=9x-3
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Step-by-step explanation:
Given :-
3x³-5x²-11x-3
To find :-
Find the zeros of Cubic Polynomial and verify that the relationship between the zeros and the coefficients?
Solution :-
Finding the zeroes :-
Given Cubic Polynomial is 3x³-5x²-11x-3
Let P(x) = 3x³-5x²-11x-3
It can be written as
=> P(x) = 3x³-8x²+3x²-8x-3x-3
=> P(x) = 3x³+3x²-8x²-8x-3x-3
=> P(x) = 3x²(x+1)-8x(x+1)-3(x+1)
=> P(x) = (x+1)(3x²-8x-3)
=> P(x) = (x+1)(3x²-9x+x-3)
=> P(x) = (x+1)[(3x(x-3)+1(x-3)]
=> P(x) = (x+1)(x-3)(3x+1)
To get zeroes of P(x) , we write it as
P(x) = 0
=> P(x) = (x+1)(x-3)(3x+1) = 0
=>x+1 = 0 or x-3 = 0 or 3x+1 = 0
=> x = -1 or x = 3 or 3x = -1
=> x = -1 or x = 3 or x = -1/3
The zeroes are -1 , -1/3 , 3
Verifying the relationship between the zeroes and the coefficients of P(x):-
P(x) = 3x³-5x²-11x-3
On comparing this with the standard Cubic Polynomial ax³+bx²+cx+d then
a = 3
b = -5
c = -11
d = -3
The zeroes of P(x) = -1 , -1/3 , 3
Let α = -1 , β = -1/3 and γ = 3
Relation-1:-
Sum of the zeroes = α+β+γ
=> (-1)+(-1/3)+(3)
=> (-3-1+9)/3
=> (9-4)/3
=> 5/3
We know that
Sum of the zeroes = α+β+γ
=> -(Coefficient of x²)/Coefficient of x³
=> -b/a
=> -(-5)/3
=> 5/3
Therefore, Sum of the zeroes = -b/a
Relation -2:-
Sum of the product of the two zeroes taken at a time = αβ+βγ+αγ
=> (-1)(-1/3) + (-1/3)(3) + (3)(-1)
=> (1/3) + (-1) + (-3)
=> (1-3-9)/3
=> (1-12)/3
=> -11/3
We know that
αβ+βγ+αγ = Coefficient of x/ Coefficient of x³
=> c/a
=> -11/3
Therefore, Sum of the product of the two zeroes taken at a time = c/a
Relation-3:-
Product of the zeroes = αβγ
=> (-1)(-1/3)(3)
=> (3/3)
=> 1
We know that
Product of the zeroes =αβγ
=> - Coefficient of x/ Coefficient of x³
=> -d/a
=> -(-3)/3
=> 3/3
=> 1
Therefore, Product of the zeroes= -d/a
Verified the relationship between the zeroes and the coefficients of the given cubic polynomial.
Answer:-
The zeroes of the given polynomial are -1 , -1/3 and 3
Used formulae:-
- The standard Cubic Polynomial is ax³+bx²+cx+d
- Sum of the zeroes =α+β+γ =
-(Coefficient of x²)/Coefficient of x³
= -b/a
- Sum of the product of the two zeroes taken at a time = αβ+βγ+αγ = Coefficient of x/ Coefficient of x³
= c/a
- Product of the zeroes = αβγ =
- Coefficient of x/ Coefficient of x³
= -d/a