find the zeros and relationship of polynomial x^2-4x+4 also verify the relationship between zeros and coefficient.
Answers
Step-by-step explanation:
Given :-
x²-4x+4
To find :-
Find the zeros and relationship of polynomial x²-4x+4 also verify the relationship between zeros and coefficients ?
Solution :-
Finding the zeroes :-
Given quadratic polynomial is x²-4x+4
Let P(x) = x²-4x+4
=> P(x) = x²-2x-2x+4
=> P(x) = x(x-2)-2(x-2)
=> P(x) = (x-2)(x-2)
To get zeroes of P(x) then we write it as P(x) = 0
=> (x-2)(x-2) = 0
=> (x-2) = 0 or (x-2) = 0
=> x = 2 or x = 2
The Zeroes of x²-4x+4 are 2 and 2
Verifying the relationship between the zeroes and the coefficients:-
Given quadratic polynomial is x²-4x+4
On comparing with the standard quadratic polynomial ax²+bx+c
a = 1
b = -4
c = 4
and
The zeroes are 2 and 2
Let α = 2 and
Let β = 2
Relation -1:-
Sum of the zeroes = 2+2 = 4
α + β = 4--------(1)
and
-b/a
=> -(-4)/1
=> 4/1
=> 4 -------------(2)
From (1) &(2)
Therefore, α + β = -b/a
Relation -2:-
Product of the zeroes = 2×2 = 4
α β = 4---------(3)
and
c/a = 4/1
=> c/a = 4--------(4)
From (3)&(4)
Therefore, α β = c/a
Answer:-
I) The zeroes of the given Quadratic Polynomial are 2 and 2
II) Verified the given relations in the given problem
Used formulae:-
→ The standard quadratic polynomial is ax²+bx+c
→ Sum of the zeroes = -b/a
→ Product of the zeroes = c/a
→ To get the zeores of a polynomial we equate the polynomial to zero
Used Method:-
→ Splitting the middle term.
Step-by-step explanation:
Solution :
Finding the zeroes :
Given quadratic polynomial is x²-4x+4 Let P(x)=x²-4x+4
=> P(x) = x²-2x-2x+4
=> P(x) = x(x-2)-2(x-2)
=> P(x) = (x-2)(x-2)
To get zeroes of P(x) then we write it as
P(x) = 0
=> (x-2)(x-2) = 0 => (x-2) = 0 or (x-2) = 0
=> x = 2 or x = 2
The Zeroes of x²-4x+4 are 2 and 2