If one root of the quadratic polynomial 3x
2+ px +4 is 2/3, then find the value of p and
the other root of the polynomial.
Answers
Step-by-step explanation:
Given:-
One root of the quadratic polynomial 3x^2+ px +4 is 2/3.
To find:-
Find the value of p and the other root of the polynomial?
Solution:-
Given quardratic polynomial P(x) = 3x^2+px+4
Given root = 2/3
If 2/3 is the root of the given Polynomial P(x) then it satisfies the given Polynomial i.e. P(2/3)=0.
=> P(2/3) = 0
=> 3(2/3)^2+p(2/3)+4 = 0
=> 3(4/9)+(2p/3)+4 = 0
=> (4/3)+(2p/3)+4 = 0
=> (4+2p+12)/3 = 0
=> (2p+16)/3 = 0
=> 2p+16 = 0×3
=>2p +16 = 0
=> 2p = -16
=> p = -16/2
=> p = -8
Therefore , p = -8
If p= -8 then the Polynomial becomes 3x^2-8x+4
=> 3x^2-6x-2x+4
=> 3x(x-2)-2(x-2)
=> (x-2)(3x-2)
To get the zeores we write it as P(x)=0
=> (x-2)(3x-2) = 0
=> x-2 = 0 or 3x-2 = 0
=> x = 2 or x = 2/3
The other root = 2
Answer:-
The value of p for the given problem is -8
The other root of the resulting Polynomial is 2
Used formulae:-
Factor Theorem:-
Let P(x) be a polynomial of the degree greater than or equal to 1 and x-a is another linear polynomial if x-a is a factor of P (x) then P(a) = 0 vice-versa.