Determine the relation between c and d if x-2 and x-1/2 be two factors of the polynomial c*x^2+5x+d. *
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Answers
Step-by-step explanation:
Given :-
x-2 and x-1/2 be two factors of the polynomial cx²+5x+d.
To find:-
Determine the relation between c and ?
Solution :-
Given quadratic polynomial =p(x) = cx²+5x+d.
Given factors of p(x) = x-2 and x-1/2
If x-2 is a factor of p(x) then it satisfies the given equation.
=> p(2) = 0
=> c(2)²+5(2)+d = 0
=> 4c+10+d = 0----------(1)
If x-1/2 is a factor of p(x) then it satisfies the given equation.
=> p(1/2) = 0
=> c(1/2)²+5(1/2)+d = 0
=> c(1/4)+(5/2)+d = 0
=> (c+10+4d)/4 = 0
=> c+4d+10 = 0--------(2)
(x-2) and (x-1/2) are the factors of p(x) then
p(2) = p(1/2)
=> 4c+10+d = c+4d+10
=> 4c+d+10-c-4d-10=0
=> (4c-c)+(d-4d)+(10-10) = 0
=> 3c+(-3d)+0 = 0
=> 3c-3d = 0
=> 3(c-d) = 0
=> c-d = 0/3
=> c-d = 0
=> c=d
Answer:-
The relationship between c and d is c = d
Used formulae:-
If 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. This is called Factor Theorem.