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Answers
Step-by-step explanation:
Given :-
(x+2a) is a factor of x⁴-2ax³+2x+2a+3
To find :-
Find the value of a ?
Solution:-
Given bi-quadratic polynomial is x⁴-2ax³+2x+2a+3
Let P(x) = x⁴-2ax³+2x+2a+3
Given factor = (x+2a)
We know that
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.
Given that
(x+2a) is a factor of P(x)
=> P(-2a) = 0
Since x+2a = 0 => x = -2a
Put x = -2a then P(-2)
=> (-2a)⁴-2a(-2a)³+2(-2a)+2a+3 = 0
=> 16a⁴-2a(-8a³)-4a+2a+3 = 0
=> 16a⁴+16a⁴-4a+2a+3 = 0
=> (16+16)a⁴+(-4+2)a +3 = 0
=> 32a⁴-2a+3 = 0
There is no real value for a for the given Polynomial.
Correction:-
If the given polynomial is x⁴+2ax³+2x+2a+3 then we get real value of a
Now P(-2a) = 0
=> (-2a)⁴+2a(-2a)³+2(-2a)+2a+3 = 0
=> 16a⁴-16a⁴-4a+2a+3 = 0
=> 0-2a+3 = 0
=> -2a +3 = 0
=> -2a = -3
=> 2a = 3
=> a = 3/2
The value of a = 3/2
Answer :-
The value of a for the given problem is 3/2
Check:-
if a = 3/2 then x+2a = x+2(3/2) = x+3
If x+3 is a factor of P(x) then P(-3) = 0
=> (-3)⁴+2(3/2)(-3)³+2(-3)+2(3/2)+3
=> 81-81-6+3+3
=> 81-81-6+6
=> 0+0
=> 0
So, x-3 is a factor of P(x) when a = 3/2
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.