When a ploynomial 2x³ +3x²+ax+b is divided by (X-2) leave remainder 2 and when divided by (x+2) leaves remainder - 2 . Find a and b.
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Answers
Step-by-step explanation:
Given :-
A ploynomial 2x³ +3x²+ax+b is divided by (x-2) leave remainder 2 and when divided by (x+2) leaves remainder - 2 .
To find :-
Find a and b ?
Solution :-
Given that :-
The Polynomial is 2x³ +3x²+ax+b
Let P(x) = 2x³ +3x²+ax+b
The divisors = (x-2) and (x+2)
Remainders = 2 and -2
We know that
By Remainder Theorem
If P(x) is divided by (x-a) then the remainder is P(a).
Now
I) If P(x) is divided by (x-2) then the remainder is P(2)
=> P(2) = 2(2)³+3(2)²+a(2)+b
=> P(2) = 2(8)+3(4)+2a+b
=> P(2) = 16+12+2a+b
=> P(2) = 28+2a+b
According to the given problem
The Polynomial is divided by (x-2) then the remainder is 2
=> 28+2a+b = 2
=> 2a+b = 2-28
=> 2a+b = -26 --------------------(1)
ii)If P(x) is divided by (x+2) then the remainder is P(-2)
=> P(-2) = 2(-2)³+3(-2)²+a(-2)+b
=> P(-2) = 2(-8)+3(4)-2a+b
=> P(-2) = -16+12-2a+b
=> P(-2) = -4-2a+b
According to the given problem
The Polynomial is divided by (x+2) then the remainder is -2
=> -4-2a+b =- 2
=> -2a+b =- 2+4
=> -2a+b = 2
=> b = 2+2a --------------------(2)
On Substituting the value of b in (1) then
=> 2a+b = -26
=> 2a+2+2a = -26
=> 4a+2 = -26
=> 4a = -26-2
=> 4a = -28
=> a = -28/4
=> a = -7
On Substituting the value of a in (2)
=> b = 2+2(-7)
=> b = 2-14
=> b = -12
Therefore, a = -7 and b = -12
Answer:-
The values of a and b are -7 and -12 respectively.
Used formulae:-
Remainder Theorem:-
Let P(x) be a polynomial of the degree greater than or equal to 1 and x-a is another linear polynomial if P(x) is divided by x-a then the remainder is P(a).