find the values of a and b ,if 2x^3 +ax^2+bx-2 has a factor (x+2) and leaves a remainder 7 when divided by 2x-3.
Answers
Step-by-step explanation:
Given :-
2x^3 +ax^2+bx-2 has a factor (x+2) and leaves a remainder 7 when divided by 2x-3.
To find :-
Find the values of a and b ?
Solution :-
Given Cubic Polynomial is 2x³+ax²+bx-2
Given factor = (x+2)
We know that
by Factor Theorem
If (x+2) is a factor of P(x) then P(-2) = 0
=> 2(-2)³+a(-2)²+b(-2)-2 = 0
=> 2(-8)+a(4)-2b-2 = 0
=> -16+4a-2b-2 = 0
=> 4a-2b-18 = 0
=> 2(2a-b-9) = 0
=> 2a-b-9 = 0/2
=> 2a-b-9 = 0
=> b = 2a-9 ------------------(1)
and
By Remainder Theorem
If P(x) is divided by x-a then the remainder is P(a)
Given that
P(x) leaves a remainder 7 when divided by 2x-3.
=> P(3/2) = 7
Since 2x-3 = 0 => x = 3/2
=> 2(3/2)³+a(3/2)²+b(3/2)-2 = 7
=> 2(27/8)+a(9/4)+(3b/2)-2 = 7
=> (27/4)+(9a/4)+(3b/2) -2 = 7
=> (27/4)+(9a/4)+(3b/2) = 7+2
=> (27/4)+(9a/4)+(3b/2) = 9
LCM of 4,4,2 = 4
=>(27+9a+6b)/4 = 9
=> 27+9a+6b = 9×4
=> 27+9a+6b = 36
=> 9a+6b = 36-27
=> 9a+6b = 9
=> 3(3a+2b) = 9
=> 3a+2b = 9/3
=> 3a +2b = 3
=> 3a +2(2a-9) = 3
=> 3a +4a-18 = 3
=> 7a -18 = 3
=> 7a = 3+18
=> 7a = 21
=> a = 21/7
=> a = 3
On Substituting the value of a in (1) then
=> b = 2(3)-9
=> b = 6-9
=> b = -3
Therefore, a = 3 and b = -3
Answer:-
The values of a and b are 3 and -3 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).
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.