When ax^2 +bx-6 is divided by x +3, the remainder is 9, Find, in terms of a only, the remainder when 2x^3 -bx^2 + 2ax-4 is divided by x-2
Answers
Step-by-step explanation:
Given :-
When ax² +bx-6 is divided by x +3, the remainder is 9.
To find :-
Find, in terms of a only, the remainder when
2x³ -bx² + 2ax-4 is divided by x-2 ?
Solution :-
Given polynomial is ax²+bx-6
Given divisor = x+3
Given remainder = 9
We know that
By Remainder Theorem,
If P(x) is divided by x+3 then the remainder = P(-3)
We have,
P(-3) = 9
=> a(-3)²+b(-3)-6 = 9
=> a(9)-3b-6 = 9
=> 9a-3b-6 = 9
=> 9a-3b = 9+6
=> 9a-3b = 15
=> 3(3a-b) = 15
=> 3a-b = 15/3
=>3a-b = 5
=> b = 3a-5 --------------------(1)
and
Given polynomial is 2x³-bx²+2ax-4
Let g(x) = 2x³-bx²+2ax-4
Given divisor = x-2
By Remainder Theorem,
If g(x) is divided by x-2 then the remainder = g(2)
=> 2(2)³-b(2)²+2a(2)-4
=> 2(8)-b(4)+4a-4
=> 16-4b+4a-4
=> 12-4b+4a
On Substituting the value of b from (1)
=> 12-4(3a-5)+4a
=> 12-12a+20+4a
=> (12+20)+(-12a+4a)
=> 32-8a
The remainder = 32-8a
Answer:-
The remainder in terms of a for the given problem is 32-8a
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).
Answer:
a is 3/4 and b is -8 thanks