find the values of a and b ,if 2x^2+ax^2-11x+b leaves a remainder 0 and 42 when divided by (x-2) and (x-3).
Answers
Let us take x-2 = 0
Then, x = 2
Given, f(x) = 2x³ + ax²-11x + b
Now, substitute the value of x in f(x),
f(2)= 2(2)³ + a(2)²-11(2) + b
= 16 + 4a-22 + b
= -6 + 4a + b
Given, remainder is 0. So, 6 + 4a+b=0
4a + b = 6 ... [equation (i)]
Now, consider (x-3)
Assume x-3=0
Then, x = 3
Given, f(x) = 2x³ + ax²-11x + b
Now, substitute the value of x in f(x), f(2) = 2(3)³ + a(3)²-11(3) + b
= 54 +9a-33 + b
= 21 +9a + b
Given, remainder is 42.
So, 21 + 9a + b = 42
9a + b = 42-21
9a + b = 21 ... [equation (ii)]
Now, subtracting equation (i) from equation (ii) we get,
(9a + b)-(4a + b) = 21-6
9a + b-4a-b = 15
5a = 15
a = 15/5
a = 3
Consider the equation (i) to find out 'b'.
4a + b = 6
4(3) + b = 6
12 + b = 6
b = 6-12
b = -6
Then, by substituting the value of a and
bf(x) = 2x³ + 3x²-11x-6
Given that remainder is 0 for, (x-2) is a factor of f(x).
So, dividing f(x) by (x-2)
Therefore, 2x³ + 3x²-11x-6 =(x-2)(2x² +7x+3)
= (x−2)(2x² + 6x +x+3)
= (x−2)(2x² + 6x+x+3)
= (x−2)(2x(x + 3) +1(x+3))
= (x-2)(x+3)(2x + 1)
Step-by-step explanation:
Correction :-
2x³+ax²-11x+b
Given :-
2x³+ax²-11x+b leaves a remainder 0 and 42 when divided by (x-2) and (x-3).
To find:-
Find the values of a and b ?
Solution :-
Given Cubic Polynomial is 2x³+ax²-11x+b
Let P(x) = 2x³+ax²-11x+b
Given divisors =(x-2) and (x-3).
Given remainders = 0 and 42
By Remainder Theorem,
If P(x) is divided by (x-2) then the remainder is P(2)
According to the given problem
P(2) = 0
=> 2(2)³+a(2)²-11(2)+b = 0
=> 2(8)+a(4)-22+b = 0
=> 16+4a-22+b = 0
=>4a+b -6 = 0
=> 4a+b = 6
=> b = 6-4a ---------------(1)
And
If P(x) is divided by (x-3) then the remainder is P(3)
According to the given problem
P(3) = 42
=> 2(3)³+a(3)²-11(3)+b = 42
=> 2(27)+a(9)-33+b = 42
=> 54+9a-33+b = 42
=> 9a+b+21 = 42
=> 9a + b = 42-21
=>9a +b = 21
=> b = 21-9a -------------(2)
From (1)&(2)
=> 21-9a = 6-4a
=> 21-6 = -4a+9a
=> 15 = 5a
=> a = 15/5
=>a = 3
On Substituting the value of a in (1) then
b = 6-4(3)
=> b = 6-12
=> b = -6
Therefore, a = 3 and b = -6
Answer:-
The values of a and b are 3 and -6 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).