When x³-2x²+ax=b is divided by x²-2x-3, the remainder is x-6. The value of a and b respectively
A. -2, -6
B. 2 and -6
C. -2 and 6
D. 2 and 6
Answers
Answer:
answer is a. -2, -6
I hope this helps
Given: When x³ - 2x² + ax = b is divided by x² - 2x - 3, the remainder is x - 6.
To find : The value of a and b
Solution :
Let p(x) = x³ - 2x² + ax - b
q (x) = x² - 2x - 3
r (x) = x - 6
Therefore,
f(x) = p (x) – r (x)
f(x) = x³ - 2x² + ax - b - (x - 6)
f(x) = x³ - 2x² + ax - b - x + 6
f(x) = x³ - 2x² + ax - x - b + 6
f(x) = x³ - 2x² + (a - 1) x - (b - 6)
q(x) = x² - 2x - 3
By middle term splitting :
= x² - 3x + x - 3
= x(x - 3) + 1(x - 3)
= (x + 1) (x - 3)
Thus, (x + 1) and (x - 3) are factor of f (x)
On putting f(-1) in f(x) :
f(-1) = 0
f(x) = x³ - 2x² + (a - 1) x - (b - 6)
f(x) = (-1)³ - 2(-1)² + (a - 1) (-1) - (b - 6)
= - 1 - 2 × 1 - a + 1 - b + 6 = 0
= - 1 - 2 - a + 1 - b + 6 = 0
= - 3 + 1 + 6 - a - b = 0
= 4 - a - b = 0
a + b = 4
a = 4 - b ……………(1)
On putting f (3) in f(x) :
f (3) = 0
f(x) = x³ - 2x² + (a - 1) x - (b - 6)
= 3³ - 2 (3)² + (a -1) 3 - b + 6 = 0
= 27 - 2 × 9 + 3a - 3 - b + 6 = 0
= 27 - 18 - 3 + 6 + 3a - b = 0
= 27 - 18 + 3 + 3a - b = 0
= 27 - 15 + 3a - b = 0
= 12 + 3a - b = 0
= 3a - b = - 12 ………….(2)
put the value of a from eq 1 in eq 2 :
= 3(4 - b) - b = - 12
= 12 - 3b - b = - 12
= - 4b = - 12 - 12
= - 4b = - 24
= b = 24/4
b = 6
Put this value of b in eq 1 :
a = 4 - b
a = 4 - 6
a = - 2
Hence the value of a is - 2 and b is 6 .
Among the given options option (C) - 2 and 6 is correct.
HOPE THIS ANSWER WILL HELP YOU…..
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