Question

If (x + 3)2 is a factor of f(x) = ex3 + kx + 6, then
find the remainder obtained when f(x) is divided
by x - 6.
(a) 1
(b) 0
(c) 5
(d) 4


Answer only if you know. (≡^∇^≡)​

Answer 1

SOLUTION

TO CHOOSE THE CORRECT OPTION

(x +3)² is a factor of f(x) = ex³ + kx + 6, then

find the remainder obtained when f(x) is divided

by x - 6.

(a) 1

(b) 0

(c) 5

(d) 4

TO DETERMINE

Here the given polynomial is

f(x) = ex³ + kx + 6

Now (x + 3)² is a factor of f(x)

So f(-3) = 0

⇒ - 27e - 3k + 6 = 0

⇒ k = 2 - 9e - - - - - (1)

Again - 3 is a root of multiplicity 2

So - 3 is also a root of f'(x)

f'(x) = 3ex² + k

Now f'(-3) = 0 gives

27e + k = 0

⇒ k = - 27e - - - - - - (2)

From Equation 1 and Equation 2 we get

$\displaystyle\sf{k = 3 \: \: and \: \: e = - \frac{1}{9} }$

Therefore

$\displaystyle\sf{f(x) = - \frac{1}{9} {x}^{3} + 3x + 6 }$

Hence by the Remainder Theorem the required Remainder when f(x) is divided by x - 6 is

$\displaystyle\sf{f(6) = - \frac{1}{9} \times {6}^{3} + (3 \times 6) + 6 }$

$\displaystyle\sf{ = - 24 + 18 + 6 }$

$= 0$

FINAL ANSWER

Hence the correct option is (b) 0

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