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 thought on “If (x + 3)2 is a factor of f(x) = ex3 + kx + 6, then<br />find the remainder obtained when f(x) is divided<br />by x – 6.<br />(a)”

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

   [tex]\displaystyle\sf{k = 3 \: \: and \: \: e = – \frac{1}{9} }[/tex]

   Therefore

   [tex]\displaystyle\sf{f(x) = – \frac{1}{9} {x}^{3} + 3x + 6 }[/tex]

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

   [tex]\displaystyle\sf{f(6) = – \frac{1}{9} \times {6}^{3} + (3 \times 6) + 6 }[/tex]

   [tex]\displaystyle\sf{ = – 24 + 18 + 6 }[/tex]

   [tex] = 0[/tex]

   FINAL ANSWER

   Hence the correct option is (b) 0

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