Question

Find the values of l and m so that l x ^4+ mx ^3+ 2x^2+ 4 is exactly divisible by

x^2 – x – 2, are respectively.

Answer 1

Given : lx⁴ + mx³ + 2x² + 4 is exactly divisible by  x² – x – 2

To find : values of l and m

Solution:

lx⁴ + mx³ + 2x² + 4 is exactly divisible by  x² – x – 2

x² – x – 2  

= x² - 2x + x - 2

= x(x - 2) + 1(x - 2)(

(x - 2)(x + 1)

Hence lx⁴ + mx³ + 2x² + 4 is exactly divisible by  (x - 2) & (x + 1)

Putting x = 2   & x = - 1

l(2)⁴ + m(2)³ + 2(2)² + 4  = 0

=> 16l + 8m + 8 + 4 =0

=> 4l + 2m = - 3     Eq1

l(-1)⁴ + m(-1)³ + 2(-1)² + 4  = 0

=>  l -  m  = - 6     Eq2

Eq1  + 2 * Eq2

=>  6l  = -3  - 12

=> l = -5/2

=> m = 7/2

l = -5/2  &  m = 7/2

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