Question

find the values of a and b for which ax^2-11x+ax+b is exactly divisible by x^2-4x-5

Answer 1

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p( x ) = ax² - 11x + ax + b

Now, factor x² - 4x - 5
x² - 5x + x - 5 = 0
x ( x - 5 ) + 1 ( x - 5 ) = 0
( x - 5 )( x + 1 ) = 0

So, x = 5 or - 1

Now,

p( -1 ) = a( - 1 )² - 11 ( - 1 ) + a( - 1 ) + b
0 = a + 11 - a + b
- 11 = b --------(1)

p( 5 ) = a( 5 )² - 11( 5 ) + a( 5 ) + b
0 = 25a - 55 + 5a + b
55 = 30a + b --------(2)

Now, (2) - (1)

55 = 30a + b
- ( - 11 = b )
=> 66 = 30a
a = 2.2

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