Question

Find the values of a and b so that the polynomials (x3-10x2+ax+b)is exactly divisible by (x-1)as well as(x-3)

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Answer 1

Hello Mate!

Let the given polynomial be p( x )

p( x ) = x^3 - 10x^2 + ax + b

Now, factor theorum states that p( x ) should be 0 where x = 1 and 3

p( x ) = 1^3 - 10(1)^2 + a(1) + b
0 = 1 - 10 + a - b
9 = a - b -------(1)

p( x ) = 3^3 - 10(3)^2 + a(3) + b
0 = 27 - 90 + 3a + b
0 = - 63 + 3a + b
63 = 3a + b --------(2)

Now, lets subtract equation (2) - (1)

63 = 3a + b
- ( 9 = a + b )
=> 54 = 2a
54/ 2 = a or a = 27

Then, keeping value if a in equation (1)

9 = a + b
9 = 27 + b
9 - 27 = b or b = - 18

Hope it helps☺!✌