The polynomial x³+ 3x² - 3, 2x³- 5x + a leave the same remainder in each cases. When divide by x-4 . Find the value of a

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Solution

Let f(x)=ax³+3x²-3

x-4 leaves a remainder 'p' when divided by p(x);

{x-4 = 0 means at x=4 the polynomial gives a reminder 'p'}
Hence

f(4)=p

So apply 4 to x in f(x),

64a+48-3 =p

64a+45=p ------->(1)

Let g(x)=2x³-5x+a

x-4 leaves a remainder 'q' when divided by g(x);

Hence

g(4)=q

So,

128-20+a=q

108+a=q ------>(2)

It is given that p=q

Equating equation 1 and equation 2,

64a+45=108+a

63a=63

a=1

So the value of 'a' is 1

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The polynomial x3+ 3x2 - 3, 2x3 - 5x + a leave the same remainder in each cases. When divide by x-4 . Find the value of a

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