The polynomial 4x^3+ ax^2+ 9x + 9, where a is a constant, is denoted by p(x). It is given that when p(x) is divided by (2x −1) the remainder is 10.  Find the value of a and hence verify that (x −3) is a factor of p(x).

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`p(x) = 4x^3+ax^2+9x+9`

It is given that when p(x) is divided by (2x-1) then the remainder is 10

Using remainder theorem;

`p(1/2)=10`

`4(1/2)^3+a(1/2)^2+9(1/2)+9=10`

`1/2+a/4+9/2+9=10`

`a/4+14=10`

`a=-16`

So the value of  a=-16

The remainder when p(x) is divided by (x-3) is given by p(3).

`p(3) = 4(3)^3+a(3)^2+9 xx 3 + 19`

`=27 xx 4 – 16 xx 9 + 27 + 9`

`=144-144`

`=0`

Since the remainder is 0, (x-3) is a factor of p(x)

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