What must be added to 3xÂ³ + xÂ² â€“ 22x + 9 so that the result is exactly divisible by 3xÂ² + 7x - 6.Give the whole soln

Let  $\\ p(x)=3x^3+x^2 -22x +9$  And  $\\ q(x)3x^2+7x-6$

When p(x) is divided by q(x) the remainder is linear expression in x.

So , let r(x) =ax+b is added to p(x) so that

p(x) + q(x ) is divisible by q(x)

Let ,f(x) =p(x) +r(x). Then,

$\\ f(x)=3x^3+x^2-22x+9+ax+b=3x^3+x^2+x(a-22)+b+9 \\ \\\Rightarrow q(x)3x^2+7x-6=(3x-2)(x+3)$

Clearly , q(x) is divisible by (x+3) and (3x-2)

i.e $\\ f(-3);and; f(2/3)=0$

NOW, $\\ \Rightarrow 3a-b=3 .........(1)\\ \\\Rightarrow 2a+3b=13 .......(2)$

From equation  (1) and (2) we get

$\\ a=2 ;and ; b=3$

Hence , $\\ r(x) =2x+3$

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