What must be added to 3x³ + x² – 22x + 9 so that the result is exactly divisible by 3x² + 7x - 6.Give the whole soln

Answers (1)

ANSWER

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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