What must be added to x3-3x2-12x+19 so that the result is exactly divisible by x2+x-6?

is divisible by  when is added to it.

Step-by-step explanation:

Let  &  

By division algorithm, when p(x) is divided by q(x), the remainder is a linear expression in x

So, let is added to so that is divisible by

Let,

⇒

⇒

We have,

Clearly, is divisible by and

{ and are factors of }

We have,

is divisible by

⇒ and are factors of

From factors theorem,

If and are factors of

then and respectively,

⇒ ⇒

⇒

⇒

Similarly,

⇒

⇒

⇒

Subtract (1) from (2)

⇒

⇒ ⇒ ⇒

Put in equation 2

⇒ × ⇒ ⇒ ⇒

∴ ⇒

Hence, is divisible by when is added to it.

Hope this helps...

step-by-step explanation: 4 cause 2+2=4

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step-by-step explanation: