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Answer 1

Answer:

2x + 5

Given:

f(x) = x³ −3x ² − 12x + 19

g(x) = x² + x − 6

By division algorithm

$\small \tt \: f(x) = g(x) \times q(x) + r(x)$

$\small \tt \: q(x) = \frac{f(x) +[−r(x)]</p><p> }{g(x)}$

If means r(x) must be added so that result is exactly divisible by x² + x − 6.

So, 2x+5 must be added to f(x), so that the result is exactly divisible by x² + x − 6.

Answer 2

step 1. divide the polynomials

step 2. after division add the remainder into x³+x²-6x.

step 3. this is because after division we got to know that how much is to be added so that the polynomial should be exactly divisible by the other one.

4. this is done by the division algorithm which states that we get the dividend when we multiply the divisor and the quotient and add the remainder in it.

f(x) = g(x)×q(x)+r(x)

Explanation:

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