# When a polynomial is divided by (x+2), the remainder is -19. When the same polynomial is divided by (x-1), the remainder is 2, how do you determine the remainder when the polynomial is divided by (x+2)(x-1)?

##### 1 Answer
Jan 10, 2016

We know that $f \left(1\right) = 2$ and $f \left(- 2\right) = - 19$ from the Remainder Theorem

#### Explanation:

Now find the remainder of polynomial f(x) when divided by (x-1)(x+2)

The remainder will be of the form Ax + B, because it is the remainder after division by a quadratic.

We can now multiply the divisor times the quotient Q ...

$f \left(x\right) = Q \left(x - 1\right) \left(x + 2\right) + A x + B$

Next, insert 1 and -2 for x ...

$f \left(1\right) = Q \left(1 - 1\right) \left(1 + 2\right) + A \left(1\right) + B = A + B = 2$

$f \left(- 2\right) = Q \left(- 2 - 1\right) \left(- 2 + 2\right) + A \left(- 2\right) + B = - 2 A + B = - 19$

Solving these two equations, we get A = 7 and B = -5

Remainder $= A x + B = 7 x - 5$