Question

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)?

Answer 1

We know that `f(1) = 2` and `f(-2)=-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(x) = Q(x-1)(x+2) + Ax + B`

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

`f(1) = Q(1-1)(1+2) + A(1) + B=A+B=2`

`f(-2) = Q(-2-1)(-2+2)+A(-2)+B=-2A+B=-19`

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

Remainder `= Ax + B = 7x-5`