Grade 9 - Remainder Factor Theorem Question

Question

The polynomial P(x) has remainders 10 and 1 when it is divided by x+2 and x-1 respectively. Find the remainder, in the form of ax+b when P(x) is divided by x^2 + x - 2.

Answer 1

If we take a quotient, Q, multiply it by the divisor, D, then add the remainder, R, we get the dividend, P(x).

 

P(x) = QD + R

 

 

Using this formula,

 

P(x) = Q(x + 2) + 10                       eq1

P(x) = Q(x - 1) + 1                         eq2

 

Solve the system to find P(x).

 

Qx + 2Q + 10 = Qx - Q + 1

 

2Q + 10 = -Q + 1

 

3Q = - 9

 

  Q = -3

 

 

Plug this value of Q into any of the equations,

 

P(x) = -3(x + 2) + 10

P(x) = -3x + 4

 

Now we can take P(x) and divide by (x² + x - 2) using long division.  But keep in mind.  You are dividing a polynomial of less degree than the degree of (x² + x - 2).