Remainder & Factor Theorem

 

 

 

Contents

 

Introduction

 

Remainder Theorem

 

Factor Theorem

 

Cubic Equations

 

Quiz 

 

 

Algebra Main Page

 

 

Remainder Theorem

The remainder theorem states that if a polynomial f(x) is divided by a linear function x - k, the remainder is f(k).

Proof:

In any division,

dividend = divisor X quotient + remainder

Let Q(x) be the quotient and R be remainder.

f(x) = (x - k) Q(x) + R

f(k) = (k - k) Q(x) + R

      = 0 + R

      = R

Note:  The degree of the remainder is always one less than the degree of the divisor.

Example 1

Find the remainder when x3 + 4x2 - 7x + 6 is divided by x - 1.

Let f(x) = x3 + 4x2 - 7x + 6
      f(1) = 13 + 4 (1)2 - 7 + 6
           = 4

 
Example 2
 
Given that the expression 2x3 + 3px2 - 4x + p has a remainder of 5 when divided by x + 2, find the value of p.
 
Let f(x) = 2x3 + 3px2 - 4x + p
      f (-2) = 2(-2)3 + 3(-2)2p - 4(-2) + p = 5
                                                13p - 8 = 5
                                                     13p = 13
                                                         p = 1

Example 3
 
If the expression ax4 + bx3 - x2 + 2x + 3 has remainder 4x + 3 when divided by x2 + x - 2, find the value of a and b.
 
Let f(x) = ax4 + bx3 - x2 + 2x + 3
 x2 + x - 2 = (x + 2)(x - 1)
 
f(-2) = a(-2)4 + b(-2)3 - (-2)2 + 2(-2) + 3 = 4(-2) + 3
                                16a - 8b - 4 - 4 + 3 = -5
                                                    2a - b = 0 --------(1)
 
f(1) = a + b - 1 + 2 + 3 = 4(1) + 3
                           a + b = 3  --------(2)
 
(1) + (2) :    3a = 3
                    a = 1
 
when a = 1, b = 2.