HOW TO USE REMAINDER THEOREM TO FIND THE REMAINDER

Question 1 :

Check whether p(x) is a multiple of g(x) or not .

p(x) = x³ - 5x² + 4x - 3 ; g(x) = x – 2

Solution :

In order to check if g(x) is a multiple of p(x), let us check if g(x) is a factor of p(x).

x - 2  =  0

x  =  2

p(x) = x³ - 5x² + 4x - 3 

p(2) = 2³ - 5(2)² + 4(2) - 3 

  =  5 - 5(4) + 8 - 3

  =  5 - 20 + 8 - 3

  =  13 -23

  =  -10 ≠ 0

(x - 2) is a not a factor of p(x). Hence it not a multiple of p(x).

Question 2 :

By remainder theorem, find the remainder when, p(x) is divided by g(x) where,

(i)  p(x) = x³ - 2x² - 4x - 1 and g(x)  =  x + 1

Solution :

x + 1  =  0

x  =  -1

p(x) = x³ - 2x² - 4x - 1

p(-1) = (-1)³ - 2(-1)² - 4(-1) - 1

p(-1)  =  -1 - 2(1) + 4 - 1

=  -1 - 2 + 4 - 1

p(-1)  =  0

The remainder is 0.

(ii)  p(x)  =  4x³ - 12x² + 14x - 3 and g(x)  =  2x - 1

Solution :

2x - 1  =  0

x  =  1/2

p(x) = 4x³ - 12x² + 14x - 3

p(1/2) = 4(1/2)³ - 12(1/2)² + 14(1/2) - 3

p(1/2)  =  (4/8) - (12/4) + 7 - 3

=  (1/2) - 3 + 7 - 3

  =  (1/2) + 1

=  3/2

Hence the remainder is 3/2.

(iii)  p(x)  =  x³ - 3x² + 4x + 50 and g(x)  =  x - 3

Solution :

x - 3  =  0

x  =  3

 p(x)  =  x³ - 3x² + 4x + 50

 p(3)  =  3³ - 3(3)² + 4(3) + 50

  =  27 - 27 + 12 + 50

  =  62

Hence the remainder is 62.

Question 3 :

Find the remainder when 3x³ - 4x² + 7x - 5 is divided by (x+3).

Solution :

p(x)  =  3x³ - 4x² + 7x - 5

x + 3  =  0

x  =  -3

p(-3)  =  3(-3)³ - 4(-3)² + 7(-3) - 5

  =  3(-27) - 4(9) - 21 - 5

  =  -81 - 36 - 21 - 5

  =  -143

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