The Remainder Theorem

Last week, we started learning Chapter 2 about The Remainder Theorem. Actually, I kinda like this subchapter because I've learnt it in my high school. This subchapter is not that difficult. You just need to pay more attention while dividing and multiplying.  In my high school, besides using long division, we use Horner's method. What is Horner's method? We'll check it out. It's easier than using Long Division. But, I'm not sure whether you can use this method on test. Well, at least this method can help you with mutliple choice questions or just check your answer.

Divide x⁵ – 4x³ + 3 by x² – 2x – 3

1) Use Long Division 

2) Use Horner's Method

Quotient = x³ + 2x² + 3x + 12

Remainder      =    (x – b)R₂  + R₁

                         =    (x – 3)(33) + (138)

                         =    33x – 99 + 138

                         =    33x – 39

This is an example in the textbook Page 86. (Advanced Functions 12 McGraw-Hill Ryerson)
We'll try to solve the question by using Horner's Method.

The volume, V, in cubic centimetres, of a rectangular box is given by V(x) = x³ + 7x² + 14x + 8.
Determine expressions for possible dimensions of the box if the height, h, in centimetres, is given by x + 2.

Solution

Horner's Method

Q(x) = lw  = x² + 5x +4

                   = (x + 1)(x + 4)

P(x) = V = x³ + 7x² + 14x + 8

              = (x + 2)(x² + 5x +4)

Expressions for the possible dimensions of the box, in centimetres, are (x + 1), (x + 2), and (x + 4).