A Bit of Algebra

 

Remainder and Factor Theorem

 The remainder theorem simply states that if a polynomial p(x) is divided by (x-a) then the remainder is p(a). From this it is obvious that if (x-a) is a factor or p(x) then the p(a), the remainder, will be 0. This is the Factor theorem.

E.g. Prove that is a factor of f(x) where f(x) =

If it is a factor, then i.e.

Long Division of Polynomials

 It's a good idea to practice a few of these. Lay columns out clearly to help avoid mistakes.

 E.g.

 

Inequalities

 Inequalities differ from equations in these important ways:

• The inequality remains true if a quantity is added to or subtracted from both sides
  
• The inequality reverses if both sides are multiplied or divided by a negative quantity
  

   This last point should be remembered when multiplying or dividing by unknown variables. Two separate answers will be produced depending whether the variable is +ve or - ve.

   E.g. If you wanted to remove the denominator in the inequality :

  

  you would have 2 different scenarios, one where (4x+3) is +ve and another where (4x+3) is - ve:
  

   When But when

   

   

  Solving Quadratic and Cubic Inequalities

   These types of questions will often involve all the things mentioned in this section as well as testing your knowledge of graphs.

  1. Prove that is a factor of
  2. Hence, solve the inequality

1. Use the factor theorem and show that when x=2,
2. Use long division to find the remaining quadratic

This will factorise to
So the roots of the equation are at

 

 With reference to the graph it is clear that the answer is

[note or not and]