Factor Theorem 6 = 2 x 3, we say 2 and 3 are factors of 6, i.e, remainder = 0 Likewise, we say (x+2) and (x – 2) are factors of So, when (x+2) is a factor of, it means the remainder is zero

Application of factor theorem: Given that has a factor of (x+2), find the values of a. If (x+2) is a factor, that means f(-2) = 0 Just solve the above equation to get values of a.

Using remainder and factor theorem together: The expression is exactly divisible by x + 3 but it leaves a remainder of 91 when divided by x – 4. Find a and b and hence find the remainder when it is divided by x + 2. Firstly, exactly divisible means x + 3 is a factor  f(-3) = 0 Next, remainder when divided by x – 4  f(4) = 91 You need to find a and b  solve the above 2 equations simultaneously !! Finally, when you have a and b, you have the expression. To find the remainder when divided by x + 2, you need to find f(-2)

Deg of Q(x)? What is the deg or highest power of Q(x)?? Answer : 2 Why? Well, LHS you have a x 3, therefore, RHS also must have an x 3 term. Since you already have (x + 1), therefore, Q(x) should be ax 2 + bx + c so deg = 2 Q(x) is deg 3 Q(x) is deg 2

By now, you should know that a quadratic expression has at most 2 factors What if it’s a cubic expression?? How many factors will it have??? Answer: At most 3 Hence, when you solve a quadratic equation, you expect to get 2 answers if you can factorise it So, when you solve a cubic equation, you expect to get 3 answers if you can factorise it

Finding factors How to find a factor of a cubic expression?? Use trial and error!! Sub values of x until you get no remainder Once you find 1 factor, you can easily find the rest! Q(x) is deg 2 because f(x) is deg 3 To find a, b and c, you can expand and compare coefficients or just do by observation. Once you find a, b and c, you can either factorise Q(x) and hence solve the equation or solve using formula (will learn later)