The remainder and factor theorems and cubic equations
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The teaching part (1 min) is free to watch. A free account opens the whole lecture: 5 worked examples, 4 practice questions, 1 real past-paper question with the answer explained, the mistakes that lose marks, the full-marks tip, a one-card summary and a quick quiz.
What this lecture covers
- A shortcut for long division
- The remainder theorem
- The factor theorem
Read the transcript of the teaching part
Last time
Before we start: last time, in Modulus equations and inequalities, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.
A shortcut for long division
Long division of a cubic is slow and easy to get wrong. But if you only want the remainder, or only want to know whether a bracket is a factor, you can skip the division. Just substitute one number.
The remainder theorem
When a polynomial p of x is divided by x minus a, the remainder is p of a. For a bracket like x plus two, a is minus two. Read the sign carefully. For a bracket like two x minus one, set it to zero. Then x is one half, and the remainder is p of one half.
The factor theorem
If p of a equals zero, there is no remainder, so x minus a is a factor. And the other way round. If x minus a is a factor, then p of a is zero. This gives you equations for unknown constants.
How often it comes up
How often Cambridge has asked each part, counted by ExamYard from every past paper in its question bank: Factor and remainder theorems.
The whole chapter: Algebra.
Chapter 3.1: Algebra
- Modulus equations and inequalities 5 min
- The remainder and factor theorems and cubic equations 6 min
- Rational-index binomial expansions, partial fractions and polynomial division 7 min
Also for A Level Maths 9709 Paper 3
Which topics come up most in A Level Maths 9709 Paper 3 past papers, counted from every paper since 2004.