Factor and Remainder Theorems

Start with the theorems

The Polynomials mind map brings the factor and remainder theorems together. It states that if (x-a) is a factor of f(x), then f(a) is zero. It also states that f(a) is the remainder when f(x) is divided by (x-a). Start here to see how substituting a value into a polynomial can help with both kinds of question.

The Polynomials flashcard deck asks what a polynomial is, how the theorems work and how polynomial division is carried out. Use it after the mind map to recall the ideas without looking at the statements.

Follow worked examples

The note Factor Theorem answer works through checking whether x+2 is a factor. It substitutes -2 into the polynomial and shows that the result is zero. Follow the calculation, then return to the mind map and connect the result to the factor theorem statement.

The note Remainder Theorem answer uses f(4) and a given remainder to find a value for a. Read it alongside the remainder theorem statement in the mind map to see how substitution is used in a different question.

Practice with questions and cards

The question slides include checking a factor, factorizing a polynomial and finding a remainder. After studying the notes, work through a question and refer back to the mind map when you need to check which theorem applies. There are no quizzes on this page, but the slides offer questions to work through and the flashcards offer prompts for recall.

These resources form the Factor and Remainder Theorems section of Polynomials and Factoring in Mathematics. Together, they give school students and teachers a way to move between theorem statements, worked calculations and practice questions. Choose a card or question to work on next.