Remainder And Factor Theorems
Overview
Dividing one polynomial by another can be slow. The remainder and factor theorems give you a shortcut so powerful it feels like cheating: you can find the remainder of a division, or test whether something is a factor, just by substituting a single number. No long division required.
In this lesson you will meet the remainder theorem, which says the remainder when a polynomial is divided by \((x-a)\) is simply \(f(a)\), and the factor theorem, its special case when that remainder is zero. Together they are the key to factorising and solving cubic equations, a staple of the higher-level syllabus.
Objectives
- Know and use the remainder and factor theorems.
- Find factors of polynomials and solve cubic equations.
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Lesson Evaluation
Congratulations on completing the lesson on Remainder And Factor Theorems. Now that youve explored the key concepts and ideas, its time to put your knowledge to the test. This section offers a variety of practice questions designed to reinforce your understanding and help you gauge your grasp of the material.
You will encounter a mix of question types, including multiple-choice questions, short answer questions, and essay questions. Each question is thoughtfully crafted to assess different aspects of your knowledge and critical thinking skills.
Use this evaluation section as an opportunity to reinforce your understanding of the topic and to identify any areas where you may need additional study. Don't be discouraged by any challenges you encounter; instead, view them as opportunities for growth and improvement.
- What is the remainder when f(x) = x^3 - 2x^2 + 5x - 1 is divided by (x - 2)? A. 0 B. 5 C. 9 D. 11 Answer: C
- By the factor theorem, (x - a) is a factor of f(x) when: A. f(0) = a B. f(a) = 0 C. f(a) = 1 D. f(1) = a Answer: B
- Is (x - 1) a factor of f(x) = x^3 - 6x^2 + 11x - 6? A. Yes, because f(1) = 0 B. No, because f(1) = 6 C. Yes, because f(0) = -6 D. No, because f(1) = 1 Answer: A
- The full factorisation of x^3 - 6x^2 + 11x - 6 is: A. (x - 1)(x - 2)(x - 3) B. (x + 1)(x + 2)(x + 3) C. (x - 1)^3 D. (x - 2)(x - 3)(x - 6) Answer: A
- What is the remainder when f(x) = x^2 + 3x + 2 is divided by (x + 2)? A. 0 B. 2 C. 6 D. 12 Answer: A
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