Class: Senior Secondary School 2 (SS2 / SSS2)
Term: First Term
Week: 4
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Algebra and Functions
Previous Lesson: Polynomials: Meaning, Operations and Division.
Topic: POLYNOMIALS
Subject Matter: Remainder theorem, factor theorem, factorization of polynomials.

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

Define the remainder theorem and the factor theorem.
State the conditions for applying the remainder theorem and factor theorem.
Apply the remainder theorem to find the remainder when a polynomial is divided by a linear factor.
Apply the factor theorem to test if a linear expression is a factor of a polynomial.

Affective Domain:

Appreciate the importance of remainder and factor theorems in simplifying polynomial expressions.
Develop interest in solving problems involving polynomial factorization.

Psychomotor Domain:

Solve problems accurately using the remainder theorem.
Solve problems accurately using the factor theorem.
Factorize polynomials completely using the factor theorem and other methods.

Collaborate with peers to solve challenging problems related to polynomial theorems.

Reference Materials

The following resources were used in planning this lesson:

Senior Secondary Schools Education Curriculum
State Unified Scheme of Work
New General Mathematics for Senior Secondary Schools 2
https://www.mathsisfun.com/algebra/polynomials-remainder-factor.html
https://www.khanacademy.org/math/algebra2/x2ec2f6f830c979a4:polynomial-division/x2ec2f6f830c979a4:remainder-and-factor-theorems/v/remainder-and-factor-theorems

Instructional Materials

The teacher will teach this lesson with the aid of:

Charts showing examples of remainder theorem.
Charts illustrating the application of the factor theorem.
Whiteboard and markers.

Rationale for the Lesson

This lesson helps pupils understand how to efficiently find remainders of polynomial divisions and identify factors without performing long division. These skills are important for solving more complex algebraic equations and are foundational for advanced mathematical concepts in science and engineering.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of basic algebraic operations, division of algebraic expressions, and solving linear equations.

Lesson Content/Board Summary

POLYNOMIALS

Introduction to Polynomials

A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, f(x) = axⁿ + bx^n-1 + … + kx + c, where n is a non-negative integer.

Remainder Theorem

The remainder theorem states that if a polynomial f(x) is divided by a linear expression (x – a), then the remainder is f(a).

Worked Examples:

Example 1: Find the remainder when f(x) = x³ – 4x² + 5x – 2 is divided by (x – 2).

Solution:
Step 1: Identify ‘a’ from the divisor (x – a). Here, x – a = x – 2, so a = 2.
Step 2: Substitute ‘a’ into the polynomial f(x).
f(2) = (2)³ – 4(2)² + 5(2) – 2
f(2) = 8 – 4(4) + 10 – 2
f(2) = 8 – 16 + 10 – 2
f(2) = 0
The remainder is 0.

Example 2: When x³ + kx² – 3x + 1 is divided by (x + 1), the remainder is 6. Find the value of k.

Solution:
Step 1: Identify ‘a’. Here, x – a = x + 1, so a = -1.
Step 2: Substitute ‘a’ into the polynomial f(x) and set it equal to the remainder.
f(-1) = (-1)³ + k(-1)² – 3(-1) + 1 = 6
Step 3: Simplify and solve for k.
-1 + k(1) + 3 + 1 = 6
-1 + k + 4 = 6
k + 3 = 6
k = 6 – 3
k = 3
The value of k is 3.

Factor Theorem

The factor theorem is a special case of the remainder theorem. It states that (x – a) is a factor of a polynomial f(x) if and only if f(a) = 0 (i.e., the remainder is 0).

Worked Examples:

Example 1: Show that (x – 1) is a factor of f(x) = x³ + 2x² – 5x + 2.

Solution:
Step 1: Identify ‘a’. Here, x – a = x – 1, so a = 1.
Step 2: Substitute ‘a’ into the polynomial f(x).
f(1) = (1)³ + 2(1)² – 5(1) + 2
f(1) = 1 + 2 – 5 + 2
f(1) = 0
Since f(1) = 0, (x – 1) is a factor of f(x).

Example 2: If (x + 2) is a factor of x³ + px² – x – 2, find the value of p.

Solution:
Step 1: Identify ‘a’. Here, x – a = x + 2, so a = -2.
Step 2: Since (x + 2) is a factor, f(-2) must be 0.
f(-2) = (-2)³ + p(-2)² – (-2) – 2 = 0
Step 3: Simplify and solve for p.
-8 + p(4) + 2 – 2 = 0
-8 + 4p = 0
4p = 8
p = 2
The value of p is 2.

Factorization of Polynomials

To factorize a polynomial using the factor theorem:

Use trial and error (testing factors of the constant term) to find a value ‘a’ for which f(a) = 0. This means (x – a) is a factor.
Once a factor (x – a) is found, divide the polynomial f(x) by (x – a) using long division or synthetic division to find the other factor (a quotient polynomial).
Factorize the quotient polynomial further if it is not irreducible.

Worked Example:

Example 1: Factorize completely f(x) = x³ – 7x + 6.

Solution:
Step 1: Find a factor using the factor theorem. Test factors of 6 (±1, ±2, ±3, ±6).
f(1) = (1)³ – 7(1) + 6 = 1 – 7 + 6 = 0.
Therefore, (x – 1) is a factor.

Step 2: Divide f(x) by (x – 1) using long division.

x² + x – 6
____________
x-1 | x³ + 0x² – 7x + 6
-(x³ – x²)
_________
x² – 7x
-(x² – x)
_______
-6x + 6
-(-6x + 6)
_________
0

The quotient is x² + x – 6.

Step 3: Factorize the quadratic quotient.
x² + x – 6 = (x + 3)(x – 2)

Step 4: Write the complete factorization.
Therefore, x³ – 7x + 6 = (x – 1)(x + 3)(x – 2).

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 3 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher asks pupils to recall how to divide algebraic expressions and what a remainder is in division. The teacher then introduces the topic of polynomials and the need for quicker methods to find remainders and factors.
Pupils’ Activity: Pupils respond to questions on algebraic division and listen attentively to the introduction.
Learning Point: Pupils recall prior knowledge and are introduced to the lesson topic and its relevance.

Step 2: Remainder Theorem

Time: 8 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains the remainder theorem, stating that when a polynomial f(x) is divided by (x – a), the remainder is f(a). The teacher demonstrates this with a simple example from the board summary.
Pupils’ Activity: Pupils listen, take notes, and ask questions for clarification.
Learning Point: Pupils understand the statement and application of the remainder theorem.

Step 3: Examples on Remainder Theorem

Time: 6 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher guides pupils through another worked example on the remainder theorem, such as finding an unknown coefficient given a remainder. The teacher encourages pupils to solve along.
Pupils’ Activity: Pupils participate in solving the example and practice applying the theorem.
Learning Point: Pupils reinforce their understanding and application of the remainder theorem.

Step 4: Factor Theorem

Time: 8 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher introduces the factor theorem as a special case of the remainder theorem, stating that (x – a) is a factor if f(a) = 0. The teacher demonstrates this with an example from the board summary.
Pupils’ Activity: Pupils listen, take notes, and compare it with the remainder theorem.
Learning Point: Pupils understand the statement and application of the factor theorem.

Step 5: Examples on Factor Theorem

Time: 6 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher guides pupils through another worked example on the factor theorem, such as finding an unknown coefficient given a factor. The teacher ensures pupils understand the conditions.
Pupils’ Activity: Pupils participate in solving the example and practice applying the theorem.
Learning Point: Pupils reinforce their understanding and application of the factor theorem.

Step 6: Factorization of Polynomials

Time: 8 minutes
Teaching Skill: Demonstration/Explanation
Teacher’s Activity: The teacher demonstrates how to factorize a polynomial completely using the factor theorem to find one factor, and then long division to find the quadratic factor, which can then be factorized further. The teacher uses the example from the board summary.
Pupils’ Activity: Pupils observe the steps, take notes, and ask questions about the factorization process.
Learning Point: Pupils learn the systematic approach to factorize polynomials using the theorems.

Step 7: Evaluation/Review

Time: 4 minutes
Teaching Skill: Questioning/Assessment
Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

State the Remainder Theorem.
What is the condition for (x – a) to be a factor of a polynomial f(x)?
If f(x) = x³ – 2x² + 3x – 1, find the remainder when f(x) is divided by (x – 1).
Show that (x + 1) is a factor of x³ + 5x² + 7x + 3.

Pupils’ Activity: Pupils answer orally and in writing.
Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 2 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key learning points on the remainder theorem, factor theorem, and polynomial factorization. The teacher then assigns homework from the textbook.
Pupils’ Activity: Pupils listen to the summary and copy the assigned homework.
Learning Point: Pupils consolidate their understanding and are given tasks for independent practice.

Lesson Keywords

Polynomials – Algebraic expressions with variables and coefficients involving only non-negative integer exponents.
Remainder Theorem – A theorem that states the remainder of polynomial division f(x) ÷ (x – a) is f(a).
Factor Theorem – A theorem that states (x – a) is a factor of f(x) if and only if f(a) = 0.
Factorization – The process of expressing a polynomial as a product of its factors.

Differentiation

For pupils who grasp concepts quickly, the teacher will provide more complex polynomial problems involving higher degrees or multiple unknown coefficients. For pupils needing more support, the teacher will provide additional guided examples and simplified problems, focusing on one theorem at a time before combining them.

Note for teachers using this lesson plan

Ensure pupils have a solid understanding of basic algebraic operations, especially substitution and solving equations. Emphasize the connection between the remainder theorem and the factor theorem. Encourage pupils to practice long division or synthetic division to find subsequent factors after using the factor theorem. Visual aids and step-by-step problem-solving demonstrations are important for clarity.