Class: Senior Secondary School 2 (SS2 / SSS2)
Term: First Term
Week: 5
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Algebra
Previous Lesson: Polynomials: Remainder Theorem, Factor Theorem and Factorization.
Topic: POLYNOMIALS
Subject Matter: Roots of cubic equation (values of x that satisfy ax^3 + bx^2 + cx + d = 0), sum of roots (α + β + δ = -b/a), sum of products of two roots (αβ + αδ + βδ = c/a), product of roots (αβδ = -d/a)

Specific Objectives

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

Cognitive Domain:

Define a cubic equation and its general form.
State the formulas relating the roots and coefficients of a cubic equation.
Identify the coefficients (a, b, c, d) from a given cubic equation.

Affective Domain:

Appreciate the importance of understanding the relationship between roots and coefficients in solving polynomial problems.
Develop interest in solving problems involving cubic equations.

Psychomotor Domain:

Compute the sum of roots of a given cubic equation.
Compute the sum of the products of the roots taken two at a time from a given cubic equation.
Compute the product of the roots of a given cubic equation.
Solve problems involving algebraic manipulations of roots of cubic equations.

Work collaboratively with peers to solve problems related to cubic equations.

Reference Materials

The following resources were used in planning this lesson:

Senior Secondary Schools Education Curriculum
State Unified Scheme of Work
Further Mathematics for Senior Secondary Schools by P.N. Okeke et al.
https://www.studywithjosh.com/documents/sss-2-further-mathematics-first-term-lesson-note

Instructional Materials

The teacher will teach this lesson with the aid of:

Charts showing the general form of a cubic equation.
Charts displaying the formulas for the sum of roots, sum of products of two roots, and product of three roots of a cubic equation.
Whiteboard and markers.

Rationale for the Lesson

This lesson helps pupils understand the fundamental properties of cubic equations, which is important for advanced algebraic manipulations. It enables them to find relationships between the roots and coefficients without solving for the individual roots, providing a powerful tool in further mathematics.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of quadratic equations, their roots, and how to identify coefficients (a, b, c) from a quadratic equation.

Lesson Content/Board Summary

POLYNOMIALS

Roots of a Cubic Equation

A cubic equation is a polynomial equation of degree 3. Its general form is:

ax³ + bx² + cx + d = 0

where ‘a’, ‘b’, ‘c’, and ‘d’ are coefficients, and ‘a’ must not be zero (a ≠ 0).

A cubic equation has three roots. Let these roots be denoted by α (alpha), β (beta), and δ (delta).

Relationship Between Roots and Coefficients of a Cubic Equation

For a cubic equation ax³ + bx² + cx + d = 0 with roots α, β, and δ, the following relationships exist:

Sum of the roots:
α + β + δ = -b/a
Sum of the products of the roots taken two at a time:
αβ + αδ + βδ = c/a
Product of the roots:
αβδ = -d/a

Worked Examples

Example 1:
Given the cubic equation 2x³ – 5x² + 7x – 3 = 0, find:

a) The sum of the roots.
b) The sum of the products of the roots taken two at a time.
c) The product of the roots.

Solution:

Step 1: Identify the coefficients from the equation ax³ + bx² + cx + d = 0.
Here, a = 2, b = -5, c = 7, d = -3.

Step 2: Apply the formulas.

a) Sum of roots (α + β + δ) = -b/a = -(-5)/2 = 5/2

b) Sum of products of two roots (αβ + αδ + βδ) = c/a = 7/2

c) Product of roots (αβδ) = -d/a = -(-3)/2 = 3/2

Example 2:
If α, β, and δ are the roots of the equation x³ – 4x² + 5x – 2 = 0, find the value of 1/α + 1/β + 1/δ.

Solution:

Step 1: Identify the coefficients.
From x³ – 4x² + 5x – 2 = 0, we have a = 1, b = -4, c = 5, d = -2.

Step 2: Find the sum of roots, sum of products of two roots, and product of roots.

Sum of roots (α + β + δ) = -b/a = -(-4)/1 = 4
Sum of products of two roots (αβ + αδ + βδ) = c/a = 5/1 = 5
Product of roots (αβδ) = -d/a = -(-2)/1 = 2

Step 3: Express the required value in terms of these relationships.

1/α + 1/β + 1/δ = (βδ + αδ + αβ) / (αβδ)

Rearranging the numerator: = (αβ + αδ + βδ) / (αβδ)

Substitute the values: = 5 / 2

Teaching Methods/Instructional Techniques

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

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher greets the pupils and asks them to recall what they know about quadratic equations and their roots, emphasizing the relationship between roots and coefficients.
Pupils’ Activity: Pupils respond by recalling previous knowledge about quadratic equations.
Learning Point: Pupils connect prior knowledge of quadratic equations to the new topic.

Step 2: Explanation of Cubic Equation

Time: 5 minutes
Teaching Skill: Explanation
Teacher’s Activity: The teacher introduces the concept of a cubic equation, its general form (ax³ + bx² + cx + d = 0), and explains that it has three roots.
Pupils’ Activity: Pupils listen attentively and write down the general form of a cubic equation.
Learning Point: Pupils understand the definition and general form of a cubic equation.

Step 3: Introduction of Roots and their Relationships

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher introduces the three roots (α, β, δ) and then presents the formulas relating these roots to the coefficients (sum of roots, sum of products of two roots, and product of roots), using charts.
Pupils’ Activity: Pupils observe the charts, listen to the explanations, and copy the formulas into their notebooks.
Learning Point: Pupils learn the three key formulas for roots and coefficients of a cubic equation.

Step 4: Worked Example 1 (Finding Relationships from Equation)

Time: 10 minutes
Teaching Skill: Demonstration/Problem Solving
Teacher’s Activity: The teacher guides pupils through Example 1 on the board, demonstrating how to identify coefficients and apply the formulas to find the sum of roots, sum of products of two roots, and product of roots.
Pupils’ Activity: Pupils follow the teacher’s steps, ask questions, and solve the example in their notebooks.
Learning Point: Pupils practice applying the formulas to find the relationships between roots and coefficients from a given cubic equation.

Step 5: Worked Example 2 (Algebraic Manipulation of Roots)

Time: 5 minutes
Teaching Skill: Demonstration/Problem Solving
Teacher’s Activity: The teacher presents Example 2, showing how to solve problems that require algebraic manipulation of the roots using the established relationships.
Pupils’ Activity: Pupils observe the steps and work through the example.
Learning Point: Pupils learn to use the relationships between roots and coefficients to solve more complex problems.

Step 6: Class Practice

Time: 5 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher provides a similar problem for pupils to solve independently or in pairs, offering guidance as needed.
Pupils’ Activity: Pupils attempt to solve the practice problem based on the examples taught.
Learning Point: Pupils reinforce their understanding and problem-solving skills.

Step 7: Evaluation/Review

Time: 5 minutes
Teaching Skill: Questioning/Assessment
Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

State the general form of a cubic equation.
For the cubic equation 3x³ + 2x² – 4x + 1 = 0, identify the values of a, b, c, and d.
Using the equation from question 2, what is the sum of its roots?
What is the product of its roots?

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 5 minutes
Teaching Skill: Summary
Teacher’s Activity: The teacher summarizes the key relationships between the roots and coefficients of a cubic equation and assigns homework: “Find the cubic equation whose roots are 1, 2, and 3.”
Pupils’ Activity: Pupils listen to the summary and copy down the homework assignment.
Learning Point: Pupils consolidate their learning and prepare for independent practice.

Lesson Keywords

Polynomials – Algebraic expressions consisting of variables and coefficients, involving only non-negative integer exponents of the variables.
Cubic equation – A polynomial equation of degree three, having the general form ax³ + bx² + cx + d = 0.
Roots – The values of the variable (x) that satisfy a polynomial equation; also known as solutions or zeros.
Coefficients – The numerical or constant part of a term in an algebraic expression (e.g., a, b, c, d in ax³ + bx² + cx + d = 0).
Sum of roots – The sum of all solutions of a polynomial equation.
Product of roots – The product of all solutions of a polynomial equation.

Differentiation

For pupils who grasp the concepts quickly, the teacher can provide more challenging problems involving inverse relationships (e.g., forming the cubic equation given its roots). For pupils who need more support, the teacher can provide additional guided practice with simpler coefficients and step-by-step assistance.

Note for teachers using this lesson plan

Ensure that pupils clearly understand the signs in the formulas (-b/a, c/a, -d/a). Emphasize the importance of correctly identifying the coefficients a, b, c, and d from the given cubic equation before applying the formulas. Encourage pupils to practice with various examples to build confidence and proficiency in solving problems related to roots of cubic equations.