Lesson Note on Polynomials: Meaning, Operations and Division for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS2)
Term: 1st Term
Week: 3
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Algebra
Previous Lesson: Roots of Quadratic Equation: Lines, Curves and Solving Problems.
Topic: POLYNOMIALS
Subject Matter: Meaning of polynomial (algebraic expression of terms with non-negative integer powers), addition of polynomials, subtraction of polynomials, multiplication of polynomials, division of polynomials by a polynomial of lesser degree.

Specific Objectives

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

Cognitive Domain:

• Define a polynomial and identify its degree and coefficients.
• State the steps involved in adding, subtracting, multiplying, and dividing polynomials.

Affective Domain:

• Appreciate the importance of polynomials in solving mathematical problems.
• Develop a systematic approach to solving problems involving polynomial operations.

Psychomotor Domain:

• Perform addition, subtraction, and multiplication of polynomials accurately.
• Divide a polynomial by another polynomial of a lesser degree using long division.

Social Domain:

• Work cooperatively with peers to solve problems related to polynomial operations.

Reference Materials

The following resources were used in planning this lesson:

• Senior Secondary Schools Education Curriculum
• State Unified Scheme of Work
• New Further Mathematics for Senior Secondary Schools by P.N. Chikezie et al.

Instructional Materials

The teacher will teach this lesson with the aid of:

• Charts showing examples of polynomials of various degrees.
• Whiteboard/chalkboard and markers/chalk.

Rationale for the Lesson

This lesson helps pupils understand algebraic expressions that are fundamental to higher mathematics. It enables them to perform basic operations on these expressions, which is essential for solving more complex equations and modeling real-world situations in various fields like engineering and economics.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of basic algebraic expressions, simplification of expressions, and the laws of indices from their junior secondary mathematics.

Lesson Content/Board Summary

POLYNOMIALS

Meaning of Polynomial

A polynomial is an algebraic expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The general form of a polynomial is:

P(x) = a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀

Where:

• a_n, a_n-1, …, a₀ are coefficients (real numbers).
• x is the variable.
• n is a non-negative integer (the degree of the polynomial).
• Degree: The highest power of the variable in the polynomial.
• Leading Coefficient: The coefficient of the term with the highest power.

Examples:

1. 3x² – 5x + 7 (Degree 2, Coefficients: 3, -5, 7)
2. x³ + 2x – 1 (Degree 3, Coefficients: 1, 0, 2, -1 (for x³, x², x, constant respectively))
3. 5x⁴ (Degree 4, Coefficient: 5)

Operations on Polynomials

Addition of Polynomials

To add polynomials, combine like terms (terms with the same variable and exponent) by adding their coefficients.

Example 1: Add (3x² + 2x – 1) and (x² – 4x + 5)

Solution:

Step 1: Write the polynomials together, grouping like terms.

(3x² + x²) + (2x – 4x) + (-1 + 5)

Step 2: Add the coefficients of the like terms.

(3+1)x² + (2-4)x + (-1+5)

4x² – 2x + 4

Subtraction of Polynomials

To subtract polynomials, distribute the negative sign to all terms of the polynomial being subtracted, then combine like terms as in addition.

Example 2: Subtract (x² – 4x + 5) from (3x² + 2x – 1)

Solution:

Step 1: Write the subtraction expression.

(3x² + 2x – 1) – (x² – 4x + 5)

Step 2: Distribute the negative sign to the terms in the second polynomial.

3x² + 2x – 1 – x² + 4x – 5

Step 3: Group and combine like terms.

(3x² – x²) + (2x + 4x) + (-1 – 5)

2x² + 6x – 6

Multiplication of Polynomials

To multiply polynomials, multiply each term of the first polynomial by every term of the second polynomial (using the distributive law), then combine like terms.

Example 3: Multiply (x + 2) by (x² – 3x + 1)

Solution:

Step 1: Multiply each term of (x + 2) by the polynomial (x² – 3x + 1).

x(x² – 3x + 1) + 2(x² – 3x + 1)

Step 2: Distribute x and 2 to the terms inside the parentheses.

(x³ – 3x² + x) + (2x² – 6x + 2)

Step 3: Combine like terms.

x³ + (-3x² + 2x²) + (x – 6x) + 2

x³ – x² – 5x + 2

Division of Polynomials

Polynomials can be divided using long division, similar to numerical long division. This method is used when dividing a polynomial by another polynomial of a lesser degree.

Example 4: Divide (x³ – 6x² + 11x – 6) by (x – 1)

Solution:

Step 1: Set up the long division problem.

x² – 5x + 6

x – 1 | x³ – 6x² + 11x – 6

Step 2: Divide the first term of the dividend (x³) by the first term of the divisor (x) to get x². Write x² above the x² term.

Step 3: Multiply x² by the entire divisor (x – 1) to get x³ – x². Write this below the dividend.

Step 4: Subtract (x³ – x²) from (x³ – 6x²). Remember to change signs and add.

x – 1 | x³ – 6x² + 11x – 6

-(x³ – x²)

——————

-5x² + 11x

Step 5: Bring down the next term (+11x).

Step 6: Repeat the process: Divide -5x² by x to get -5x. Write -5x in the quotient.

Step 7: Multiply -5x by (x – 1) to get -5x² + 5x. Write this below.

Step 8: Subtract (-5x² + 5x) from (-5x² + 11x).

x – 1 | x³ – 6x² + 11x – 6

-(x³ – x²)

——————

-5x² + 11x

-(-5x² + 5x)

—————-

6x – 6

Step 9: Bring down the next term (-6).

Step 10: Repeat: Divide 6x by x to get 6. Write +6 in the quotient.

Step 11: Multiply 6 by (x – 1) to get 6x – 6. Write this below.

Step 12: Subtract (6x – 6) from (6x – 6). The remainder is 0.

x – 1 | x³ – 6x² + 11x – 6

-(x³ – x²)

——————

-5x² + 11x

-(-5x² + 5x)

—————-

6x – 6

-(6x – 6)

———-

0

Therefore, (x³ – 6x² + 11x – 6) ÷ (x – 1) = x² – 5x + 6.

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 an algebraic expression is and to give examples of simple algebraic expressions.

Pupils’ Activity: Pupils respond by defining algebraic expressions and giving examples like x+2, 2y-5, etc.

Learning Point: Pupils recall prior knowledge of algebraic expressions.

Step 2: Meaning of Polynomials

Time: 6 minutes

Teaching Skill: Explanation/Definition

Teacher’s Activity: The teacher defines a polynomial, explains its general form, and uses charts to show examples of polynomials, highlighting their degree and coefficients.

Pupils’ Activity: Pupils listen attentively, copy notes, and identify the degree and coefficients of given polynomial examples.

Learning Point: Pupils understand the definition, degree, and coefficients of a polynomial.

Step 3: Addition of Polynomials

Time: 6 minutes

Teaching Skill: Demonstration/Problem Solving

Teacher’s Activity: The teacher demonstrates how to add polynomials by combining like terms, providing a worked example on the board.

Pupils’ Activity: Pupils observe the steps, ask questions for clarity, and practice solving similar problems.

Learning Point: Pupils learn how to add polynomials by combining like terms.

Step 4: Subtraction of Polynomials

Time: 6 minutes

Teaching Skill: Demonstration/Problem Solving

Teacher’s Activity: The teacher explains how to subtract polynomials, emphasizing the distribution of the negative sign, and works through an example on the board.

Pupils’ Activity: Pupils follow the steps shown by the teacher and attempt practice questions.

Learning Point: Pupils learn to subtract polynomials by distributing the negative sign and combining like terms.

Step 5: Multiplication of Polynomials

Time: 6 minutes

Teaching Skill: Demonstration/Problem Solving

Teacher’s Activity: The teacher demonstrates the multiplication of polynomials using the distributive law, solving an example on the board.

Pupils’ Activity: Pupils pay attention to the application of the distributive law and solve practice problems.

Learning Point: Pupils learn how to multiply polynomials using the distributive law.

Step 6: Division of Polynomials

Time: 8 minutes

Teaching Skill: Demonstration/Problem Solving

Teacher’s Activity: The teacher guides pupils through the process of polynomial long division using a step-by-step worked example on the board.

Pupils’ Activity: Pupils carefully follow the long division steps and attempt to solve a similar division problem.

Learning Point: Pupils understand and can perform polynomial long division.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. What is a polynomial?
2. Identify the degree and coefficients of the polynomial 5x⁴ – 2x³ + 7x – 9.
3. Add (2x² + 5x – 3) and (x² – 3x + 1).
4. Subtract (4x – 2) from (3x² + 6x – 5).
5. Multiply (x – 3) by (x + 4).

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 2 minutes

Teaching Skill: Summarizing

Teacher’s Activity: The teacher summarizes the key points of the lesson, reinforcing the understanding of polynomials and their operations. The teacher also gives homework assignments from the textbook.

Pupils’ Activity: Pupils listen to the summary and copy down the homework.

Learning Point: Pupils consolidate their learning and prepare for further practice.

Lesson Keywords

• Polynomial – An algebraic expression with non-negative integer exponents.
• Degree – The highest power of the variable in a polynomial.
• Coefficient – The numerical factor of a term in a polynomial.
• Term – A single number or variable, or numbers and variables multiplied together.
• Like terms – Terms that have the same variables raised to the same power.

Differentiation

For pupils who are struggling, the teacher will provide additional simpler examples and one-on-one guidance during practice sessions. Advanced pupils will be given more complex polynomial problems involving higher degrees or multiple variables to challenge their understanding and problem-solving skills.

Note for teachers using this lesson plan

Ensure that pupils have a strong grasp of basic algebra and the laws of indices before introducing polynomial operations. Emphasize the importance of combining like terms carefully during addition and subtraction, and distributing correctly during multiplication and division. Encourage pupils to show all steps clearly, especially for polynomial long division, to minimize errors.