Chapter 2: Polynomials

2.1 Introduction

Polynomial in one variable: An expression p(x) where the highest power of x is its degree.
Degree of a polynomial: The highest power of the variable in p(x).
Examples of polynomials by degree:
- Degree 1: 4x + 2
- Degree 2: 2y² – 3y + 4
- Degree 3: 5x³ – 4x² + x – 2
- Degree 6 polynomial example: A polynomial in u such as 7u⁶ – 3u⁴ + 2u² + u – 1. (Original text contained transcription errors in this specific example)
Non-polynomial expressions: Expressions like 1/(x – 1), √x + 2, x + 1/(x + 2) are not polynomials.
Linear polynomial: A polynomial of degree 1.
- General form: ax + b, where a ≠ 0.
- Examples: 2x – 3, (1/2)x – √3, 3z + 4, etc.
Quadratic polynomial: A polynomial of degree 2.
- Origin: 'Quadratic' is derived from 'quadrate', meaning 'square'.
- General form: ax² + bx + c, where a, b, c are real numbers and a ≠ 0.
- Examples: 2x² + 3x – 2/5, y² – 2, etc.
Cubic polynomial: A polynomial of degree 3.
- General form: ax³ + bx² + cx + d, where a, b, c, d are real numbers and a ≠ 0.
- Examples: 2 – x³, x³, 3x², 3 – x² + x³.
Value of a polynomial p(x) at x = k: Denoted by p(k), it is the real number obtained by replacing x with k in p(x).
- Example: For p(x) = x² – 3x – 4, p(2) = 2² – 3(2) – 4 = –6.
Zero of a polynomial p(x): A real number k is a zero of p(x) if p(k) = 0.
- Example: For p(x) = x² – 3x – 4, p(–1) = 0 and p(4) = 0, so –1 and 4 are its zeroes.
Finding zeroes of a linear polynomial: If k is a zero of p(x) = ax + b, then ak + b = 0, which means k = –b/a.
- Formula: Zero = –(Constant term) / (Coefficient of x).
Chapter focus: This chapter explores the relationship between zeroes and coefficients for various polynomials, and covers the division algorithm for polynomials.

2.2 Geometrical Meaning of the Zeroes of a Polynomial

Definition recap: A real number k is a zero of the polynomial p(x) if p(k) = 0.
Purpose: To understand the geometrical meaning of zeroes through graphical representations of linear, quadratic, and cubic polynomials.

2.2.1 Geometrical Meaning for Linear Polynomials

Graph of y = ax + b (a ≠ 0): A straight line.
Zero of the polynomial: It is the x-coordinate of the point where the graph of y = ax + b intersects the x-axis.
- Example: For y = 2x + 3, the graph intersects the x-axis at (–3/2, 0). The zero is –3/2.
Conclusion for linear polynomials: A linear polynomial ax + b (a ≠ 0) has exactly one zero, corresponding to its x-intercept.

2.2.2 Geometrical Meaning for Quadratic Polynomials

Graph of y = ax² + bx + c (a ≠ 0): A parabola.
- Shape: Opens upwards if a > 0, and opens downwards if a < 0.
Zeroes of the polynomial: These are the x-coordinates of the points where the parabola intersects the x-axis.
- Example: For y = x² – 3x – 4, the graph intersects the x-axis at x = –1 and x = 4, which are its zeroes.
Three cases for quadratic polynomial zeroes (geometrically):
Maximum number of zeroes: A quadratic polynomial (degree 2) can have at most two zeroes.

2.2.3 Geometrical Meaning for Cubic Polynomials

Zeroes of the polynomial: These are the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.
- Example: For y = x³ – 4x, the graph intersects the x-axis at x = –2, 0, and 2, which are its zeroes.
Maximum number of zeroes: A cubic polynomial (degree 3) can have at most three zeroes.

2.2.4 General Remark on Zeroes

Polynomial of degree n: The graph of y = p(x) intersects the x-axis at at most n points.
Conclusion: A polynomial p(x) of degree n has at most n zeroes.

2.2.5 Example Problems (Finding Number of Zeroes from Graph)

Method: The number of zeroes is determined by how many times the graph intersects the x-axis.
Fig 2.9 (i): 1 zero.
Fig 2.9 (ii): 2 zeroes.
Fig 2.9 (iii): 3 zeroes.
Fig 2.9 (iv): 1 zero.
Fig 2.9 (v): 1 zero.
Fig 2.9 (vi): 4 zeroes.

2.2.6 Exercise 2.1 (Number of Zeroes from Graph)

Task: Find the number of zeroes of p(x) for given graphs (refer to Fig. 2.10 in the textbook).

2.3 Relationship between Zeroes and Coefficients of a Polynomial

2.3.1 Introduction and Linear Polynomials

Linear polynomial ax + b: Its zero is –b/a.
Objective: To explore if a similar relationship exists between zeroes and coefficients for quadratic and cubic polynomials.

2.3.2 Quadratic Polynomials: Relation between Zeroes and Coefficients

Factoring example: For p(x) = 2x² – 8x + 6 = 2(x – 1)(x – 3), the zeroes are 1 and 3.
Observation: The sum (1+3=4) and product (1×3=3) of zeroes are related to coefficients (–(–8)/2 = 4 and 6/2 = 3).
Derivation: If α and β are zeroes of ax² + bx + c (a ≠ 0), then p(x) = k(x – α)(x – β). Comparing coefficients gives the relationships.
Formulas for quadratic polynomial ax² + bx + c:
- Sum of zeroes (α + β): –(Coefficient of x) / (Coefficient of x²) = –b/a.
- Product of zeroes (αβ): (Constant term) / (Coefficient of x²) = c/a.

2.3.2.1 Example 2: Find Zeroes and Verify for x² + 7x + 10

Polynomial: x² + 7x + 10 = (x + 2)(x + 5).
Zeroes: –2 and –5.
Verification:
- Sum of zeroes: (–2) + (–5) = –7. Formula: –b/a = –7/1 = –7. (Verified)
- Product of zeroes: (–2) × (–5) = 10. Formula: c/a = 10/1 = 10. (Verified)

2.3.2.2 Example 3: Find Zeroes and Verify for x² – 3

Polynomial: x² – 3 = (x – √3)(x + √3).
Zeroes: √3 and –√3.
Verification:
- Sum of zeroes: √3 + (–√3) = 0. Formula: –b/a = –0/1 = 0. (Verified)
- Product of zeroes: (√3) × (–√3) = –3. Formula: c/a = –3/1 = –3. (Verified)

2.3.2.3 Example 4: Find Quadratic Polynomial from Sum and Product of Zeroes

Given: Sum of zeroes (α + β) = –3, Product of zeroes (αβ) = 2.
From formulas: –b/a = –3 and c/a = 2.
Assuming a = 1: Then b = 3 and c = 2.
Quadratic polynomial: x² + 3x + 2.
General form: Any polynomial fitting these conditions is k(x² + 3x + 2) for real k.

2.3.3 Cubic Polynomials: Relation between Zeroes and Coefficients

For cubic polynomial ax³ + bx² + cx + d: If α, β, γ are its zeroes.
Formulas for cubic polynomial:
- Sum of zeroes (α + β + γ): –b/a.
- Sum of products of zeroes taken two at a time (αβ + βγ + γα): c/a.
- Product of zeroes (αβγ): –d/a.

2.3.3.1 Example 5: Verify Zeroes and Relationship for Cubic Polynomial

Polynomial: p(x) = 3x³ – 5x² – 11x – 3. (Coefficients: a=3, b=–5, c=–11, d=–3)
Given zeroes: 3, –1, –1/3.
Verification of zeroes: p(3)=0, p(–1)=0, p(–1/3)=0. (Confirmed)
Verification of relationships:
- Sum of zeroes: 3 + (–1) + (–1/3) = 5/3. Formula: –b/a = –(–5)/3 = 5/3. (Verified)
- Sum of products (two at a time): (3)(–1) + (–1)(–1/3) + (–1/3)(3) = –3 + 1/3 – 1 = –11/3. Formula: c/a = –11/3. (Verified)
- Product of zeroes: (3)(–1)(–1/3) = 1. Formula: –d/a = –(–3)/3 = 1. (Verified)

2.3.4 Exercise 2.2 (Find Zeroes/Polynomials and Verify)

Task 1: Find zeroes of given quadratic polynomials and verify relationships with coefficients.
Task 2: Find quadratic polynomials given the sum and product of their zeroes.

2.4 Summary

Polynomial types by degree:
- Degree 1: Linear polynomial.
- Degree 2: Quadratic polynomial.
- Degree 3: Cubic polynomial.
Quadratic polynomial general form: ax² + bx + c, where a, b, c are real numbers and a ≠ 0.
Zeroes of a polynomial p(x) geometrically: These are precisely the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.
Maximum number of zeroes:
- Quadratic polynomial: At most 2 zeroes.
- Cubic polynomial: At most 3 zeroes.
- General polynomial of degree n: At most n zeroes.
Relationship for quadratic polynomial (if α, β are zeroes of ax² + bx + c):
- Sum of zeroes: α + β = –b/a.
- Product of zeroes: αβ = c/a.
Relationship for cubic polynomial (if α, β, γ are zeroes of ax³ + bx² + cx + d):
- Sum of zeroes: α + β + γ = –b/a.
- Sum of products of zeroes taken two at a time: αβ + βγ + γα = c/a.
- Product of zeroes: αβγ = –d/a.