MAT 111: Algebra Comprehensive University Study Guide

MAT 111: ALGEBRA - COURSE OVERVIEW AND NUMBER SYSTEMS

Instructor: Dr. Taiwo. O. SANGODAPO Affiliation: Department of Mathematics, University of Ibadan Session: 2021/2022 Session

Course Outline

1. Polynomials and Rational Functions

2. Principle of Mathematical Induction

3. Permutation, Combination and the Binomial Theorem

4. Sequences and Series

5. Complex Numbers

6. Matrices and Determinants

An Excursion to Number Systems

There are many types of number systems essential for algebraic study:

1. The Natural Numbers ($N$):    The set of counting numbers: $N = \{1, 2, 3, \dots\}$. This set represents positive whole numbers. The operations of addition ($+$) and multiplication ($\times$) are well-defined on $N$.

2. The Set of Integers ($Z$):    Includes positive and negative whole numbers along with zero: $Z = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$. The operations of addition ($+$), multiplication ($\times$), and subtraction ($-$) are well-defined. Note that $N \subset Z$.

3. The Set of Rational Numbers ($Q$):    Defined as $Q = \{\frac{a}{b} : a, b \in Z, b \neq 0\}$. This is the set of all fractions. The relationship between sets is $N \subset Z \subset Q$. Addition, subtraction, multiplication, and division are well-defined on $Q$.

4. Irrational Numbers:    Numbers that cannot be expressed as fractions, such as $\sqrt{2}$, $\pi$, or $\sqrt{5}$. The union of rational and irrational numbers forms the Real Number System ($R$).

5. The Complex Number System ($C$):    Consider the equation $x^2 + 1 = 0 \implies x = \pm \sqrt{-1}$. Because the square root of a negative number has no solution in $R$, the complex system is defined as $C = \{a + ib : a, b \in R\}$, where $i = \sqrt{-1}$.

POLYNOMIALS AND RATIONAL FUNCTIONS

Introduction to Polynomials

A real polynomial $f(x)$ in variable $x$ of degree $n$ is represented by: $f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$ where:

• $a_n \neq 0$

• $a_n$ is the leading coefficient.

• $a_0$ is the constant term.

Classification by Degree

• Degree 1 ($n=1$): Linear function.

• Degree 2 ($n=2$): Quadratic function.

• Degree 3 ($n=3$): Cubic function.

• Degree 4 ($n=4$): Quartic function.

• Degree 5 ($n=5$): Quintic function.

Basic Arithmetic Operations

Operations of addition, subtraction, multiplication, and division apply to polynomials.

Example: If $f(x) = 3x^3 + 4x^2 + 5x - 1$ and $g(x) = 6x^2 - 7x + 5$:

1. Addition: $f(x) + g(x) = 3x^3 + (4+6)x^2 + (5-7)x + (-1+5) = 3x^3 + 10x^2 - 2x + 4$.

2. Subtraction: $f(x) - g(x) = 3x^3 + (4-6)x^2 + (5 - (-7))x + (-1-5) = 3x^3 - 2x^2 + 12x - 6$.

3. Multiplication: $f(x) \cdot g(x) = (3x^3 + 4x^2 + 5x - 1)(6x^2 - 7x + 5)$. Results in: $18x^5 + 3x^4 + 17x^3 - 21x^2 + 32x - 5$.

Division of Polynomials

A polynomial $f(x)$ can be divided by $g(x)$ provided the degree of $g(x) \leq$ degree of $f(x)$. Here, $g(x)$ is the divisor ($g(x) \neq 0$).

General Form: $f(x) = d(x) \times q(x) + r(x)$ Where $d(x)$ is the divisor, $q(x)$ is the quotient, and $r(x)$ is the remainder.

Long Division Examples:

1. Divide $x^3 + 3x^2 - x + 1$ by $x - 2$:    - Quotient: $x^2 + 5x + 9$    - Remainder: $19$

2. Divide $5x^3 + 14x^2 - 6x - 8$ by $2x^2 + x - 3$:    - Quotient: $\frac{5}{2}x + \frac{23}{4}$    - Remainder: $-\frac{17}{4}x + \frac{37}{4}$

3. Divide $x^3 - 2x^2 + 3x - 6$ by $x + 2$:    - Quotient: $x^2 - 4x + 11$    - Remainder: $-28$

Remainder and Factor Theorems

Remainder Theorem

If a polynomial $f(x)$ of degree $\geq 1$ is divided by $ax + b$, the remainder is $f(-\frac{b}{a})$.

Example 1: Find the remainder when $f(x) = x^3 - 3x^2 + x - 5$ is divided by $2x - 1$.

• Set $2x - 1 = 0 \implies x = \frac{1}{2}$.

• $f(\frac{1}{2}) = (\frac{1}{2})^3 - 3(\frac{1}{2})^2 + \frac{1}{2} - 5 = \frac{1}{8} - \frac{3}{4} + \frac{1}{2} - 5 = -\frac{41}{8}$.

• Remainder is $-5.125$.

Example 2: Find the remainder when $f(x) = 6x^3 + 2x^2 - 1$ is divided by $x + 1$.

• Set $x + 1 = 0 \implies x = -1$.

• $f(-1) = 6(-1)^3 + 2(-1)^2 - 1 = -6 + 2 - 1 = -5$.

Factor Theorem

If $ax + b$ is a factor of $f(x)$, then $f(-\frac{b}{a}) = 0$, and conversely.

Example: Show $x - 2$ is a factor of $x^4 - 15x^2 + 10x + 24$.

• $f(2) = 2^4 - 15(2)^2 + 10(2) + 24 = 16 - 60 + 20 + 24 = 0$.

• Therefore, $x - 2$ is a factor.

LINEAR, QUADRATIC AND CUBIC EQUATIONS

Linear Equations

General form: $y = ax + b$ (Degree 1). Geometrically, this is a straight line.

• Slope: $a$

• y-intercept: $b$

• x-intercept (root): $-\frac{b}{a}$ Every linear equation has exactly one root.

Simultaneous Linear Equations in Two Variables

General form: $ax + by = e_1$ $cx + dy = e_2$

Methods for solution: Substitution, Elimination, Cramer’s Rule, and Graphical methods.

Types of Solutions

1. Unique Solution (Type I): Lines intersect at one point (e.g., $3x + y = 6$ and $x - y = 10 \implies x=4, y=-6$).

2. Infinite Solutions (Type II): Both equations represent the same line (e.g., $4x + 2y = 8$ and $12x + 6y = 24$).

3. No Solution (Type III): Lines are parallel and never intersect (e.g., $x + 3y = 7$ and $3x + 9y = 5$).

Quadratic Equations

General form: $y = ax^2 + bx + c$ (Degree 2). Geometrically, this is a parabola (convex or concave).

The Quadratic Formula

By completing the square on $ax^2 + bx + c = 0$: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Nature of Roots (Discriminant)

The discriminant is $D = b^2 - 4ac$.

1. $D > 0$: Two distinct real roots.

2. $D = 0$: Two equal real roots (one repeated root).

3. $D < 0$: Imaginary or complex roots (no real roots).

Sum and Product of Roots

If $\alpha$ and $\beta$ are roots of $ax^2 + bx + c = 0$:

• Sum: $\alpha + \beta = -\frac{b}{a}$

• Product: $\alpha\beta = \frac{c}{a}$ The equation can be written as: $x^2 - (\alpha + \beta)x + \alpha\beta = 0$.

Useful Identities:

1. $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$

2. $\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)$

3. $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$

Cubic Equations

General form: $ax^3 + bx^2 + cx + d = 0$. If $\alpha$, $\beta$, and $\gamma$ are roots:

• $\alpha + \beta + \gamma = -\frac{b}{a}$

• $\alpha\beta + \alpha\gamma + \beta\gamma = \frac{c}{a}$

• $\alpha\beta\gamma = -\frac{d}{a}$

INEQUALITIES AND ABSOLUTE VALUES

Representation of Inequalities

Inequalities (e.g., "$x$ is greater than or equal to 5") can be shown in three ways:

1. Symbolic Notation: $x \geq 5$

2. Interval Notation: $[5, \infty)$

3. Number Line: A solid circle at 5 with an arrow pointing right.

Interval Types

• Open Interval ($a, b$): $a < x < b$ (endpoints not included).

• Closed Interval [$a, b$]: $a \leq x \leq b$ (endpoints included).

• Half-Open/Half-Closed [$a, b$) or ($a, b$]: One endpoint excluded. Note: Infinity ($\infty$) is not a number and is always represented with an open bracket.

Properties of Inequalities

Let $a, b, c, d \in R$:

1. If $a < b$ and $b < c$, then $a < c$.

2. If $a < b$, then $a + c < b + c$.

3. Multiplication/Division by $c > 0$ preserves the inequality sign.

4. Multiplication/Division by $c < 0$ reverses the inequality sign (e.g., if $a < b$, then $ac > bc$).

Quadratic Inequalities

Methods for solving $x^2 + 2x - 3 > 0$:

1. Analytical Method: Factor to $(x-1)(x+3) > 0$. Solve for Cases: ($+$ and $+$) or ($-$ and $-$). Results: $x > 1$ or $x < -3$.

2. Sign Table Method: Identify critical values ($-3, 1$). Test values in segments $(-\infty, -3)$, $(-3, 1)$, and $(1, \infty)$.

3. Graphical Method: Plot the parabola and find regions where $y > 0$.

Absolute Values (Modulus)

Definition: $|x|$ represents the distance from the origin. $|x| = \begin{cases} x & \text{if } x > 0 \\ -x & \text{if } x < 0 \\ 0 & \text{if } x = 0 \end{cases}$

Properties:

1. $|-x| = |x|$

2. $|xy| = |x||y|$

3. $|x| < a \iff -a < x < a$

4. $|x| > a \iff x > a \text{ or } x < -a$

Example: Solve $|2x - 1| < 7$ $-7 < 2x - 1 < 7 \implies -6 < 2x < 8 \implies -3 < x < 4$.

RATIONAL FUNCTIONS AND PARTIAL FRACTIONS

Rational Functions

A rational function is defined as $f(x) = \frac{p(x)}{q(x)}$ where $p, q$ are polynomials and $q(x) \neq 0$.

• Domain: Entire real set except values where the denominator $q(x) = 0$.

• Range: Set of possible $y$ values; found by expressing $x$ in terms of $y$.

• Zeros: Found by setting the numerator $p(x) = 0$.

Partial Fractions

Resolution into partial fractions decomposes a compound rational function.

Proper vs. Improper Fractions

• Proper: Degree of numerator < Degree of denominator.

• Improper: Degree of numerator $\geq$ Degree of denominator. Must use long division first.

Rules for Partial Fractions

1. Type I (Linear Non-Repeated):    $\frac{px+q}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$

2. Type II (Linear Repeated):    $\frac{f(x)}{(x-a)^k} = \frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \dots + \frac{A_k}{(x-a)^k}$

3. Type III (Non-Factorizable Quadratic):    $\frac{f(x)}{(x-a)(x^2 + bx + c)} = \frac{A}{x-a} + \frac{Bx + C}{x^2 + bx + c}$

PRINCIPLE OF MATHEMATICAL INDUCTION

The Principle

To prove a statement $P(n)$ is true for all positive integers $n$:

1. Basis Step: Show $P(1)$ is true.

2. Inductive Hypothesis: Assume $P(k)$ is true for some arbitrary integer $k$.

3. Inductive Step: Prove that $P(k) \implies P(k+1)$.

Example: Sum of Series Prove $1 \cdot 2 + 2 \cdot 3 + \dots + n(n+1) = \frac{n(n+1)(n+2)}{3}$.

• For $n=1$: $1(2) = 2$. RHS: $\frac{1(2)(3)}{3} = 2$. True.

• Assume for $n=k$: $\sum_{i=1}^k i(i+1) = \frac{k(k+1)(k+2)}{3}$.

• For $n=k+1$: $\frac{k(k+1)(k+2)}{3} + (k+1)(k+2) = (k+1)(k+2) [\frac{k}{3} + 1] = \frac{(k+1)(k+2)(k+3)}{3}$. True.

Example: Divisibility Prove 9 is a factor of $5^{2n} + 3n - 1$.

• For $n=1$: $5^2 + 3(1) - 1 = 25 + 2 = 27$. $27 = 9 \times 3$. True.

• Step: Show $5^{2(k+1)} + 3(k+1) - 1$ is a multiple of 9 using hypothesis $5^{2k} = 9N - 3k + 1$.

PERMUTATIONS AND COMBINATIONS

Permutations ($nPr$)

An arrangement of objects in a definite order.

• Arrangement of $n$ distinct objects: $n! = n \times (n-1) \times \dots \times 1$.

• Arrangement of $r$ objects from $n$ different objects: $nPr = \frac{n!}{(n-r)!}$.

Special Cases

1. Permutations with Repetitions: If $r_1, r_2 \dots r_k$ are counts of identical objects, total arrangements = $\frac{n!}{r_1! r_2! \dots r_k!}$.    - e.g., MATHEMATICS (11 letters: 2M, 2A, 2T): $\frac{11!}{2!2!2!}$.

2. Circular Arrangements: Number of ways to arrange $n$ objects in a circle is $(n-1)!$.

Combinations ($nCr$)

A selection of objects where order does not matter. $nCr = \binom{n}{r} = \frac{n!}{r!(n-r)!}$

Properties:

• $nCr = nC_{n-r}$

• $nCr + nC_{r+1} = n+1 C_{r+1}$

• $n! = 0$ if $n < 0$.

THE BINOMIAL THEOREM

Positive Integral Index

$(a + b)^n = \sum_{r=0}^n nC_r a^{n-r} b^r$

• The term $T_{r+1} = nC_r a^{n-r} b^r$ is the general term.

• The expansion has $n+1$ terms.

• Coefficients are found using Pascal's Triangle.

Negative and Fractional Indices

If $|x| < 1$ and $q$ is any real number: $(1 + x)^q = 1 + qx + \frac{q(q-1)}{2!}x^2 + \frac{q(q-1)(q-2)}{3!}x^3 + \dots$

• The expansion is infinite if $q$ is not a positive integer.

• Range of Validity: The series converges only if $|x| < 1$.

SEQUENCES AND SERIES

Arithmetical Progression (A.P.)

Sequence: $a, a+d, a+2d, \dots$

• n-th term ($a_n$): $a + (n-1)d$

• Sum of first n terms ($s_n$): $s_n = \frac{n}{2}[2a + (n-1)d]$

• Arithmetic Mean of $a$ and $c$: $\frac{1}{2}(a + c)$.

Geometrical Progression (G.P.)

Sequence: $a, ar, ar^2, \dots$

• n-th term ($a_n$): $ar^{n-1}$

• Sum ($s_n$): $s_n = \frac{a(1-r^n)}{1-r}$ for $r < 1$; $s_n = \frac{a(r^n-1)}{r-1}$ for $r > 1$.

• Geometric Mean of $a$ and $c$: $\pm \sqrt{ac}$.

Harmonic Progression (H.P.)

A sequence where terms are the reciprocals of an A.P.

• n-th term: $a_n = \frac{1}{a + (n-1)d}$

• Harmonic Mean of $a$ and $c$: $\frac{2ac}{a+c}$.

Series Types and Finite Sums

1. Telescoping Series: $\sum_{k=1}^n [f(k) - f(k+1)] = f(1) - f(n+1)$.

2. Standard Sums:    - $\sum_{k=1}^n k = \frac{1}{2}n(n+1)$    - $\sum_{k=1}^n k^2 = \frac{1}{6}n(n+1)(2n+1)$    - $\sum_{k=1}^n k^3 = \frac{1}{4}n^2(n+1)^2$

COMPLEX NUMBERS

Algebra of Complex Numbers

Let $z = x + iy$, where $x$ is the real part and $y$ is the imaginary part.

• Addition: $(x_1 + iy_1) + (x_2 + iy_2) = (x_1+x_2) + i(y_1+y_2)$.

• Conjugate: $\bar{z} = x - iy$. Note: $z\bar{z} = x^2 + y^2$.

• Division: Multiply numerator and denominator by the conjugate of the denominator.

Argand Diagram and Polar Form

• Modulus ($r$): $|z| = \sqrt{x^2 + y^2}$.

• Argument ($\theta$): $\arg(z) = \tan^{-1}(\frac{y}{x})$.

• Polar Form: $z = r(\cos(\theta) + i\sin(\theta))$.

De Moivre’s Theorem

For any integer $n$: $[r(\cos(\theta) + i\sin(\theta))]^n = r^n(\cos(n\theta) + i\sin(n\theta))$

Roots of Complex Numbers: $\sqrt[n]{z} = \sqrt[n]{r} \left[ \cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right) \right]$ for $k = 0, 1, \dots, n-1$.

MATRICES AND DETERMINANTS

Matrix Algebra

• Equality: Two matrices are equal if they have the same size and identical corresponding entries.

• Multiplication ($AB$): Defined if columns of $A =$ rows of $B$. In general, $AB \neq BA$.

• Transpose ($A^T$): Rows become columns and columns become rows.

Special Matrices

• Identity Matrix ($I_n$): Diagonal entries are 1, others are 0.

• Invertible Matrix: $A$ is invertible if there exists a unique $A^{-1}$ such that $AA^{-1} = I$.

• Symmetric: $A = A^T$.

• Skew-Symmetric: $A^T = -A$.

Rank and Systems of Equations

• Row Reduced Echelon Form (RREF): Standard form for solving linear systems via elementary row operations.

• Rank ($R_A$): Number of non-zero rows in RREF.

• AX=B:   1. If $\text{rank}(A) < \text{rank}([A|B])$: Inconsistent (No solution).   2. If $\text{rank}(A) = \text{rank}([A|B]) = n$: Unique solution.   3. If $\text{rank}(A) = \text{rank}([A|B]) < n$: Infinite solutions.

Determinants and Cramer’s Rule

• Determinant ($|A|$): A scalar value. For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $|A| = ad - bc$.

• Adjoint Method: $A^{-1} = \frac{1}{|A|} \text{adj}(A)$, where $\text{adj}(A)$ is the transpose of the cofactor matrix.

• Cramer’s Rule: For a system $AX=B$, $x_i = \frac{|B_i|}{|A|}$, where $B_i$ is $A$ with the $i$-th column replaced by $B$.