Comprehensive Algebra I Study Guide

Foundations and Definitions of Algebra

• Conceptual Overview: Algebra acts as a system for generalizing arithmetic. It utilizes variables (letters representing numbers), formulas, and operations to analyze relationships between values and solve problems, ranging from practical business applications to theoretical puzzles.
• Historical Context:
  • Babylonians (2000 B.C.): Solved three-term quadratic equations.
  • Egyptians: Focused on linear equations. The Rhind Mathematical Papyrus (preserved in the British Museum) contains early problems where the unknown was designated by the word "aha". An example problem is $x + \frac{x}{7} = 19$, yielding $x = 16\, \frac{5}{8}$.
  • Hindus (6th–7th Century A.D.): Brahmagupta provided solutions to quadratic equations and early interpretations of $0$. They were among the first to treat irrational numbers as actual numbers.
  • Al-Khowarizmi (A.D. 825): Wrote the first algebra textbook in Baghdad.
  • Diophantus: Often called the "father of algebra," he was the first to use symbols systematically to abbreviate mathematical thoughts.
• Number Sets:
  • Natural Numbers (Counting Numbers): Start at $1$ and increase by ones ($1, 2, 3, 4, \dots$).
  • Whole Numbers: Natural numbers plus zero ($0, 1, 2, 3, \dots$).
  • Integers: Positive and negative whole numbers and zero ($\dots, -3, -2, -1, 0, 1, 2, 3, \dots$).
  • Rational Numbers: Any number that can be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$. Their decimal forms either terminate (e.g., $3.4$) or repeat (e.g., $0.164164\dots$).
  • Irrational Numbers: Numbers that cannot be written as fractions; their decimals are non-repeating and non-terminating (e.g., $\pi$, $\sqrt{2}$).
  • Real Numbers: The set containing all rational and irrational numbers.
  • Prime Numbers: Whole numbers greater than $1$ divisible only by $1$ and themselves. The only even prime is $2$.
  • Composite Numbers: Numbers that are not prime and can be divided by at least one number other than $1$ and itself.
• Algebraic Terminology:
  • Expression: A combination of values and operations (e.g., $2x^2 + 4x$).
  • Term: Groups of factors connected by multiplication/division but separated by addition/subtraction (e.g., $3xy + 5x - 6$ has three terms).
  • Equation: Uses an equal sign ($=$) to show a relationship of equality between two expressions.
  • Variable: A letter (typically from the end of the alphabet like $x, y, z$) representing an unknown number.
  • Constant: A value that never changes in an equation; often represented by the first letters of the alphabet ($a, b, c$).
  • Coefficient: A number written in front of a variable (e.g., in $3x$, $3$ is the coefficient).

Basic Operations and Properties

• Types of Operations:
  • Binary Operations: Involve two numbers (Addition, Subtraction, Multiplication, Division).
  • Non-Binary Operations: Require only one number (Square roots, Absolute value, Factorials).
• Special Operations:
  • Absolute Value ($|a|$): Represents the undirected distance from zero. $|a| = a$ if $a \ge 0$; $|a| = -a$ if $a < 0$.
  • Factorial ($n!$): The product of $n$ and every positive integer smaller than it ($n! = n(n-1)(n-2)\dots 3 \cdot 2 \cdot 1$). By definition, $0! = 1$.
  • Greatest Integer Function: Changes a non-integer real number to the greatest integer it exceeds (rounding up).
• Rules for Signed Numbers:
  • Addition (Same Signs): Find the sum and keep the sign. $(+a) + (+b) = +(a+b)$; $(-a) + (-b) = -(a+b)$.
  • Addition (Different Signs): Find the difference of the absolute values; the sign is determined by the number farther from zero.
  • Subtraction: Change the minus to plus and change the sign of the following number. $(+a) - (-b) = (+a) + (+b)$.
  • Multiplication/Division: Like signs yield positive results; different signs yield negative results.
  • Even-Odd Rule: In a string of multiplication/division, an even number of negatives results in a positive; an odd number results in a negative.
• Algebraic Properties:
  • Commutative Property: The order of numbers does not affect the result in addition or multiplication ($a + b = b + a$; $a \cdot b = b \cdot a$). It does not apply to subtraction or division.
  • Associative Property: The grouping of numbers does not affect the result ($a + (b + c) = (a + b) + c$; $a(bc) = (ab)c$). It does not apply to subtraction or division.
• Operations with Zero:
  • $0 + a = a$; $a - 0 = a$; $0 - a = -a$.
  • $a \cdot 0 = 0$.
  • $0 \div a = 0$.
  • Division by Zero: Numbers cannot be divided by zero; the result is undefined.

Working with Fractions, Decimals, and Exponents

• Fraction Types:
  • Proper Fractions: Numerator is smaller than the denominator (value $< 1$).
  • Improper Fractions: Numerator is larger than the denominator (value $> 1$).
  • Mixed Numbers: Consist of a whole number and a proper fraction.
• Operations on Fractions:
  • Reduction: A fraction is in lowest terms if the numerator and denominator are Relatively Prime (GCF is $1$).
  • Common Denominators: Required for adding/subtracting. To find the Least Common Denominator (LCD), check multiples of the largest denominator until the smaller denominator divides in evenly.
  • Multiplication: Multiply numerators together and denominators together. Change mixed numbers to improper fractions first.
  • Division: Multiply the first fraction by the reciprocal (flip) of the second fraction.
• Rounding Numbers: Determine the target place; look one digit to the right. If it is $\ge 5$, increase the target digit by $1$. Else, leave it.
• Decimal to Fraction Conversion:
  • Terminating: Put digits over a power of $10$ matching the number of places (e.g., $0.36 = \frac{36}{100}$).
  • Repeating: Put repeating digits over a denominator of as many nines as there are repeating digits (e.g., $0.126126\dots = \frac{126}{999}$).
• Laws of Exponents:
  • Multiplication: Same bases, add exponents ($x^a \cdot x^b = x^{a+b}$).
  • Division: Same bases, subtract exponents ($x^a \div x^b = x^{a-b}$).
  • Power of Zero: $x^0 = 1$ (provided $x \neq 0$).
  • Negative Exponents: $x^{-a} = \frac{1}{x^a}$.
  • Power of a Power: $(x^n)^m = x^{nm}$.
  • Fractional Exponents: $\sqrt[n]{a} = a^{1/n}$; $\sqrt[n]{a^m} = a^{m/n}$.
• Scientific Notation: Written as $N \times 10^a$, where $1 \le N < 10$ and $a$ is an integer. Positive $a$ moved the decimal left; negative $a$ moved it right.

The Principles of Factoring and Distribution

• Distribution Rules:
  • Basic: $a(b + c) = ab + ac$.
  • Polynomial x Polynomial: Multiply each term in the first factor by each term in the second factor.
  • FOIL Method (First, Outer, Inner, Last): Specific to binomial multiplication $(a+b)(c+d)$.
• Special Products:
  • Perfectly Squared Binomial: $(a + b)^2 = a^2 + 2ab + b^2$.
  • Sum and Difference Product: $(a + b)(a - b) = a^2 - b^2$.
  • Sum/Difference of Two Cubes:
    • $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.
    • $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.
• Factoring Techniques:
  • Greatest Common Factor (GCF): The largest value that divides all terms ($ab + ac + ad = a(b + c + d)$).
  • Factoring by Grouping: Used for four or more terms; group into equal sets and pull out common binomial factors.
  • unFOIL: Systematic guessing and checking for trinomials in form $ax^2 + bx + c$.
  • Remainder Theorem: The remainder from dividing a polynomial $P(x)$ by $x + a$ is $P(-a)$.
  • Synthetic Division: A shortcut for polynomial division using only coefficients and the opposite of the constant in a linear divisor.
  • Rational Root Theorem: Possible rational solutions are $\pm \frac{\text{factors of the constant}}{\text{factors of the lead coefficient}}$.

Solving Equations and Inequalities

• Linear Equations: Standard form is $ax + b = c$. Variables are raised to the first power. Solve by reversing the order of operations: subtract/add first, then multiply/divide.
• Quadratic Equations: Standard form is $ax^2 + bx + c = 0$.
  • Methods: Square-root rule (if $b=0$), factoring (using the Multiplication Property of Zero), or the Quadratic Formula.
  • Multiplication Property of Zero (MPZ): If $pq = 0$, then $p = 0$ or $q = 0$.
  • Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• Cubic and Higher-Degree Equations: Max number of solutions is equal to the degree. Methods include finding integer factors, GCF extraction, grouping, or synthetic division.
• Quadratic-Like Equations: Equations in form $ax^{2n} + bx^n + c = 0$. Use substitution (e.g., $q = x^n$) to reduce to quadratic form.
• Radical Equations: Isolate the radical and raise both sides to the power of the root. Check for Extraneous Solutions (answers that solve the derived equation but not the original).
• Inequalities:
  • Notation: $<, >, \le, \ge$. In interval notation, $( )$ indicates non-inclusive limits, and $[ ]$ indicates inclusive limits.
  • Rule of Negatives: If you multiply or divide an inequality by a negative number, the direction of the symbol must be reversed.
  • Quadratic Inequalities: Move all terms to one side, factor, find "critical numbers" where the expression equals 0, and test the signs in intervals on a number line.
  • Rational Inequalities: Critical numbers include those making the numerator zero AND those making the denominator zero (denominator zeros are always non-inclusive).
  • Absolute-Value Inequalities:
    • $|ax + b| > c$ is equivalent to $ax + b > c$ OR $ax + b < -c$.
    • $|ax + b| < c$ is equivalent to $-c < ax + b < c$.

Applications and Geometry Formulas

• Measurement Conversions:
  • $12\, \text{inches} = 1\, \text{foot}$.
  • $3\, \text{feet} = 1\, \text{yard}$.
  • $5,280\, \text{feet} = 1\, \text{mile}$.
• Geometry Formulas:
  • Perimeter: Square ($P = 4s$); Rectangle ($P = 2l + 2w$); Triangle ($P = s_1 + s_2 + s_3$); Circle Circumference ($C = 2\pi r = \pi d$).
  • Area: Square ($A = s^2$); Rectangle ($A = lw$); Triangle ($A = \frac{1}{2}bh$); Heron’s Formula for any triangle ($A = \sqrt{s(s-a)(s-b)(s-c)}$ where $s$ is semi-perimeter); Circle ($A = \pi r^2$).
  • Volume: Prism/Box ($V = lwh$); Cylinder ($V = \pi r^2h$); Pyramid ($V = \frac{1}{3}Bh$); Cone ($V = \frac{1}{3}\pi r^2h$); Sphere ($V = \frac{4}{3}\pi r^3$).
  • Pythagorean Theorem: In any right triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.
• Story Problem Types:
  • Distance: $d = rt$ (Distance = rate $\times$ time).
  • Mixture: Strength $\times$ Quantity + Strength $\times$ Quantity = Final Strength $\times$ Total Quantity.
  • Interest:
    • Simple Interest: $I = Prt$.
    • Compound Interest: $A = P(1 + \frac{r}{n})^{nt}$.
  • Taxes/Discounts:
    • Total Price = $\text{Price} \times (1 + \text{tax rate})$.
    • Discounted Price = $\text{Original Price} \times (1 - \text{discount rate})$.
• Probability and Counting:
  • Permutations: Order matters. $P(n, r) = \frac{n!}{(n-r)!}$.
  • Combinations: Order does not matter. $C(n, r) = \frac{n!}{r!(n-r)!}$.

Graphing and Coordinate Geometry

• Coordinate Plane: Divided by two axes (x-axis and y-axis) into four numbered quadrants (I, II, III, IV counterclockwise from upper right). The Origin is $(0,0)$.
• Points: Represented as Ordered Pairs $(x, y)$.
• Lines:
  • Intercepts: x-intercept where $y=0$; y-intercept where $x=0$.
  • Slope ($m$): A measure of steepness. $m = \frac{y_2 - y_1}{x_2 - x_1}$.
  • Horizontal Lines: $y = c$ (slope $= 0$).
  • Vertical Lines: $x = c$ (slope is undefined).
  • Slope-Intercept Form: $y = mx + b$ (where $m$ is slope and $b$ is y-intercept).
  • Parallel Lines: Slopes are equal ($m_1 = m_2$).
  • Perpendicular Lines: Slopes are negative reciprocals ($m_1 = -\frac{1}{m_2}$).
  • Intersections: Found by substitution or graphing to find where coordinates satisfy both equations.
• Parabolas: Graphs of quadratic equations. The Vertex is the extreme point.
  • Multiplying $x^2$ by a number $> 1$ makes it steeper; proper fractions make it flatter.
  • $y = x^2 + k$ moves vertex up/down.
  • $y = (x - h)^2$ moves vertex right ($h > 0$).

The Part of Tens

• Famous Equations:
  • Relativity: $E = mc^2$ (Energy, mass, speed of light).
  • The Value of $e$: Euler's number ($\approx 2.718$). Formula: $(1 + \frac{1}{x})^x$ as $x \rightarrow \infty$.
  • Newton's Gravity: $F = G \frac{m_1 m_2}{d^2}$.
  • Euler’s Identity: $e^{i\pi} + 1 = 0$.
  • Fermat’s Last Theorem: $a^n + b^n = c^n$ has no integer solutions for $n > 2$.
• Ways to Avoid Pitfalls:
  • Middle Terms: Don't forget $2ab$ in $(a+b)^2$.
  • Radicals: You cannot split $\sqrt{a+b}$.
  • Exponents: Never multiply bases together (e.g., $2^3 \cdot 2^4 = 2^7$, not $4^7$).