Introduction to Algebra

Fundamentals of Algebra What is Algebra? Algebra is the branch of mathematics in which we use symbols—like $x$, $y$, or $z$—to represent numbers and unknown quantities. Rather than working with specific numbers, algebra allows us to work with general symbols that can stand for any number. This is what makes algebra so powerful: when you prove a relationship using symbols, that relationship holds true no matter what specific numbers you substitute in later. Think of it this way: if you discover that $3x + 5 = 11$ when $x = 2$, you've solved one specific problem. But if you develop a method to solve $3x + 5 = 11$ symbolically—showing that $x = 2$ works for any similar equation of this form—then you've created a general tool. Algebraic Expressions An algebraic expression is a combination of numbers, variables, and arithmetic operations. Here are some examples: $3x + 7$ $2x^2 - 5x + 3$ $\frac{a + b}{c}$ Each expression represents a quantity that depends on the values of the variables. The power of algebraic expressions is that they can be simplified, manipulated, and rearranged using universal arithmetic rules. For example, $2x + 3x = 5x$ is true regardless of what number $x$ represents. The Role of Variables A variable is a symbol that holds a place for a number. Variables are useful because they allow a single formula or expression to describe countless different situations. For instance, the formula $A = \pi r^2$ describes the area of any circle, not just one particular circle. The variable $r$ can take on any positive value. Variables can represent: Unknown values we need to find (like solving $x + 3 = 7$) Arbitrary numbers that vary across different situations General quantities in formulas that apply broadly Symbolic Manipulation Symbolic manipulation is the process of rearranging and simplifying algebraic expressions and equations using mathematical rules. This skill is fundamental to algebra because it lets us move from a specific problem to a general solution. When you master symbolic manipulation, you can: Simplify complex expressions Solve equations to find unknown values Derive formulas that work for entire classes of problems For example, learning how to solve $2x + 3 = 11$ symbolically teaches you a method you can apply to $2x + 3 = 25$, $2x + 3 = 100$, or any equation of that form. Solving Equations and Inequalities Linear Equations A linear equation is an equation where the variable appears only to the first power. The simplest and most common form is $ax + b = c$, where $a$, $b$, and $c$ are constants. Example: Solve $2x + 3 = 11$ To solve a linear equation, isolate the variable by undoing operations in reverse order: Subtract 3 from both sides: $2x = 8$ Divide both sides by 2: $x = 4$ The key principle is: whatever operation you perform on one side of the equation, you must perform on the other side. This keeps the equation balanced and maintains the equality. Quadratic Equations A quadratic equation has the form $ax^2 + bx + c = 0$, where $a \neq 0$. These equations are more complex than linear equations because they can have up to two solutions. There are several methods to solve quadratic equations: Method 1: Factoring If the quadratic can be factored into the form $(x - p)(x - q) = 0$, then the solutions are $x = p$ and $x = q$. Example: $x^2 - 5x + 6 = 0$ factors as $(x - 2)(x - 3) = 0$, so $x = 2$ or $x = 3$. Method 2: Completing the Square This method rewrites the left side as a perfect square. For example, $x^2 + 6x + 9 = (x + 3)^2$. By completing the square, you transform the equation into a form you can solve by taking square roots. Method 3: The Quadratic Formula The quadratic formula works for any quadratic equation: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ This formula tells you the exact solutions. The expression $b^2 - 4ac$ is called the discriminant: If the discriminant is positive, there are two real solutions If the discriminant is zero, there is one real solution If the discriminant is negative, there are no real solutions Example: For $2x^2 + 3x - 2 = 0$, we have $a = 2$, $b = 3$, $c = -2$. $$x = \frac{-3 \pm \sqrt{9 - 4(2)(-2)}}{2(2)} = \frac{-3 \pm \sqrt{9 + 16}}{4} = \frac{-3 \pm 5}{4}$$ So $x = \frac{1}{2}$ or $x = -2$. Inequalities An inequality is a statement that one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. Examples include $2x - 5 > 3$ or $x + 1 \leq 7$. Solving inequalities follows the same steps as solving equations, with one critical difference: When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. Example: Solve $-3x + 2 < 8$ Subtract 2 from both sides: $-3x < 6$ Divide both sides by $-3$ (and reverse the inequality): $x > -2$ Notice the $<$ became $>$ because we divided by a negative number. The solution to an inequality is typically a range of values rather than a single number. In this case, any number greater than $-2$ satisfies the inequality. <extrainfo> Higher-Degree Polynomial Equations Polynomial equations of degree three or higher (cubic, quartic, etc.) often cannot be solved using simple algebraic methods like factoring or formulas. Instead, they typically require: Factoring (if possible, by finding rational roots first) Synthetic division (to test potential roots) Numerical approximation methods (for complex or unsolvable cases) These techniques are more advanced and may not be central to a foundational algebra course. </extrainfo> Functions Definition and Notation A function is a rule that assigns exactly one output to each input from a specified set. Think of a function as a machine: you put in a number, and the machine produces exactly one output. Functions are denoted as $f(x)$, which reads as "$f$ of $x$." Here, $x$ is the input variable and $f(x)$ represents the output. For example, if $f(x) = 2x + 1$, then $f(3) = 2(3) + 1 = 7$. The key requirement for a function is that each input produces exactly one output. An equation like $y^2 = x$ is not a function because one input ($x = 4$) produces two outputs ($y = 2$ and $y = -2$). Representations of Functions Functions can be represented in three main ways: 1. Algebraic Formula The most common representation: $f(x) = 3x^2 - 2x + 5$ 2. Table of Ordered Pairs A list of input-output pairs: | $x$ | 1 | 2 | 3 | 4 | |-----|---|---|---|---| | $f(x)$ | 6 | 11 | 18 | 27 | 3. Graph in the Coordinate Plane A visual representation where the horizontal axis is the input ($x$) and the vertical axis is the output ($f(x)$ or $y$). All three representations describe the same function—just in different formats. Domain and Range The domain of a function is the set of all permissible input values—that is, all values of $x$ for which the function is defined. The range of a function is the set of all possible output values that the function can produce. Examples: For $f(x) = \sqrt{x}$, the domain is $x \geq 0$ (you can't take the square root of a negative number), and the range is also $y \geq 0$ (square roots produce non-negative outputs). For $f(x) = \frac{1}{x}$, the domain is all real numbers except $x = 0$ (division by zero is undefined), and the range is all real numbers except $y = 0$ (the function never produces zero). When you're given a real-world problem, the domain is often restricted by the context. For instance, if $h(t)$ represents the height of a ball in flight, the domain is typically restricted to $t \geq 0$ since time cannot be negative. Functional Behavior: Growth, Intercepts, and Symmetry Understanding how a function behaves helps you interpret graphs and predict outcomes. Growth Growth describes whether a function's outputs increase or decrease as inputs increase: A function is increasing on an interval if $f(a) < f(b)$ whenever $a < b$ A function is decreasing on an interval if $f(a) > f(b)$ whenever $a < b$ For linear functions like $f(x) = 2x + 1$, the constant slope tells you the growth pattern immediately. For more complex functions, you often need to examine the graph or use calculus. Intercepts Intercepts are points where the graph crosses the axes: The $x$-intercept (or zero of the function) occurs when $f(x) = 0$. This is where the graph crosses the $x$-axis. The $y$-intercept occurs when $x = 0$, which is the output $f(0)$. These intercepts are useful because they give you key reference points on the graph. Symmetry Some functions have special symmetry properties: An even function satisfies $f(-x) = f(x)$, meaning the graph is symmetric about the $y$-axis. Example: $f(x) = x^2$ (since $(-x)^2 = x^2$) An odd function satisfies $f(-x) = -f(x)$, meaning the graph is symmetric about the origin. Example: $f(x) = x^3$ (since $(-x)^3 = -x^3$) Recognizing symmetry helps you sketch graphs and understand function behavior. Polynomials and Their Properties What is a Polynomial? A polynomial is an algebraic expression consisting of a sum of terms, where each term is a constant multiplied by a variable raised to a non-negative integer exponent. General form: $an x^n + a{n-1} x^{n-1} + \cdots + a1 x + a0$ Examples: $5x^3 + 2x^2 - 7x + 4$ (a polynomial) $3x^2 - 2x + \sqrt{5}$ (a polynomial; coefficients can be irrational) $2x^{-1} + 3$ (NOT a polynomial; exponents must be non-negative) $\sqrt{x} + 1$ (NOT a polynomial; $\sqrt{x} = x^{1/2}$ uses a non-integer exponent) Standard Form and Degree The standard form of a polynomial arranges terms from highest to lowest exponent. For example: $$4x^3 + 2x^2 - 7x + 5$$ The degree of a polynomial is the highest exponent of its variable. In the example above, the degree is 3 (this would be called a cubic polynomial). The leading coefficient is the coefficient of the highest-degree term. In $4x^3 + 2x^2 - 7x + 5$, the leading coefficient is 4. Understanding degree and leading coefficient is important because they influence the polynomial's overall behavior and graph shape. Zeroes and Factoring A zero (or root) of a polynomial is a value of the variable that makes the polynomial equal to zero. For instance, if $f(x) = x^2 - 5x + 6$, then $x = 2$ is a zero because $f(2) = 4 - 10 + 6 = 0$. Factoring a polynomial means expressing it as a product of lower-degree polynomials. Factoring reveals the zeroes directly because if a polynomial factors as $(x - 2)(x - 3)$, then the zeroes are $x = 2$ and $x = 3$ (the values that make each factor zero). Example: $x^2 - 5x + 6 = (x - 2)(x - 3)$, so the zeroes are $x = 2$ and $x = 3$. Finding zeroes is essential because they reveal where the graph crosses the $x$-axis and give insight into the polynomial's overall structure. End Behavior The end behavior of a polynomial describes what happens to the $y$-values as $x$ approaches positive or negative infinity. End behavior is determined by: The degree of the polynomial The sign of the leading coefficient (positive or negative) For even-degree polynomials (like degree 2, 4, 6): If the leading coefficient is positive, the graph rises on both ends (looks like an upside-down U) If the leading coefficient is negative, the graph falls on both ends (looks like a regular U) For odd-degree polynomials (like degree 1, 3, 5): If the leading coefficient is positive, the graph falls on the left and rises on the right If the leading coefficient is negative, the graph rises on the left and falls on the right Understanding end behavior helps you sketch polynomial graphs and predict long-term behavior. Systems of Equations What is a System of Equations? A system of equations consists of two or more equations involving the same variables that you consider simultaneously. You're looking for the value(s) of the variables that satisfy all equations at once. Example: $$\begin{cases} 2x + y = 7 \\ x - y = 2 \end{cases}$$ The solution is the ordered pair $(x, y)$ that makes both equations true. In this example, $x = 3$ and $y = 1$ works: $2(3) + 1 = 7$ ✓ and $3 - 1 = 2$ ✓ The Substitution Method The substitution method solves one equation for one variable, then substitutes that expression into the other equation(s). Step-by-step for the example above: Solve the second equation for $x$: $x = y + 2$ Substitute into the first equation: $2(y + 2) + y = 7$ Simplify: $2y + 4 + y = 7 \Rightarrow 3y = 3 \Rightarrow y = 1$ Back-substitute: $x = 1 + 2 = 3$ The substitution method works well when one equation can be easily solved for one variable. The Elimination Method The elimination method adds or subtracts equations (or multiplied versions of them) to cancel out one variable, leaving an equation with only one unknown. Using the same example: Write both equations: $2x + y = 7$ $x - y = 2$ Notice that the $y$ terms already have opposite coefficients (+1 and -1). Add the equations: $(2x + y) + (x - y) = 7 + 2$ $3x = 9$ $x = 3$ Substitute back into either original equation to find $y = 1$ The key insight is that if you multiply equations by constants, you can create opposite coefficients for any variable, allowing you to eliminate it. Consistency and Types of Solutions Not all systems have solutions. Understanding what solutions look like is critical: Consistent vs. Inconsistent: A consistent system has at least one solution (the equations are compatible) An inconsistent system has no solution (the equations contradict each other) Types of Consistent Systems: Independent system: exactly one solution (the equations represent different lines that intersect at one point) Dependent system: infinitely many solutions (the equations represent the same line or are scalar multiples of each other) Example of an inconsistent system: $$\begin{cases} x + y = 5 \\ x + y = 7 \end{cases}$$ No values of $x$ and $y$ satisfy both equations simultaneously, so the system is inconsistent. Example of a dependent system: $$\begin{cases} x + y = 5 \\ 2x + 2y = 10 \end{cases}$$ The second equation is just 2 times the first, so they describe the same line. Every point on the line is a solution, giving infinitely many solutions. <extrainfo> Matrix Methods Matrix methods, such as Gaussian elimination, represent a linear system as an augmented matrix and use row operations to simplify it to a form where solutions are apparent. For the system: $$\begin{cases} 2x + y = 7 \\ x - y = 2 \end{cases}$$ You'd write the augmented matrix: $$\begin{bmatrix} 2 & 1 & | & 7 \\ 1 & -1 & | & 2 \end{bmatrix}$$ Then perform row operations to reduce this to a simpler form. This method is systematic and particularly useful for larger systems, but it requires careful bookkeeping of row operations. </extrainfo> Exponential and Logarithmic Relationships Exponential Functions An exponential function has the form: $$f(x) = a \cdot b^x$$ where $a$ is an initial value (often called the initial amount), $b$ is the base (a positive constant, typically $b > 1$), and $x$ is the exponent. Key characteristics: If $b > 1$, the function grows exponentially (like population growth or compound interest) If $0 < b < 1$, the function decays exponentially (like radioactive decay) The graph passes through the point $(0, a)$ because $f(0) = a \cdot b^0 = a$ Example: $f(x) = 100 \cdot 2^x$ models a quantity that doubles with each unit increase in $x$. At $x = 0$, you have 100 units. At $x = 1$, you have 200. At $x = 2$, you have 400. Properties of Exponents Understanding exponent properties is essential for working with exponential functions. Product Rule: When multiplying powers with the same base, add the exponents. $$b^m \cdot b^n = b^{m+n}$$ Example: $2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128$ Quotient Rule: When dividing powers with the same base, subtract the exponents. $$\frac{b^m}{b^n} = b^{m-n}$$ Example: $\frac{2^5}{2^2} = 2^{5-2} = 2^3 = 8$ Power of a Power Rule: When raising a power to a power, multiply the exponents. $$(b^m)^n = b^{mn}$$ Example: $(2^3)^2 = 2^{3 \cdot 2} = 2^6 = 64$ Zero Exponent: Any non-zero number raised to the power zero equals one. $$b^0 = 1$$ Negative Exponent: A negative exponent indicates a reciprocal. $$b^{-n} = \frac{1}{b^n}$$ Example: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ These properties allow you to simplify exponential expressions and solve exponential equations. Logarithmic Functions A logarithm is the inverse of an exponential function. If you're solving an exponential equation like $2^x = 8$, you need logarithms. The logarithmic function is written as: $$\logb(y) = x$$ This reads as "log base $b$ of $y$ equals $x$" and means "what power must you raise $b$ to in order to get $y$?" The relationship between exponential and logarithmic forms is: $$\logb(y) = x \iff b^x = y$$ Examples: $\log2(8) = 3$ because $2^3 = 8$ $\log{10}(1000) = 3$ because $10^3 = 1000$ $\log3(1) = 0$ because $3^0 = 1$ Key Properties: $\logb(1) = 0$ (since $b^0 = 1$) $\logb(b) = 1$ (since $b^1 = b$) $\logb(x \cdot y) = \logb(x) + \logb(y)$ (product rule) $\logb\left(\frac{x}{y}\right) = \logb(x) - \logb(y)$ (quotient rule) $\logb(x^n) = n \logb(x)$ (power rule) These properties make logarithms invaluable for solving exponential equations. <extrainfo> Change-of-Base Formula The change-of-base formula allows you to convert logarithms with one base to logarithms with a different base: $$\logb(y) = \frac{\logk(y)}{\logk(b)}$$ where $k$ is any positive base, typically 10 (common logarithm) or $e$ (natural logarithm). This is useful because calculators typically only have buttons for $\log{10}$ and $\loge$ (denoted $\ln$). If you need $\log3(7)$, you'd compute: $$\log3(7) = \frac{\log{10}(7)}{\log{10}(3)} \approx \frac{0.845}{0.477} \approx 1.77$$ Applications Exponential and logarithmic relationships model real-world phenomena: Population growth: $P(t) = P0 \cdot e^{rt}$ where $r$ is the growth rate Radioactive decay: $N(t) = N0 \cdot \left(\frac{1}{2}\right)^{t/h}$ where $h$ is the half-life Compound interest: $A(t) = P\left(1 + \frac{r}{n}\right)^{nt}$ Understanding these models helps interpret real-world data and make predictions. </extrainfo>