Comprehensive Study Guide: College Algebra Core Topics and Practice

Polynomial and Rational Equations

Solving Polynomial Equations

Polynomial equations are equations involving expressions with powers of x. Solving these equations often involves factoring, using the quadratic formula, or other algebraic techniques.

• Quadratic Equation: An equation of the form $ax^2 + bx + c = 0$.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Factoring: Expressing a polynomial as a product of its factors to find its zeros.

• Example: Solve $x^2 - 5x + 7x - 35 = 0$ by combining like terms and factoring.

Solving Radical Equations

Radical equations contain variables inside a root. To solve, isolate the radical and raise both sides to the appropriate power.

• Example: $\sqrt{x^2 - 4} - \sqrt{x - 2} = 2$

• Isolate one radical, square both sides, and solve for x. Always check for extraneous solutions.

Solving Polynomial and Rational Inequalities

These inequalities involve finding the set of x values that satisfy a polynomial or rational expression being greater or less than zero.

• Test Intervals: Find zeros, plot them on a number line, and test intervals between zeros.

• Example: Solve $x^2 + 4x < 8$ by rearranging to $x^2 + 4x - 8 < 0$ and factoring.

Absolute Value Inequalities

Solving Absolute Value Inequalities

Absolute value inequalities involve expressions like $|x - a| < b$ or $|x - a| > b$. They are solved by considering the definition of absolute value.

• For $|x - a| < b$: $-b < x - a < b$

• For $|x - a| > b$: $x - a < -b$ or $x - a > b$

• Example: $|2x + 3| < 8$ leads to $-8 < 2x + 3 < 8$

• Express solutions in set-builder, interval notation, and on a number line.

Algebraic Functions

Definition and Identification of Functions

A function is a relation where each input (x) has exactly one output (y). To determine if an equation is a function, check if for every x there is only one y.

• Vertical Line Test: A graph represents a function if no vertical line intersects it more than once.

• Example: $y = \sqrt{x - 3}$ is a function, but $x = y^2$ is not (since one x can have two y values).

Symmetry of Functions

Functions can be symmetric with respect to the y-axis (even), the origin (odd), or neither.

• Even Function: $f(-x) = f(x)$ (symmetric about the y-axis)

• Odd Function: $f(-x) = -f(x)$ (symmetric about the origin)

• Example: $y = x^2$ is even; $y = x^3$ is odd.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain.

• Example: $f(x) = \begin{cases} x^2 - 1 & \text{if } x < 1 \\ 2x + 1 & \text{if } x \geq 1 \end{cases}$

• State the domain and range using interval notation.

Domain and Range

The domain is the set of all possible input values (x), and the range is the set of all possible output values (y).

• Express domain and range in set-builder and interval notation.

• Example: For $f(x) = \sqrt{x - 2}$, domain is $[2, \infty)$.

Operations with Functions

Functions can be added, subtracted, multiplied, divided, and composed.

• Composition: $(f \circ g)(x) = f(g(x))$

• Example: If $f(x) = x^2$ and $g(x) = x + 1$, then $(f \circ g)(x) = (x + 1)^2$

Inverse Functions

The inverse of a function reverses the roles of x and y. Not all functions have inverses.

• Finding the Inverse: Solve $y = f(x)$ for $x$ in terms of $y$, then swap $x$ and $y$.

• Example: $f(x) = 2x + 3$; inverse is $f^{-1}(x) = \frac{x - 3}{2}$

Quadratic Functions and Graphs

Standard Form and Vertex

The standard form of a quadratic is $y = ax^2 + bx + c$. The vertex is the maximum or minimum point.

• Vertex Formula: $x = -\frac{b}{2a}$

• Axis of Symmetry: The vertical line $x = -\frac{b}{2a}$

• Example: For $y = x^2 - 10x + 13$, vertex at $x = 5$

Graphing Circles

The equation of a circle in standard form is $(x - h)^2 + (y - k)^2 = r^2$.

• Center: $(h, k)$

• Radius: $r$

• Example: $x^2 + (y - 1)^2 = 9$ has center $(0, 1)$ and radius $3$

Polynomial Division and Zeros

Long Division and Synthetic Division

Polynomials can be divided using long division or synthetic division to find quotients and remainders.

• Long Division: Similar to numerical long division, but with polynomials.

• Synthetic Division: A shortcut for dividing by linear factors of the form $x - c$.

Finding Zeros and Factor Theorem

The zeros of a polynomial are the values of x that make the polynomial zero. The Factor Theorem states that if $f(c) = 0$, then $(x - c)$ is a factor.

• Remainder Theorem: The remainder of $f(x)$ divided by $x - c$ is $f(c)$.

• Example: If $f(3) = 0$, then $x - 3$ is a factor of $f(x)$.

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions have the form $f(x) = a \cdot b^x$, where $a \neq 0$ and $b > 0$, $b \neq 1$.

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$ for $a > 0$

• Horizontal Asymptote: $y = 0$

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The general form is $f(x) = \log_b x$.

• Domain: $(0, \infty)$

• Range: $(-\infty, \infty)$

• Key Properties: $\log_b (xy) = \log_b x + \log_b y$, $\log_b (x^k) = k \log_b x$

Solving Exponential and Logarithmic Equations

• To solve $a^x = b$, take the logarithm of both sides: $x = \log_a b$

• To solve $\log_b x = c$, rewrite as $x = b^c$

• Example: $2^x = 8 \implies x = 3$

Applications

Variation Problems

Variation describes how one quantity changes with another. Types include direct, inverse, and joint variation.

• Direct Variation: $y = kx$

• Inverse Variation: $y = \frac{k}{x}$

• Joint Variation: $y = kxz$

• Example: If $y$ varies directly as $x$ and $y = 10$ when $x = 2$, then $k = 5$ and $y = 5x$

Exponential Growth and Decay

Exponential growth and decay are modeled by $A = A_0 e^{kt}$, where $k > 0$ for growth and $k < 0$ for decay.

• Half-life: The time it takes for a substance to decay to half its original amount.

• Example: $A = 100 e^{-0.01t}$ models decay with a rate of $-0.01$

Summary Table: Types of Variation

Type	Equation	Description
Direct	$y = kx$	y increases as x increases
Inverse	$y = \frac{k}{x}$	y decreases as x increases
Joint	$y = kxz$	y varies with both x and z

Additional info:

• Some problems require graphing and visual analysis, such as determining symmetry and function properties from graphs.

• Piecewise and step functions are included, requiring knowledge of domain, range, and interval notation.

• Application problems cover compound interest, population growth, and radioactive decay, which are standard in College Algebra.