Precalculus Exam 3 Review: Key Concepts and Practice Problems

Complex Numbers

Addition and Subtraction of Complex Numbers

Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined by $i^2 = -1$. To add or subtract complex numbers, combine the real parts and the imaginary parts separately.

• Key Point: $(a + bi) + (c + di) = (a + c) + (b + d)i$

• Example: $(-5 + 7i) + (-8 - 11i) = (-5 - 8) + (7 - 11)i = -13 - 4i$

Multiplication of Complex Numbers

To multiply complex numbers, use the distributive property (FOIL method) and simplify using $i^2 = -1$.

• Key Point: $(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$

• Example: $(5 + 3i)(2 - i) = 5 imes 2 + 5 imes (-i) + 3i imes 2 + 3i imes (-i) = 10 - 5i + 6i - 3i^2 = 10 + i + 3 = 13 + i$

Division of Complex Numbers (Standard Form)

To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator and simplify.

• Key Point: $rac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}$

• Example: $\frac{3}{2 - i}$ Multiply numerator and denominator by the conjugate $(2 + i)$: $\frac{3}{2 - i} \times \frac{2 + i}{2 + i} = \frac{3(2 + i)}{(2 - i)(2 + i)} = \frac{6 + 3i}{4 + 2i - 2i - i^2} = \frac{6 + 3i}{4 - (-1)} = \frac{6 + 3i}{5} = \frac{6}{5} + \frac{3}{5}i$

Quadratic Equations

Solving Quadratics Using the Quadratic Formula

The quadratic formula solves equations of the form $ax^2 + bx + c = 0$.

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

• Example: For $2x^2 + 2x + 3 = 0$, $a = 2$, $b = 2$, $c = 3$.

Polynomial Division

Long Division of Polynomials

Polynomial long division is used to divide a polynomial by another polynomial, similar to numerical long division.

• Key Point: Divide the highest degree term, multiply, subtract, and repeat.

• Example: Divide $x^5 - x^4 - 2x^3 + 2x + 3$ by $x^2 + 1$.

Functions and Their Properties

Function Evaluation

To evaluate a function, substitute the given input value into the function's formula.

• Key Point: If $f(x)$ is defined, $f(a)$ is found by replacing $x$ with $a$.

• Example: If $f(x)$ is graphed, use the graph to find $f(-1)$ or $f(0)$.

Domain and Range

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

• Key Point: For rational functions, exclude values that make the denominator zero.

• Example: For $f(x) = \frac{3}{x-1}$, domain is $x \neq 1$.

Composite Functions

A composite function $f(g(x))$ is formed by substituting $g(x)$ into $f(x)$.

• Key Point: $f(g(x))$ means apply $g$ first, then $f$.

• Example: If $f(x) = \frac{3}{x-1}$ and $g(x) = \frac{2x-1}{x}$, then $f(g(x)) = \frac{3}{\frac{2x-1}{x} - 1}$

Relations and One-to-One Functions

Domain, Range, and One-to-One

A relation is a set of ordered pairs. A function is one-to-one if each output is paired with only one input.

• Key Point: A function is one-to-one if no y-value is repeated for different x-values.

• Example: For the relation {(-3, -1), (3, 4), (-5, -7), (5, -1)}, list domain and range, and check for one-to-one property.

Inverse Functions

Graphing and Finding Inverses

The inverse of a function "undoes" the original function. The graph of the inverse is a reflection across the line $y = x$.

• Key Point: To find the inverse algebraically, solve for $x$ in terms of $y$ and swap variables.

• Example: For $f(x) = \frac{2x+1}{1-x}$, solve for $x$ in terms of $y$ to find $f^{-1}(x)$.

Inequalities

Solving Polynomial and Rational Inequalities

To solve inequalities, find the zeros of the expression, test intervals, and use a sign chart to determine where the inequality holds.

• Key Point: For $x^2 - 7x^2 - 18x > 0$, factor and solve for intervals.

• Key Point: For $\frac{3x^2 + 10x - 8}{x-1} \geq 0$, find critical points and test intervals.

Polynomial Functions and Roots

Graph Analysis

Analyzing the graph of a polynomial function helps determine where the function is positive, negative, or zero.

• Key Point: Identify intervals where $f(x) < 0$ or $f(x) > 0$ from the graph.

Possible Roots of Polynomials

Use the Rational Root Theorem and Descartes' Rule of Signs to determine possible positive, negative, and rational roots.

• Key Point: Rational Root Theorem: Possible rational roots are $\pm \frac{\text{factors of constant}}{\text{factors of leading coefficient}}$.

• Key Point: Descartes' Rule of Signs: The number of sign changes in $f(x)$ gives possible positive roots; in $f(-x)$ gives possible negative roots.

• Example: For $f(x) = 9x^4 - 16x^3 + 60x^2 - 64x - 22$, list possible roots.

Complex Roots

Complex roots of polynomials with real coefficients occur in conjugate pairs.

• Key Point: If $a + bi$ is a root, then $a - bi$ is also a root.

• Example: If $-2i$ is a root of $f(x) = x^4 + 4x^3 + 24x^2 + 16x + 80$, then $2i$ is also a root. Use polynomial division or factoring to find remaining roots.

Tables

Summary Table: Rational Root Theorem

The following table summarizes the Rational Root Theorem for a polynomial $a_nx^n + ... + a_0$:

Possible Rational Roots	How to Find
$\pm \frac{\text{factors of } a_0}{\text{factors of } a_n}$	List all factors of the constant term and leading coefficient, then form all possible fractions.

Summary Table: Descartes' Rule of Signs

Type of Root	How to Determine
Positive Real Roots	Count sign changes in $f(x)$
Negative Real Roots	Count sign changes in $f(-x)$

Additional info: Some explanations and examples have been expanded for clarity and completeness.