Precalculus Study Guide: Key Concepts and Problem Types

Fundamental Concepts of Algebra

Order of Operations

Order of operations is a foundational principle in algebra that determines the sequence in which mathematical operations should be performed to accurately evaluate expressions.

• PEMDAS: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

• Example: Evaluate $3 + 2 \times (5^2 - 4)$.

Evaluating Expressions with Variables

To evaluate an expression, substitute the given values for each variable and follow the order of operations.

• Example: If $x = 2$, $y = 3$, evaluate $2x + y^2$.

Factoring Completely

Factoring is the process of expressing a polynomial as a product of its factors. Complete factoring involves breaking down the polynomial until all factors are irreducible over the integers.

• Example: Factor $x^2 - 5x + 6$ completely: $(x - 2)(x - 3)$.

Simplifying Rational Exponent Expressions

Rational exponents represent roots and powers. Simplifying involves rewriting expressions using exponent rules.

• Example: Simplify $x^{3/2} = (x^{1/2})^3 = (\sqrt{x})^3$.

Equations and Inequalities

Solving Word Problems (Price Reduction)

Translate real-world scenarios into algebraic equations and solve for the unknown.

• Example: If a $\$50$ item is reduced by $20\%$, the sale price is $\$50 - 0.20 \times \$50 = \$40$.

Absolute Value Inequalities

Absolute value inequalities involve expressions like $|x| < a$ or $|x| > a$. Solve by considering both positive and negative cases.

• Example: $|x - 3| < 5$ leads to $-5 < x - 3 < 5$.

Complex Numbers

Dividing Complex Numbers

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

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

• Example: $\frac{1 + 2i}{3 - i}$

Polynomial and Rational Functions

Solving Polynomial Equations by Factoring

Set the polynomial equal to zero, factor, and use the Zero-Product Principle to find solutions.

• Zero-Product Principle: If $ab = 0$, then $a = 0$ or $b = 0$.

• Example: $x^2 - 4 = 0 \Rightarrow (x - 2)(x + 2) = 0$

Solving Rational Equations

Rational equations contain fractions with polynomials. Clear denominators and solve for the variable.

• Example: $\frac{x}{x+1} = 2$

Solving Radical Equations

Isolate the radical, raise both sides to the appropriate power, and solve. Check for extraneous solutions.

• Example: $\sqrt{x + 3} = 5$

Analytic Geometry

Midpoint of a Segment

The midpoint formula finds the point halfway between two coordinates $(x_1, y_1)$ and $(x_2, y_2)$.

• Formula: $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Example: Between $(1, 2)$ and $(3, 4)$: $(2, 3)$

Distance Between Two Points

Use the distance formula to find the length of the segment joining two points.

• Formula: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Equation of a Line

Lines can be written in slope-intercept form, point-slope form, or standard form.

• Slope-intercept form: $y = mx + b$

• Point-slope form: $y - y_1 = m(x - x_1)$

Functions and Graphs

Domain of a Function

The domain is the set of all possible input values (x-values) for which the function is defined.

• Example: For $f(x) = \frac{1}{x}$, domain is $x \neq 0$.

Composite Functions

Composite functions combine two functions: $(f \circ g)(x) = f(g(x))$.

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

Vertex of a Parabola

The vertex is the maximum or minimum point of a parabola. For $y = ax^2 + bx + c$, the vertex is at $x = -\frac{b}{2a}$.

• Formula: $x = -\frac{b}{2a}$

• Example: For $y = 2x^2 - 4x + 1$, vertex at $x = 1$

Polynomial Zeros and Theorems

Rational Zero Theorem

Lists all possible rational zeros of a polynomial based on its coefficients.

• Possible zeros: $\pm \frac{\text{factors of constant term}}{\text{factors of leading coefficient}}$

Descartes’s Rule of Signs

Predicts the number of positive and negative real zeros by counting sign changes in the polynomial’s coefficients.

• Positive zeros: Number of sign changes in $f(x)$

• Negative zeros: Number of sign changes in $f(-x)$

Finding Zeros Using Synthetic Division

Synthetic division is a shortcut for dividing polynomials by linear factors to test possible zeros.

• Example: Test $x = 2$ as a zero for $x^3 - 4x^2 + 5x - 2$

Factor Theorem

If $f(c) = 0$, then $(x - c)$ is a factor of $f(x)$.

• Example: If $f(2) = 0$, then $(x - 2)$ is a factor.

Rational and Radical Inequalities

Solving Rational Inequalities

Set the rational expression greater or less than zero, find critical points, and test intervals.

• Example: $\frac{x - 1}{x + 2} > 0$

Variation Problems

Direct, Inverse, and Joint Variation

Variation problems describe relationships where one variable changes in response to another.

• Direct variation: $y = kx$

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

• Joint variation: $y = kxz$

Functions: Inverses

Finding the Inverse of a Function

To find the inverse, swap $x$ and $y$ and solve for $y$.

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

Systems of Equations

Solving Systems of Two Linear Equations

Systems can be solved by substitution, elimination, or graphing.

• Example: $x + y = 5$, $x - y = 1$

Solving Systems of Three Linear Equations

Use substitution, elimination, or matrix methods to solve for three variables.

• Example: $x + y + z = 6$, $2x - y + z = 3$, $x + 2y - z = 4$