College Algebra Study Guide: Factoring, Complex Numbers, Quadratic Equations, and More

Section P.5 – Factoring

Factoring Techniques

Factoring is a fundamental algebraic skill used to simplify expressions and solve equations. The process involves expressing a polynomial as a product of its factors.

• Greatest Common Factor (GCF): Identify and factor out the largest common factor from all terms in a polynomial.

• Factoring by Grouping: Group terms with common factors and factor each group, then factor the entire expression.

• Factoring Trinomials: Express trinomials of the form $ax^2 + bx + c$ as a product of two binomials.

• Difference of Two Squares: Recognize and factor expressions of the form $a^2 - b^2 = (a + b)(a - b)$.

Example: Factor $x^2 - 9$.

Solution: $x^2 - 9 = (x + 3)(x - 3)$

Section 1.4 – Complex Numbers Review

Operations with Complex Numbers

Complex numbers extend the real number system and are written in the form $a + bi$, where $i$ is the imaginary unit ($i^2 = -1$).

• Addition and Subtraction: Combine like terms: $(a + bi) + (c + di) = (a + c) + (b + d)i$.

• Multiplication: Use distributive property and $i^2 = -1$ to simplify: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$.

• Square Roots of Negative Numbers: $\\sqrt{-a} = i\sqrt{a}$ for $a > 0$.

Example: $(2 + 3i) + (4 - 2i) = 6 + i$

Section 1.5 – Solving Quadratic Equations

Methods for Solving Quadratic Equations

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. There are several methods to solve them:

• Factoring: Set the equation to zero and factor the quadratic expression.

• Square Root Property: If $x^2 = a$, then $x = \pm\sqrt{a}$.

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

Example: Solve $x^2 - 5x + 6 = 0$ by factoring.

Solution: $(x - 2)(x - 3) = 0 \implies x = 2, 3$

Section 1.6 – Polynomial, Radical, and Absolute Value Equations

Solving Various Types of Equations

This section covers solving more complex equations, including those with higher-degree polynomials, radicals, and absolute values.

• Polynomial Equations: Use factoring or the Rational Root Theorem to find solutions.

• Radical Equations: Isolate the radical and square both sides to eliminate it, then solve the resulting equation.

• Equations with Rational Exponents: Rewrite exponents as roots and solve accordingly.

• Quadratic in Form: Some equations can be rewritten as quadratics in a new variable.

• Absolute Value Equations: Set up two cases: $|A| = B$ implies $A = B$ or $A = -B$.

Example: Solve $|x - 3| = 5$.

Solution: $x - 3 = 5 \implies x = 8$ or $x - 3 = -5 \implies x = -2$

Section 1.7 – Interval Notation and Inequalities

Expressing Solutions and Solving Inequalities

Interval notation is a concise way to describe sets of numbers, and inequalities are solved using similar techniques as equations, with attention to the direction of the inequality when multiplying or dividing by negatives.

• Interval Notation: Use parentheses for open intervals and brackets for closed intervals. Example: $(a, b)$, $[a, b]$.

• Linear Inequalities: Solve as you would equations, but reverse the inequality sign when multiplying/dividing by a negative.

• Quadratic Inequalities: Find the zeros, test intervals, and determine where the inequality holds.

Example: Solve $x + 2 > 5$ and express the solution in interval notation.

Solution: $x > 3$, so the solution is $(3, \infty)$