College Algebra: Essential Concepts and Skills Study Guide

Inequalities and Number Properties

Inequality Symbols and Their Meanings

Inequality symbols are used to compare the relative size of numbers or expressions. Understanding these symbols is fundamental in algebra.

• < (less than): Indicates that one value is smaller than another. Example: $-8 < 5$ means "-8 is less than 5."

• ≤ (less than or equal to): Indicates that one value is smaller than or equal to another. Example: $x \leq 7$ means "x is less than or equal to 7."

• > (greater than): Indicates that one value is larger than another. Example: $x > 4$ means "x is greater than 4."

• ≥ (greater than or equal to): Indicates that one value is larger than or equal to another. Example: $x \geq -9$ means "x is greater than or equal to -9."

Strict inequalities (<, >) do not include equality, while nonstrict inequalities (≤, ≥) do include equality.

• $x > 0$: x is a positive number

• $x < 0$: x is a negative number

• $x = 0$: x is neutral

• $x \geq 0$: x is a nonnegative number

• $x \leq 0$: x is a nonpositive number

Evaluating Algebraic Expressions

Substitution and Absolute Value

To evaluate an algebraic expression, substitute the given values for the variables and perform the indicated operations.

• Absolute value $|a|$ is the distance of a number a from zero on the number line.

Example: If $x = 3$ and $y = -2$, evaluate $-2|x + 5y| - \frac{|x|}{y}$ and $|3x - 12y|$.

Domain of a Variable

Definition and Determination

The domain of a variable is the set of all possible values that the variable can assume, typically restricted by the context of the problem (such as avoiding division by zero).

• To find the domain, identify values that would make the expression undefined (e.g., denominators equal to zero).

Example: Determine the domain of $\frac{x^2 - 9}{x + 5}$.

Laws of Exponents

Exponent Rules and Applications

Exponents represent repeated multiplication. The following rules apply for real numbers $a, b$ and integers $m, n$:

Rule	Formula	Example
Product Rule	$a^m \cdot a^n = a^{m+n}$	$x^5 \cdot x^2 = x^7$
Quotient Rule	$\frac{a^m}{a^n} = a^{m-n}$	$\frac{x^5}{x^2} = x^3$
Power Rule	$(a^m)^n = a^{mn}$	$(x^2)^5 = x^{10}$
Product to a Power	$(ab)^n = a^n b^n$	$(2x^3y^4)^5 = 2^5 x^{15} y^{20}$
Quotient to a Power	$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$	$\left(\frac{x^2}{y^3}\right)^4 = \frac{x^8}{y^{12}}$
Zero Exponent	$a^0 = 1$ (if $a \neq 0$)	$5^0 = 1$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$x^{-3} = \frac{1}{x^3}$

Example: Simplify $(-3)^2$ and $-3^2$.

Additional info: $(-3)^2 = 9$ while $-3^2 = -9$ because of the order of operations.

Polynomials

Definition and Standard Form

A polynomial in one variable is an algebraic expression of the form $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$, where $a_n \neq 0$ and $n$ is a nonnegative integer.

• Leading coefficient: The coefficient of the term with the highest degree.

• Degree: The highest power of the variable in the polynomial.

• Standard form: Terms are written in descending order of degree.

Example	Name	Standard Form	Degree	Leading Coefficient
$4x^2 - 3x$	Quadratic	$4x^2 - 3x$	2	4
$8y^3 - 2$	Cubic	$8y^3 - 2$	3	8

Adding and Subtracting Polynomials

Combine like terms (terms with the same variable and exponent) to add or subtract polynomials.

• Like terms: Terms with the same variable raised to the same power.

Example: $(7x^2 - 7x + 6x^2 + 4) + (3x + 2)$

Multiplying Polynomials

Use the distributive property, FOIL method (for binomials), or box method to multiply polynomials. Apply exponent rules as needed.

• FOIL: First, Outer, Inner, Last (for multiplying two binomials).

Example: $(5x + 7y)(5x - 7y)$

Dividing Polynomials

Divide polynomials using long division or synthetic division (when the divisor is linear).

Example: Divide $(7x^2 - 7x + 6x^2 + 4)$ by $(3x + 2)$.

Factoring Polynomials

Factoring Techniques

Factoring is the process of writing a polynomial as a product of its factors. Recognize patterns such as:

• Difference of squares: $a^2 - b^2 = (a + b)(a - b)$

• Sum/difference of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$, $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

• Trinomials: $x^2 + bx + c$ can be factored using the X-method or by inspection.

• Perfect square trinomials: $a^2 + 2ab + b^2 = (a + b)^2$

Example: Factor $x^2 - 4x + 4$.

Rational Expressions

Reducing to Lowest Terms

A rational expression is a quotient of two polynomials. Simplify by factoring numerator and denominator and canceling common factors.

Example: Simplify $\frac{2x^2y^3 + 5xy - 3y}{y^2 - 2xy^3}$.

Multiplying and Dividing Rational Expressions

• To multiply: Factor, then multiply numerators and denominators, cancel common factors.

• To divide: Multiply by the reciprocal of the divisor.

Example: $\frac{3}{14} \div \frac{12}{7} = \frac{3}{14} \times \frac{7}{12}$

Adding and Subtracting Rational Expressions

• Find a common denominator (usually the least common denominator, LCD).

• Rewrite each fraction with the LCD, then add or subtract numerators.

Example: $\frac{8}{x+4} - \frac{3}{x-6}$

Simplifying Complex Fractions

• Treat numerator and denominator separately.

• Combine fractions in the numerator and denominator.

• Use the KFC (Keep, Flip, Change) method for division.

Example: $\frac{2x+5}{x-2} \div \frac{x^2}{x+3}$

Radicals and Rational Exponents

Simplifying Radicals

The principal nth root of a real number $a$ is $\sqrt[n]{a} = b$ means $b^n = a$.

• Radicand: The number under the radical sign.

• Index: The degree of the root (e.g., 2 for square root, 3 for cube root).

Example: $\sqrt[3]{-81} = -3$ because $(-3)^3 = -81$.

Rationalizing Denominators

To rationalize the denominator, rewrite the expression so that the denominator contains no radicals.

Example: $\frac{-\sqrt{3}}{\sqrt{8}}$

Simplifying Expressions with Rational Exponents

Recall that $a^{m/n} = \sqrt[n]{a^m}$.

Example: Simplify $\left(\frac{a^3}{b^2}\right)^{-2/3}$.

Summary Table: Laws of Exponents

Rule	Formula
Product Rule	$a^m \cdot a^n = a^{m+n}$
Quotient Rule	$\frac{a^m}{a^n} = a^{m-n}$
Power Rule	$(a^m)^n = a^{mn}$
Product to a Power	$(ab)^n = a^n b^n$
Quotient to a Power	$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
Zero Exponent	$a^0 = 1$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$

Key Takeaways

• Mastery of inequalities, exponents, polynomials, and rational expressions is foundational for success in College Algebra.

• Always check for restrictions on the domain, especially when variables appear in denominators or under even roots.

• Factoring and simplifying are recurring skills throughout algebraic problem solving.