College Algebra Fundamentals: Equations, Expressions, and Polynomials

Equations, Inequalities, and Problem Solving

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable is not raised to any power other than one. Solving these equations involves isolating the variable on one side.

• Key Point 1: To solve $x + a = b$, subtract $a$ from both sides: $x = b - a$.

• Key Point 2: Always check your solution by substituting back into the original equation.

• Example: Solve $h + 8 = 14$. Subtract 8: $h = 14 - 8 = 6$.

Translating Word Problems into Equations

Many algebraic problems require converting a verbal statement into a mathematical equation.

• Key Point 1: Identify the unknown and assign a variable.

• Key Point 2: Express relationships using algebraic expressions.

• Example: "Three times the sum of a number and 4 is equal to three times the difference of the number and 7." Let $x$ be the number: $3(x + 4) = 3(x - 7)$.

Exponents and Polynomials

Properties of Exponents

Exponents indicate repeated multiplication. Understanding their properties is essential for simplifying expressions.

• Key Point 1: Product Rule: $a^m imes a^n = a^{m+n}$

• Key Point 2: Power Rule: $(a^m)^n = a^{mn}$

• Key Point 3: Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

• Example: Simplify $(x^2)^3 = x^{2 \times 3} = x^6$.

Polynomials: Degree and Classification

Polynomials are algebraic expressions consisting of terms with variables raised to whole number powers. The degree is the highest exponent.

• Key Point 1: The degree of $x^4 + 2x^2 + 3$ is 4.

• Key Point 2: Classification:

  Type	Definition	Example
  Monomial	One term	$5x^2$
  Binomial	Two terms	$x^2 + 3x$
  Trinomial	Three terms	$x^2 + 3x + 2$

• Example: $w^2 + v^2 + x^2 + y^2$ is a polynomial of degree 2 and is not a monomial, binomial, or trinomial.

Factoring and Simplifying Expressions

Combining Like Terms

Like terms have the same variable raised to the same power. Combine them by adding or subtracting their coefficients.

• Key Point 1: $3x + 5x = 8x$

• Key Point 2: $2a^2 - 4a^2 = -2a^2$

• Example: $4x^2 + 3x^2 = 7x^2$

Order of Operations

Follow PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

• Key Point 1: Simplify inside parentheses first.

• Key Point 2: Apply exponents before multiplication.

• Example: $2 + 3 \times 4 = 2 + 12 = 14$

Rational Expressions and Fractions

Simplifying Fractions

Reduce fractions to lowest terms by dividing numerator and denominator by their greatest common factor (GCF).

• Key Point 1: $\frac{12}{18} = \frac{2}{3}$ (divide both by 6)

• Key Point 2: To add or subtract fractions, use a common denominator.

• Example: $\frac{5}{8} + \frac{3}{8} = \frac{8}{8} = 1$

Multiplying and Dividing Fractions

Multiply numerators and denominators directly. For division, multiply by the reciprocal.

• Key Point 1: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$

• Key Point 2: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

• Example: $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$

Classification of Numbers

Number Sets

Numbers are classified into several sets based on their properties.

Set	Definition	Example
Natural Numbers	Counting numbers	1, 2, 3, ...
Whole Numbers	Natural numbers plus zero	0, 1, 2, ...
Integers	Whole numbers and negatives	-2, -1, 0, 1, 2
Rational Numbers	Numbers expressible as $\frac{a}{b}$	$\frac{1}{2}$, $-3$
Irrational Numbers	Cannot be written as $\frac{a}{b}$	$\sqrt{2}$, $\pi$
Real Numbers	All rational and irrational numbers	Any point on the number line

• Example: $-8$ is an integer, rational, and real number.

Additional Algebraic Concepts

Opposites and Reciprocals

The opposite (additive inverse) of a number is what you add to get zero. The reciprocal (multiplicative inverse) is what you multiply to get one.

• Key Point 1: Opposite of $a$ is $-a$.

• Key Point 2: Reciprocal of $a$ (where $a \neq 0$) is $\frac{1}{a}$.

• Example: Opposite of $-17$$is$17$; reciprocal of$\frac{2}{3}$is$\frac{3}{2}$.

Evaluating and Simplifying Expressions

Substitute values for variables and perform indicated operations.

• Key Point 1: Substitute carefully and follow order of operations.

• Key Point 2: Simplify expressions by combining like terms and applying exponent rules.

• Example: Evaluate $x^2$ for $x = -5$: $(-5)^2 = 25$.

Tables and Visual Representations

Fractional Representation of Shaded Figures

Visual models can be used to represent fractions. The fraction is the number of shaded parts over the total number of equal parts.

• Key Point 1: If 5 out of 8 parts are shaded, the fraction is $\frac{5}{8}$.

Summary Table: Key Algebraic Properties

Property	Rule	Example
Distributive	$a(b + c) = ab + ac$	$2(x + 3) = 2x + 6$
Associative	$(a + b) + c = a + (b + c)$	$(2 + 3) + 4 = 2 + (3 + 4)$
Commutative	$a + b = b + a$	$3 + 5 = 5 + 3$