College Algebra Essentials: Fundamental Concepts of Algebra (Ch. P) Study Guide

Algebraic Expressions

Definitions and Components

An algebraic expression is a combination of numbers and variables connected by mathematical operations such as addition, subtraction, multiplication, and division. Understanding the parts of an algebraic expression is essential for manipulating and evaluating them.

• Variable: A letter (commonly x, y, etc.) representing an unknown value that can vary.

• Coefficient: The numerical factor multiplying a variable; its value can vary and is usually placed before the variable.

• Constant: A number without variables; its value does not change and is usually at the end of the expression.

Example: In the expression $2x + 5$, 2 is the coefficient, x is the variable, and 5 is the constant.

Equations vs. Expressions

An equation is formed when an equality symbol (=) is placed between two expressions. An expression does not have an equality sign.

Evaluating Algebraic Expressions

To evaluate an expression, substitute given values for the variables and use the order of operations (PEMDAS):

• Parentheses

• Exponents

• Multiplication and Division

• Addition and Subtraction

Example: Evaluate $2x + 5$ when $x = 3$:

• Substitute: $2(3) + 5 = 6 + 5 = 11$

Exponents in Expressions

General Form and Rules

Exponents represent repeated multiplication. The base is the number being multiplied, and the exponent (or power) indicates how many times the base is used as a factor.

• $a \cdot a \cdot \ldots \cdot a$ (n times) $= a^n$

• Example: $4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 = 4^5$

Order of Operations with Exponents

When evaluating expressions with exponents, follow PEMDAS, giving priority to parentheses and exponents before multiplication/division and addition/subtraction.

Simplifying Algebraic Expressions

Combining Like Terms

To simplify expressions, reduce the number of terms by combining like terms. Like terms have the same variable(s) raised to the same power(s).

• Term: Parts of expressions separated by + or - signs.

• Example: $2x + 3 + 4(x + 2)$ simplifies by distributing and combining like terms.

Steps to Simplify:

1. Distribute constants/variables through parentheses.

2. Group like terms together.

3. Combine like terms by adding/subtracting.

Example: $-3(5 - x) + 10 - 7x$

• Distribute: $-15 + 3x + 10 - 7x$

• Combine: $-15 + 10 = -5$, $3x - 7x = -4x$

• Result: $-5 - 4x$

Rules of Exponents

Exponent Laws

Exponent rules are used to simplify expressions when like terms cannot be combined. The main rules are:

Name	Rule
Product Rule	$a^m \cdot a^n = a^{m+n}$
Quotient Rule	$\frac{a^m}{a^n} = a^{m-n}$
Power Rule	$(a^m)^n = a^{m \cdot n}$
Power of a Product	$(a \cdot b)^n = a^n \cdot b^n$
Zero Exponent	$a^0 = 1$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$

Example: $2x^3 \cdot 7x^2 = 14x^{3+2} = 14x^5$

Zero & Negative Exponents

Zero exponents yield 1, and negative exponents represent reciprocals.

• $x^0 = 1$

• $x^{-2} = \frac{1}{x^2}$

Simplifying Expressions with Exponents

Use the checklist:

• No powers raised to other powers

• No parentheses

• No same bases multiplied or divided

• No zero or negative exponents

• All numbers with exponents evaluated

Radical Expressions

Square Roots and nth Roots

The reverse of squaring a number is taking the square root. Positive real numbers have two roots: a principal (positive) root and a negative root.

• $\sqrt{36} = 6$ (principal root)

• $-\sqrt{36} = -6$ (negative root)

• $\sqrt{-36}$ is imaginary

Radical Symbol: $\sqrt{}$ denotes the principal root. The term inside is the radicand.

Even vs. Odd Roots

The nth root of a number is the reverse of raising it to the nth power. The index n is written at the top-left of the radical symbol.

• Even roots of negative numbers are imaginary.

• Odd roots of negative numbers are real and negative.

• If the exponent of the radicand matches the index, they cancel: $\sqrt[n]{a^n} = a$

Simplifying Radical Expressions

Expanding and Factoring Radicals

When radicals are not perfect powers, rewrite them as products where one factor is a perfect power.

• $\sqrt{180} = \sqrt{36 \cdot 5} = 6\sqrt{5}$

Radicals with Variables

Apply the same rules to variables inside radicals. Split the radicand into factors where one is a perfect power.

• $\sqrt{x^4} = x^2$

• $\sqrt{8x^3} = \sqrt{8} \cdot \sqrt{x^3} = 2\sqrt{2} \cdot x\sqrt{x}$

Radicals with Fractions

Radicals can be split or combined over fractions:

• $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

Adding & Subtracting Radicals

Combine like radicals (same radicand and index) by adding/subtracting their coefficients.

• $2\sqrt{x} + 4\sqrt{x} = 6\sqrt{x}$

• Unlike radicals must be simplified before combining.

Rationalizing Denominators

Rationalizing with One Term

Radicals should not remain in the denominator. Multiply numerator and denominator by a value that eliminates the radical.

• $\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

Rationalizing with Conjugates

For denominators with two terms, multiply by the conjugate (change the sign between terms) to eliminate radicals.

• $(a + b)(a - b) = a^2 - b^2$

• $\frac{1}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3}$

Rational Exponents

Radicals as Exponents

Radical expressions can be rewritten as exponents with fractional powers.

• $\sqrt[n]{a^m} = a^{\frac{m}{n}}$

• $\sqrt{a} = a^{\frac{1}{2}}$

• $\sqrt[3]{a} = a^{\frac{1}{3}}$

Introduction to Polynomials

Definitions and Classification

A polynomial is an algebraic expression where variables have only whole number exponents (no negatives or fractions).

• Monomial: One term

• Binomial: Two terms

• Trinomial: Three terms

Example: $6x^2 + 3x + 5x^3$ is a trinomial.

Standard Form, Degree, and Leading Coefficient

Polynomials are written in descending order of exponents, with like terms combined. The degree is the highest exponent, and the leading coefficient is the coefficient of the term with the highest degree.

• Example: $5x^2 - 3x^3 + 2x - 7 + x^3$ becomes $-2x^3 + 5x^2 + 2x - 7$ (degree 3, leading coefficient -2)

Adding & Subtracting Polynomials

Add or subtract polynomials by combining like terms. Distribute signs into parentheses before combining.

• Example: $(3x^2 + 2x + 4) - (5x + 10 - x^2)$

• Distribute: $3x^2 + 2x + 4 - 5x - 10 + x^2$

• Combine: $4x^2 - 3x - 6$

Multiplying Polynomials

FOIL Method

FOIL is used to multiply two binomials. It stands for First, Outer, Inner, Last terms.

• $(x + 2)(x + 3)$

• First: $x \cdot x = x^2$

• Outer: $x \cdot 3 = 3x$

• Inner: $2 \cdot x = 2x$

• Last: $2 \cdot 3 = 6$

• Combine: $x^2 + 5x + 6$

Distributive Property

For polynomials with more than two terms, distribute each term of the shorter expression across the longer one.

• $x(x^2 + x - 2) = x^3 + x^2 - 2x$

Special Product Formulas

Some products can be simplified using formulas:

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

• Square of Binomial: $(a + b)^2 = a^2 + 2ab + b^2$

• Cube of Binomial: $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

Factoring Polynomials

Factoring Out the Greatest Common Factor (GCF)

Always look for the GCF in all terms and factor it out first.

• Example: $2x^2 + 6 = 2(x^2 + 3)$

Factor by Grouping

If no common factor exists for all terms, group terms into pairs and factor each group.

• Example: $x^2 - 5x + 7x + 35$

• Group: $(x^2 - 5x) + (7x + 35)$

• Factor: $x(x - 5) + 7(x + 5)$

Factoring Using Formulas

Use special product formulas to factor polynomials:

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

• Perfect Square Trinomials: $a^2 + 2ab + b^2 = (a + b)^2$

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

AC Method for Factoring Trinomials

For trinomials $ax^2 + bx + c$, use the AC method:

1. List factors of $a \cdot c$.

2. Find two factors whose sum is $b$.

3. If $a = 1$, factor as $(x + p)(x + q)$.

4. If $a \neq 1$, rewrite and factor by grouping.

Example: Factor $x^2 + 5x + 6$:

• Factors of 6: 2 and 3

• Sum: $2 + 3 = 5$

• Result: $(x + 2)(x + 3)$

Example: Factor $2x^2 + 7x + 6$:

• Factors of $2 \cdot 6 = 12$: 3 and 4

• Sum: $3 + 4 = 7$

• Rewrite: $2x^2 + 3x + 4x + 6$

• Group: $(2x^2 + 3x) + (4x + 6)$

• Factor: $x(2x + 3) + 2(2x + 3) = (2x + 3)(x + 2)$

Summary Table: Special Product and Factoring Formulas

Formula Name	Multiplication Formula	Factoring Formula
Difference of Squares	$(a + b)(a - b) = a^2 - b^2$	$a^2 - b^2 = (a + b)(a - b)$
Perfect Square Trinomial	$(a + b)^2 = a^2 + 2ab + b^2$	$a^2 + 2ab + b^2 = (a + b)^2$
Sum of Cubes	$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$	$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
Difference of Cubes	$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$	$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Additional info: Some content was inferred and expanded for completeness, including full definitions, step-by-step examples, and summary tables for exponent and factoring rules.