Fundamental Concepts of Algebra: Study Guide

P.1 – Algebraic Expressions and Real Numbers

Order of Operations

Order of operations is a fundamental rule for evaluating mathematical expressions. It ensures consistency and accuracy in calculations.

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

• Example: To evaluate $13 - \frac{20}{5 \times 2}$, first compute $5 \times 2 = 10$, then $\frac{20}{10} = 2$, finally $13 - 2 = 11$.

Sets and Names of Numbers

Numbers are classified into various sets based on their properties.

• Natural Numbers: $\{1, 2, 3, ...\}$

• Whole Numbers: $\{0, 1, 2, 3, ...\}$

• Integers: $\{..., -2, -1, 0, 1, 2, ...\}$

• Rational Numbers: Numbers that can be written as $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

• Irrational Numbers: Numbers that cannot be written as fractions, e.g., $\sqrt{2}$, $\pi$.

• Real Numbers: All rational and irrational numbers.

Inequalities

Inequalities compare the relative size of numbers or expressions.

• Symbols: $<$ (less than), $>$ (greater than), $\leq$ (less than or equal to), $\geq$ (greater than or equal to).

• Example: $x > 3$ means $x$ is any number greater than 3.

Absolute Values

The absolute value of a number is its distance from zero on the number line, always non-negative.

• Definition: $|x| = x$ if $x \geq 0$, $|x| = -x$ if $x < 0$.

• Example: $| -5 | = 5$

Properties of Real Numbers

These properties are essential for simplifying and manipulating algebraic expressions.

• Commutative Property: $a + b = b + a$, $ab = ba$

• Associative Property: $(a + b) + c = a + (b + c)$, $(ab)c = a(bc)$

• Distributive Property: $a(b + c) = ab + ac$

• Identity Property: $a + 0 = a$, $a \times 1 = a$

• Inverse Property: $a + (-a) = 0$, $a \times \frac{1}{a} = 1$ (for $a \neq 0$)

P.2 – Exponents and Scientific Notation

Exponents

Exponents represent repeated multiplication of a base number.

• Definition: $a^n$ means $a$ multiplied by itself $n$ times.

• Properties:

  • $a^m \times a^n = a^{m+n}$

  • $\frac{a^m}{a^n} = a^{m-n}$

  • $(a^m)^n = a^{mn}$

  • $(ab)^n = a^n b^n$

  • $a^0 = 1$ (for $a \neq 0$)

• Example: $x^3 y^5 \times x^4 y^2 = x^{3+4} y^{5+2} = x^7 y^7$

Common Mistakes

• Incorrect Distribution: $(a + b)^2 \neq a^2 + b^2$; correct: $(a + b)^2 = a^2 + 2ab + b^2$

• Multiplication: $3^2 \times 3^3 = 3^{2+3} = 3^5$

• Negative Exponents: $a^{-n} = \frac{1}{a^n}$

• Example: $25^{-1} = \frac{1}{25}$

Scientific Notation

Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10. Note: This section is skipped in the provided materials.

P.3 – Radicals and Rational Exponents

Simplifying Roots

Roots are the inverse operation of exponents. Simplifying roots involves expressing them in their simplest form.

• Square Root: $\sqrt{a}$ is a number that, when squared, gives $a$.

• Cube Root: $\sqrt[3]{a}$ is a number that, when cubed, gives $a$.

• Example: $\sqrt{98} = \sqrt{2 \times 49} = 7\sqrt{2}$

• Example: $\sqrt[3]{16} = 2$ (since $2^3 = 8$; $16$is not a perfect cube, so$\sqrt[3]{16}$ is irrational)

Rationalizing Denominators

Expressions are not considered simplified if the denominator contains a radical or an irrational number.

• Rationalizing: Multiply numerator and denominator by a suitable value to eliminate radicals from the denominator.

• Example: $\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

Rational Exponents

Rational exponents are another way to represent roots.

• Definition: $a^{\frac{p}{q}} = \sqrt[q]{a^p}$

• Example: $25^{3/2} = (\sqrt{25})^3 = 5^3 = 125$

• Example: $32^{7/5} = (\sqrt[5]{32})^7 = 2^7 = 128$

Converting Between Exponents and Radicals

• Exponent to Radical: $x^{1/3} = \sqrt[3]{x}$

• Radical to Exponent: $\sqrt[5]{x^2} = x^{2/5}$

• Example: $x^{2/5} = \sqrt[5]{x^2}$

Simplifying with Rational Exponents

• Example: $27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9$

• Example: $x^{5/2} y^{2/3} = (\sqrt{x^5})(\sqrt[3]{y^2})$

P.4 – Polynomials

Basics of Polynomials

A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication.

• General Form: $a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$

• Degree: The highest power of the variable.

• Example: $3x^2 + 2x - 5$ is a polynomial of degree 2.

Additional info: Detailed study of polynomials will be covered in Chapter 3.

P.5 – Factoring

Factoring Overview

Factoring is rewriting an expression as a product of simpler expressions (factors).

• Greatest Common Factor (GCF): Factor out the largest common factor from all terms.

• Example: $35x^5 - 14x^{11} + 7x^4$; GCF is $7x^4$, so $7x^4(5x - 2x^7 + 1)$

• Grouping: Used when terms can be grouped to factor by pairs.

• Example: $ar - br + as - bs = (ar + as) - (br + bs) = a(r + s) - b(r + s) = (a - b)(r + s)$

• Trinomials: Factor expressions of the form $ax^2 + bx + c$.

• Example: $x^2 + 5x - 6 = (x + 6)(x - 1)$

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

• Example: $49x^2 - 64y^2 = (7x + 8y)(7x - 8y)$

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

• Example: $x^3 - 8 = (x - 2)(x^2 + 2x + 4)$

P.6 – Rational Expressions

Finding the Domain

The domain of a rational expression is all real numbers except those that make the denominator zero.

• Example: $g(x) = \frac{3}{(x + 1)(x - 5)}$; domain excludes $x = -1$ and $x = 5$.

Simplifying Rational Expressions

Simplifying involves factoring numerators and denominators and canceling common factors.

• Example: $\frac{x^2 - 2x - 3}{x^2 - x - 6} = \frac{(x - 3)(x + 1)}{(x - 3)(x + 2)} = \frac{x + 1}{x + 2}$ (for $x \neq 3$)

• Example: $\frac{a^3 + a^2}{a^2 + a}$; factor numerator and denominator: $a^2(a + 1)/a(a + 1) = a$ (for $a \neq -1$ and $a \neq 0$)

HTML Table: Types of Factoring

The following table summarizes common factoring methods:

Method	When to Use	Example
GCF	All terms share a common factor	$6x^2 + 9x = 3x(2x + 3)$
Grouping	Four terms, can be grouped	$x^3 + x^2 + x + 1 = (x^3 + x^2) + (x + 1)$
Trinomials	Quadratic expressions	$x^2 + 5x + 6 = (x + 2)(x + 3)$
Difference of Squares	Two terms, both perfect squares	$x^2 - 16 = (x + 4)(x - 4)$
Sum/Difference of Cubes	Two terms, both perfect cubes	$x^3 + 8 = (x + 2)(x^2 - 2x + 4)$

Additional info: Factoring is foundational for solving equations, simplifying expressions, and understanding polynomial functions.