Introduction to Algebra

• Algebra involves symbols and manipulation rules.
• Symbols, known as variables, represent quantities without fixed values.
• Algebra addresses mathematical problems featuring unknown quantities.

Basic Operations

• Addition combines terms, like ( a + b ).
• Subtraction finds the difference between terms, such as ( a - b ).
• Multiplication involves repeated addition or scaling, like ( a \cdot b ) or ( ab ).
• Division splits into equal parts, such as ( \frac{a}{b} ) or ( a \div b ).
• Exponentiation raises a number to a power, like ( a^n ).
• Radicals find the root of a number, such as ( \sqrt{a} ).

Variables and Constants

• Variables, typically letters like ( x, y, z ), represent unknown or changeable values.
• Constants are fixed values that remain unchanged, examples include ( 2, -5, \pi ).

Expressions and Equations

• An algebraic expression combines variables, constants, and operations, such as ( 3x + 2y - 5 ).
• An equation shows the equality between two expressions, like ( 3x + 2 = 5 ).
• Equations are solved to determine the value(s) of variables that satisfy the equation.

Solving Linear Equations

• Linear equations have variables raised to the first power.
• Isolate the variable on one side using inverse operations to solve.
• Equality is maintained by adding or subtracting the same value from both sides.
• Multiplying or dividing both sides by the same non-zero value preserves equality.

Example of Solving a Linear Equation

• In the equation ( 2x + 3 = 7 ), subtracting 3 from both sides gives ( 2x = 4 ).
• Dividing both sides by 2 yields ( x = 2 ).

Systems of Linear Equations

• A system comprises two or more linear equations sharing the same variables.
• System solutions must satisfy all equations simultaneously.
• Solution methods include substitution, elimination, and graphing.

Solving Systems by Substitution

• Isolate one variable in one equation.
• Substitute the expression into the other equation.
• Solve for the remaining variable.
• Substitute the found value back in to solve for the other variable.

Solving Systems by Elimination

• Multiply equations by constants to make coefficients of one variable opposites.
• Add the equations to eliminate a variable.
• Solve for the remaining variable.
• Substitute the found value to solve for the eliminated variable.

Polynomials

• Polynomials consist of variables and coefficients, using addition, subtraction, multiplication, and non-negative integer exponents.
• Example: ( 3x^2 - 2x + 1 )

Terms in a Polynomial

• Monomials are single terms, like ( 5x^3 ).
• Binomials consist of two terms, like ( 2x + 3 ).
• Trinomials consist of three terms, like ( x^2 - 4x + 7 ).

Degree of a Polynomial

• A term's degree equals the exponent of its variable.
• A polynomial's degree is the highest degree of its terms.
• In ( 4x^3 - 2x^2 + x - 5 ), the degree is 3.

Operations with Polynomials

• Addition combines like terms (same variable and exponent).
• Subtraction distributes the negative sign and combines like terms.
• Multiplication uses the distributive property across all terms.

Factoring Polynomials

• Factoring simplifies a polynomial into a product of simpler polynomials or monomials.
• Techniques include GCF, special product formulas, and factoring quadratic trinomials.

Factoring out the GCF

• Identify the greatest common factor (GCF) in all polynomial terms.
• Factor out the GCF from each term.
• Example: ( 6x^2 + 9x = 3x(2x + 3) )

Special Product Formulas

• Difference of Squares: ( a^2 - b^2 = (a + b)(a - b) )
• Perfect Square Trinomial: ( a^2 + 2ab + b^2 = (a + b)^2 )
• Perfect Square Trinomial: ( a^2 - 2ab + b^2 = (a - b)^2 )

Quadratic Equations

• Quadratic equations are degree-two polynomial equations in the form ( ax^2 + bx + c = 0 ), where ( a \neq 0 ).

Solving Quadratic Equations

• Factoring involves factoring the quadratic expression and setting each factor to zero.
• Quadratic Formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )
• Completing the Square: Manipulate the equation to create a perfect square trinomial.

The Quadratic Formula

• Given ( ax^2 + bx + c = 0 ), the solutions for ( x ) are: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )
• The discriminant, ( b^2 - 4ac ), determines root nature: real/distinct, real/equal, or complex.

Rational Expressions

• Rational expressions are fractions with polynomials in the numerator and denominator.
• Example: ( \frac{x^2 - 1}{x + 2} )

Simplifying Rational Expressions

• Factor both numerator and denominator.
• Cancel out common factors.
• Example: ( \frac{x^2 - 4}{x + 2} = \frac{(x + 2)(x - 2)}{x + 2} = x - 2 )

Operations with Rational Expressions

• Multiplication involves multiplying the numerators and denominators.
• Division involves multiplying by the reciprocal of the divisor.
• Addition/Subtraction requires a common denominator, then add or subtract numerators.

Exponents and Radicals

• Exponents indicate the number of times a base multiplies itself, such as ( a^n ).
• Radicals indicate the root of a number, such as ( \sqrt{a} ).

Rules of Exponents

• Product of Powers: ( a^m \cdot a^n = a^{m+n} )
• Quotient of Powers: ( \frac{a^m}{a^n} = a^{m-n} )
• Power of a Power: ( (a^m)^n = a^{mn} )
• Power of a Product: ( (ab)^n = a^n b^n )
• Power of a Quotient: ( (\frac{a}{b})^n = \frac{a^n}{b^n} )
• Zero Exponent: ( a^0 = 1 ) (if ( a \neq 0 ))
• Negative Exponent: ( a^{-n} = \frac{1}{a^n} )

Rules of Radicals

• Product Rule: ( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} )
• Quotient Rule: ( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} )

Rationalizing the Denominator

• Eliminates radicals from a fraction's denominator.
• Multiply numerator and denominator by a conjugate expression.
• To rationalize ( \frac{1}{\sqrt{2}} ), multiply by ( \frac{\sqrt{2}}{\sqrt{2}} ) to get ( \frac{\sqrt{2}}{2} )

Functions

• A function relates inputs to permissible outputs, with each input having exactly one output.
• Notation: ( f(x) ), where ( x ) is the input and ( f(x) ) is the output.

Domain and Range

• Domain: all possible input values (( x )).
• Range: all possible output values (( f(x) )).

Types of Functions

• Linear Functions: ( f(x) = mx + b )
• Quadratic Functions: ( f(x) = ax^2 + bx + c )
• Polynomial Functions: ( f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_0 )
• Rational Functions: ( f(x) = \frac{P(x)}{Q(x)} ), where ( P(x) ) and ( Q(x) ) are polynomials
• Exponential Functions: ( f(x) = a^x )
• Logarithmic Functions: ( f(x) = \log_b(x) )

Graphing Functions

• Plot points by using ( x ) values to find corresponding ( f(x) ) values.
• Connect the points to visualize.
• Note intercepts, slope, vertex (quadratics), and asymptotes (rational functions).

Inequalities

• Inequalities compare expressions using symbols like ( <, >, \leq, \geq ).
• Solving finds the set of values satisfying the inequality.

Solving Linear Inequalities

• Similar to solving equations, but:
• Reverse the inequality sign when multiplying/dividing by a negative number.

Absolute Value

• Denoted as ( |x| ), it is the distance from zero.
• ( |x| = x ) if ( x \geq 0 )
• ( |x| = -x ) if ( x < 0 )

Solving Absolute Value Equations

• Isolate the absolute value expression.
• Create two equations: one with the positive value and one with the negative value.
• Solve both resulting equations.