Introduction to Algebra

• Algebra involves symbols and rules to manipulate them.
• Symbols in algebra represent quantities without fixed values, known as variables.

Basic Concepts

• A variable is a symbol, often a letter, for an unknown or changeable value.
• A constant is a value that remains the same.
• An expression combines variables, constants, and arithmetic operations.
• An equation states that two expressions are equal.
• A coefficient is a number multiplying a variable.
• A term is a single number or variable, or numbers and variables multiplied together.
• An operator is a symbol for a mathematical operation, such as +, -, ×, or ÷.

Algebraic Expressions

• When combining like terms, terms with the same variable raised to the same power can be combined by adding or subtracting their coefficients.
• The distributive property, a(b + c) = ab + ac, is used to multiply a single term by an expression in parentheses.

Equations

• A linear equation is one where the highest power of the variable is 1.
• A quadratic equation is one where the highest power of the variable is 2.
• Solving equations involves finding the value(s) of the variable(s) that make the equation true.

Solving Linear Equations

• Use inverse operations to isolate the variable on one side of the equation.
• The addition property of equality states that adding the same number to both sides of an equation does not change the equality.
• The subtraction property of equality states that subtracting the same number from both sides of an equation does not change the equality.
• The multiplication property of equality states that multiplying both sides of an equation by the same non-zero number does not change the equality.
• The division property of equality states that dividing both sides of an equation by the same non-zero number does not change the equality.

Solving Quadratic Equations

• Factoring involves expressing the quadratic expression as a product of two linear factors.
• The quadratic formula, for an equation ax² + bx + c = 0, gives solutions for x as x = (-b ± √(b² - 4ac)) / (2a).
• Completing the square is a method to rewrite the quadratic equation in the form (x + p)² = q, solvable by taking the square root of both sides.

Inequalities

• Inequalities use symbols like <, >, ≤, or ≥, instead of an equals sign.
• Solving inequalities is similar to solving equations; however, multiplying or dividing by a negative number reverses the inequality sign.

Functions

• A function relates inputs to outputs, with each input corresponding to exactly one output.
• The domain is the set of all possible input values (x-values) for a function.
• The range is the set of all possible output values (y-values) for a function.
• Functions are often denoted by f(x), where x is the input and f(x) is the output.

Linear Functions

• A linear function's graph is a straight line.
• The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.
• Slope represents the rate of change, calculated as the change in y divided by the change in x (rise over run).
• The y-intercept is where the line crosses the y-axis (where x = 0).

Systems of Equations

• A system of equations consists of two or more equations with the same variables.
• Solving systems of equations means finding the variable values that satisfy all equations simultaneously.
• The substitution method involves solving one equation for one variable and substituting that expression into the other equation.
• The elimination method involves adding or subtracting the equations to eliminate one variable.
• The graphing method involves graphing both equations and finding the point of intersection.

Polynomials

• Polynomials consist of variables and coefficients, using addition, subtraction, multiplication, and non-negative integer exponents.
• The degree is the highest power of the variable in the polynomial.
• Standard form involves writing the polynomial with terms in descending order of degree.
• Adding and subtracting polynomials involves combining like terms.
• Multiplying polynomials involves using the distributive property to multiply each term of one polynomial by each term of the other polynomial.

Factoring Polynomials

• Factoring polynomials expresses a polynomial as a product of simpler polynomials or factors.
• Common factoring involves finding the greatest common factor (GCF) of all terms and factoring it out.
• Difference of squares is expressed as a² - b² = (a + b)(a - b).
• Perfect square trinomials are expressed as a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)².

Rational Expressions

• Rational expressions are fractions where the numerator and denominator are polynomials.
• Simplifying rational expressions involves factoring both the numerator and denominator and cancelling out any common factors.
• Multiplying rational expressions involves multiplying the numerators and multiplying the denominators.
• Dividing rational expressions involves multiplying by the reciprocal of the divisor.
• Adding and subtracting rational expressions involves finding a common denominator and then adding or subtracting the numerators.

Exponents and Radicals

• An exponent indicates how many times a number is multiplied by itself.
• A radical represents the root of a number, such as a square root or cube root.
• Rules of exponents include:
  • a^m * a^n = a^(m+n)
  • a^m / a^n = a^(m-n)
  • (a^m)^n = a^(m*n)
  • (ab)^n = a^n * b^n
  • (a/b)^n = a^n / b^n
  • a^0 = 1 (if a ≠ 0)
  • a^(-n) = 1 / a^n
• Simplifying radicals involves factoring out perfect square (or cube, etc.) factors from under the radical sign.
• Rationalizing the denominator involves multiplying the numerator and denominator by a radical expression to eliminate the radical from the denominator.

Word Problems

• Translate word problems into algebraic equations or inequalities.
• Identify the unknowns and assign variables to them.
• Use given information to create equations relating the variables.
• Solve the equations to find the values of the unknowns.
• Check the solutions to ensure they make sense in the problem's context.

Graphing

• Graphing on a Cartesian coordinate system involves plotting points (x, y) on a plane with two axes.
• The x-axis is horizontal, and the y-axis is vertical.
• The origin is the point where the axes intersect (0, 0).
• Linear equations graph as straight lines.
• Quadratic equations graph as parabolas.

Absolute Value

• The absolute value of a number is its distance from zero.
• Denoted by |x|, the absolute value of x is always non-negative.
• When solving absolute value equations, consider both positive and negative cases; if |x| = a, then x = a or x = -a.
• When solving absolute value inequalities, consider both positive and negative cases, paying attention to the inequality's direction.

Logarithms

• A logarithm is the exponent to which a base must be raised to produce a number.
• Notation: log_b(x) = y means b^y = x, where b is the base, x is the argument, and y is the logarithm.
• Properties of logarithms include:
  • log_b(MN) = log_b(M) + log_b(N)
  • log_b(M/N) = log_b(M) - log_b(N)
  • log_b(M^p) = p * log_b(M)
  • log_b(1) = 0
  • log_b(b) = 1
• The common logarithm is the logarithm to the base 10 (log_10(x), often written as log(x)).
• The natural logarithm is the logarithm to the base e (log_e(x), often written as ln(x)), where e is approximately 2.71828.

Matrices

• A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
• Matrix addition and subtraction are performed element-wise, only if the matrices have the same dimensions.
• Matrix multiplication is more complex; the number of columns in the first matrix must equal the number of rows in the second matrix.
• Scalar multiplication involves multiplying each element of a matrix by a constant (scalar).

Complex Numbers

• Complex numbers are expressible as a + bi, where a and b are real numbers, and i is the imaginary unit, satisfying i² = -1.
• The real part of a complex number is a.
• The imaginary part of a complex number is b.
• Complex numbers can be added, subtracted, multiplied, and divided using the properties of i.
• The complex conjugate of a + bi is a - bi; the product of a complex number and its conjugate is always a real number.