Introduction to Algebra

• Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.

Variables and Expressions

• A variable is a symbol (usually a letter) that represents a value that can change.
• An expression is a combination of variables, numbers, and operations (like addition, subtraction, multiplication, and division).
• Algebraic expressions can be simplified by combining like terms.
• Like terms are terms that have the same variable raised to the same power.
• For example, 3x + 5x simplifies to 8x because 3x and 5x are like terms.

Equations and Inequalities

• An equation is a statement that two expressions are equal.
• Equations are solved to find the value(s) of the variable(s) that make the equation true.
• An inequality is a statement that compares two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).
• Inequalities are solved to find the range of values of the variable(s) that make the inequality true.

Solving Linear Equations

• A linear equation is an equation in which the highest power of the variable is 1.
• To solve a linear equation, isolate the variable on one side of the equation by performing the same operation on both sides.
• The goal is to get the variable by itself (e.g., x = some value).
• Common operations include addition, subtraction, multiplication, and division.
• Example: Solve 2x + 3 = 7.
• Subtract 3 from both sides: 2x = 4.
• Divide both sides by 2: x = 2.

Solving Linear Inequalities

• Solving linear inequalities is similar to solving linear equations, but there is one important difference.
• When multiplying or dividing both sides of an inequality by a negative number, you must reverse the inequality sign.
• Example: Solve -3x < 9.
• Divide both sides by -3 (and reverse the inequality sign): x > -3.

Systems of Equations

• A system of equations is a set of two or more equations with the same variables.
• The solution to a system of equations is the set of values for the variables that make all equations true simultaneously.
• Systems of equations can be solved using several methods:
  • Substitution
  • Elimination
  • Graphing

Substitution Method

• Solve one equation for one variable in terms of the other variable(s).
• Substitute the expression into the other equation(s) to eliminate that variable.
• Solve the resulting equation(s) for the remaining variable(s).
• Substitute the values back into one of the original equations to find the value of the variable that was eliminated.
• Example: Solve the system:
  • y = x + 1
  • 2x + y = 5
• Substitute (x + 1) for y in the second equation: 2x + (x + 1) = 5.
• Simplify and solve for x: 3x + 1 = 5 => 3x = 4 => x = 4/3.
• Substitute x = 4/3 back into the first equation: y = (4/3) + 1 => y = 7/3.
• The solution is x = 4/3, y = 7/3.

Elimination Method

• Multiply one or both equations by a constant so that the coefficients of one of the variables are opposites.
• Add the equations together to eliminate that variable.
• Solve the resulting equation for the remaining variable.
• Substitute the value back into one of the original equations to find the value of the variable that was eliminated.
• Example: Solve the system:
  • x + y = 6
  • x - y = 2
• Add the two equations together: 2x = 8.
• Solve for x: x = 4.
• Substitute x = 4 back into the first equation: 4 + y = 6 => y = 2.
• The solution is x = 4, y = 2.

Graphing Method

• Graph each equation on the same coordinate plane.
• The solution to the system is the point(s) where the graphs intersect.
• This method is most useful for linear equations.

Polynomials

• A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• Examples: x^2 + 3x - 5, 4y^3 - 2y + 1, 7 (a constant polynomial).
• Non-examples: x^(1/2) + 1 (fractional exponent), 2/x (variable in the denominator).

Operations with Polynomials

• Adding and Subtracting Polynomials: Combine like terms.
• Multiplying Polynomials: Use the distributive property.
• Dividing Polynomials: Use long division or synthetic division.

Factoring Polynomials

• Factoring is the process of writing a polynomial as a product of simpler polynomials.
• Common factoring techniques:
  • Greatest Common Factor (GCF)
  • Difference of Squares: a^2 - b^2 = (a + b)(a - b)
  • Perfect Square Trinomials: a^2 + 2ab + b^2 = (a + b)^2, a^2 - 2ab + b^2 = (a - b)^2
  • Factoring by Grouping.
  • Quadratic Trinomials: ax^2 + bx + c

Quadratic Equations

• A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Solving Quadratic Equations

• Factoring: Factor the quadratic expression and set each factor equal to zero.
• Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / (2a)
• Completing the Square: Rewrite the equation in the form (x + p)^2 = q and solve for x.

The Quadratic Formula

• The quadratic formula provides a general solution for any quadratic equation.
• Given ax^2 + bx + c = 0, the solutions for x are: x = (-b ± √(b^2 - 4ac)) / (2a).
• The discriminant (b^2 - 4ac) determines the nature of the roots:
• If b^2 - 4ac > 0: Two distinct real roots.
• If b^2 - 4ac = 0: One real root (a repeated root).
• If b^2 - 4ac < 0: Two complex roots.

Completing the Square

• Completing the square is a method to rewrite a quadratic equation in the form (x + p)^2 = q.
• Steps:
• Divide the equation by 'a' if a ≠ 1.
• Move the constant term to the right side of the equation.
• Add (b/2)^2 to both sides of the equation.
• Factor the left side as a perfect square.
• Take the square root of both sides and solve for x.

Rational Expressions

• A rational expression is a fraction where the numerator and denominator are polynomials.

Simplifying Rational Expressions

• Factor the numerator and denominator.
• Cancel out any common factors.

Operations with Rational Expressions

• Adding and Subtracting: Find a common denominator and combine the numerators.
• Multiplying: Multiply the numerators and multiply the denominators.
• Dividing: Multiply by the reciprocal of the divisor.

Exponents and Radicals

• Exponents indicate repeated multiplication (e.g., x^3 = x * x * x).
• Radicals (like square roots) are the inverse operation of exponents.

Rules of Exponents

• x^a * x^b = x^(a+b)
• (x^a)^b = x^(a*b)
• x^a / x^b = x^(a-b)
• x^0 = 1 (if x ≠ 0)
• x^(-a) = 1 / x^a

Simplifying Radicals

• Find the largest perfect square (or cube, etc.) factor of the radicand (the number under the radical sign).
• Use the property √(a * b) = √a * √b to separate the radical.
• Simplify the perfect square factor.
• Example: √72 = √(36 * 2) = √36 * √2 = 6√2.

Rationalizing the Denominator

• Eliminate radicals from the denominator of a fraction.
• Multiply the numerator and denominator by a suitable expression that will eliminate the radical in the denominator.
• If the denominator is a single radical (e.g., √a), multiply by √a / √a.
• If the denominator is a binomial with a radical (e.g., a + √b), multiply by its conjugate (a - √b) / (a - √b).