Algebra Class: Factoring and Equations

Factoring

• Definition: Breaking down an expression into a product of simpler factors.
• Common Methods:
  • Greatest Common Factor (GCF): Identify and factor out the largest common factor.
  • Difference of Squares: ( a^2 - b^2 = (a + b)(a - b) ).
  • Trinomials: For quadratic ( ax^2 + bx + c ), find factors of ( ac ) that add up to ( b ).
  • Factoring by Grouping: Group terms to factor out common binomials.

Linear Equations

• Standard Form: ( Ax + By = C ), where A, B, C are constants.
• Slope-Intercept Form: ( y = mx + b ) (m = slope, b = y-intercept).
• Finding Solutions:
  • Graphically: Intersection points of lines.
  • Algebraically: Solve for y or x.
• Parallel and Perpendicular Lines:
  • Parallel: Same slope (m).
  • Perpendicular: Slopes are negative reciprocals.

Polynomials

• Definition: An expression of the form ( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 ).
• Degree: Highest exponent of variable (n).
• Types:
  • Monomial: One term (e.g., ( 3x^2 )).
  • Binomial: Two terms (e.g., ( x^2 + 4 )).
  • Trinomial: Three terms (e.g., ( x^2 + 5x + 6 )).
• Operations: Addition, subtraction, multiplication, and division.

Inequalities

• Definition: Statements that compare expressions using ( <, >, \leq, \geq ).
• Solving Inequalities: Similar to equations, but reverse the inequality sign when multiplying/dividing by a negative number.
• Graphing:
  • Use a number line to represent solutions.
  • Open circles for strict inequalities (( <, > )) and closed circles for inclusive (( \leq, \geq )).

Functions and Graphs

• Definition of a Function: A relation where each input has a unique output.
• Notation: ( f(x) ) represents the output corresponding to input ( x ).
• Types of Functions:
  • Linear: Graph is a straight line (e.g., ( f(x) = mx + b )).
  • Quadratic: Graph is a parabola (e.g., ( f(x) = ax^2 + bx + c )).
  • Exponential: Rapid growth or decay (e.g., ( f(x) = a \cdot b^x )).
• Transformations:
  • Vertical Shift: Changes ( k ) in ( f(x) + k ).
  • Horizontal Shift: Changes ( h ) in ( f(x - h) ).
  • Reflection: Across x-axis by negating ( f(x) ).

Each subtopic in algebra plays a crucial role in solving mathematical problems and understanding higher-level concepts.

Factoring

• Definition involves converting an expression into simpler factors that multiply together.
• Methods include identifying the Greatest Common Factor (GCF) for simplification.
• Difference of Squares follows the formula ( a^2 - b^2 = (a + b)(a - b) ) for factoring.
• Trinomials in the form ( ax^2 + bx + c ) require finding two numbers that multiply to ( ac ) and add to ( b ).
• Factoring by Grouping involves rearranging terms to extract common binomial factors.

Linear Equations

• Expressed in Standard Form as ( Ax + By = C ), where A, B, and C are constants.
• Slope-Intercept Form is ( y = mx + b ) where ( m ) represents slope and ( b ) represents the y-intercept.
• Solutions can be determined graphically by identifying the intersection points of lines or algebraically by isolating variables.
• Parallel lines share equal slopes, while perpendicular lines have slopes that are negative reciprocals.

Polynomials

• Defined as ( a_nx^n + a_{n-1}x^{n-1} +...+ a_1x + a_0 ), where each term is a multiple of a variable raised to a non-negative integer.
• The degree of a polynomial is determined by the highest exponent present.
• Types include:
  • Monomial: One term (Example: ( 3x^2 )).
  • Binomial: Two terms (Example: ( x^2 + 4 )).
  • Trinomial: Three terms (Example: ( x^2 + 5x + 6 )).
• Polynomials can undergo operations like addition, subtraction, multiplication, and division.

Inequalities

• Inequalities are comparisons expressed using symbols ( <, >, \leq, \geq ).
• Solving these involves similar steps to equations, with a key difference: reversing the inequality sign when multiplying or dividing by a negative number.
• Graphical representation uses a number line where open circles denote strict inequalities and closed circles indicate inclusive inequalities.

Functions and Graphs

• A function is defined as a relation where each input corresponds to exactly one output.
• Function notation ( f(x) ) signifies the output value for a given input ( x ).
• Types of functions include:
  • Linear: Represented graphically as straight lines, exemplified by equations like ( f(x) = mx + b ).
  • Quadratic: Displayed as parabolas, typically described by ( f(x) = ax^2 + bx + c ).
  • Exponential: Characterized by rapid changes, represented as ( f(x) = a \cdot b^x ).
• Transformations of functions can involve shifts:
  • Vertical Shift: Changing ( k ) in ( f(x) + k ).
  • Horizontal Shift: Modifying ( h ) in ( f(x - h) ).
  • Reflection: Flipping across the x-axis by negating ( f(x) ).

Test your knowledge on factoring expressions and solving linear equations in this algebra quiz. Covering methods like GCF, difference of squares, and the properties of polynomials, you'll solidify your understanding of key concepts. Prepare to tackle various problem types, from graphical to algebraic solutions.