Algebra Class on Equations and Polynomials

Systems of Equations

• Definition: A set of two or more equations with the same variables.
• Types:
  • Consistent: At least one solution exists.
  • Inconsistent: No solution exists.
  • Dependent: Infinite solutions (equations represent the same line).
• Methods to Solve:
  • Graphical Method: Plot both equations to find intersection points.
  • Substitution Method: Solve one equation for one variable, substitute into the other.
  • Elimination Method: Add or subtract equations to eliminate one variable.

Polynomials

• Definition: An expression consisting of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication.
• Standard Form: Written as ( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 ) where ( a_n ) are coefficients and ( n ) is the degree.
• Types:
  • Monomial: One term (e.g., ( 3x^2 )).
  • Binomial: Two terms (e.g., ( x^2 + 3x )).
  • Trinomial: Three terms (e.g., ( x^2 + 3x + 2 )).
• Operations:
  • Addition, subtraction, multiplication, and division (long division or synthetic division).

Linear Equations

• Definition: An equation of the first degree, typically in the form ( ax + b = 0 ), where ( a ) and ( b ) are constants.
• Graph: A straight line on a Cartesian plane.
• Slope-Intercept Form: ( y = mx + b ) where ( m ) is the slope and ( b ) is the y-intercept.
• Standard Form: ( Ax + By = C ) where ( A, B, C ) are integers.

Quadratic Functions

• Definition: A polynomial function of degree 2, typically in the form ( f(x) = ax^2 + bx + c ).
• Graph: A parabola that opens upwards (if ( a > 0 )) or downwards (if ( a < 0 )).
• Vertex: The highest or lowest point on the graph, found using ( x = -\frac{b}{2a} ).
• Roots: Solutions to the equation ( ax^2 + bx + c = 0 ), found using:
  • Factoring: If possible.
  • Quadratic Formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).

Factoring Techniques

• Factoring Out the GCF: Identify and factor out the greatest common factor first.
• Factoring by Grouping: Group terms in pairs and factor out common factors.
• Difference of Squares: Recognize and factor expressions like ( a^2 - b^2 = (a + b)(a - b) ).
• Trinomials: For ( ax^2 + bx + c ), find two numbers that multiply to ( ac ) and add to ( b ).
• Perfect Squares: Recognize ( a^2 + 2ab + b^2 = (a + b)^2 ) and ( a^2 - 2ab + b^2 = (a - b)^2 ).

Systems of Equations

• A collection of two or more equations sharing the same variables.
• Types include:
  • Consistent: At least one solution exists.
  • Inconsistent: No solutions exist.
  • Dependent: Infinitely many solutions where both equations represent the same line.
• Common methods for solving include:
  • Graphical Method: Visual representation to identify intersection points.
  • Substitution Method: Rearrange one equation to express a variable, insert into the other equation.
  • Elimination Method: Manipulate equations to remove a variable by addition or subtraction.

Polynomials

• Composed of variables raised to non-negative integer powers, connected through addition, subtraction, and multiplication.
• Standard form is represented as ( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 ), where ( a_n ) denotes coefficients and ( n ) indicates the degree.
• Types include:
  • Monomial: Single term, e.g., ( 3x^2 ).
  • Binomial: Two terms, e.g., ( x^2 + 3x ).
  • Trinomial: Three terms, e.g., ( x^2 + 3x + 2 ).
• Operations involve addition, subtraction, multiplication, and division (including long and synthetic division).

Linear Equations

• First-degree equations, typically expressed as ( ax + b = 0 ).
• Represented graphically as straight lines on the Cartesian plane.
• Common forms:
  • Slope-Intercept Form: ( y = mx + b ) where ( m ) signifies the slope and ( b ) is the y-intercept.
  • Standard Form: Written as ( Ax + By = C ), with ( A, B, C ) being integer coefficients.

Quadratic Functions

• Polynomial functions of degree 2, generally written as ( f(x) = ax^2 + bx + c ).
• Their graphs form parabolas that open upwards for ( a > 0 ) and downwards for ( a < 0 ).
• The Vertex is the peak or trough of the parabola, calculated using ( x = -\frac{b}{2a} ).
• Roots or solutions of the equation ( ax^2 + bx + c = 0 ) can be found through:
  • Factoring, when applicable.
  • Quadratic Formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).

Factoring Techniques

• Factoring Out the GCF: Start by identifying and removing the greatest common factor.
• Factoring by Grouping: Organize terms into pairs and extract common factors from those groups.
• Difference of Squares: Recognize and factor expressions like ( a^2 - b^2 ) into ( (a + b)(a - b) ).
• Trinomials: For expressions of the type ( ax^2 + bx + c ), identify two numbers that multiply to ( ac ) and add to ( b ).
• Perfect Squares: Identify forms like ( a^2 + 2ab + b^2 = (a + b)^2 ) and ( a^2 - 2ab + b^2 = (a - b)^2 ).