Algebra I: Linear Equations

• Algebra I is a foundational course in mathematics, focusing on understanding and manipulating mathematical expressions which introduces variables, equations, and graphs, establishing a basis for more advanced topics.

Linear Equations

• A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.
• The standard form of a linear equation is often expressed as ax + b = c, where a, b, and c are constants, and x is the variable.
• Solving linear equations involves isolating the variable to find its value, typically done by performing the same operations on both sides of the equation to maintain equality.
• Common operations include addition, subtraction, multiplication, and division.
• Equations can have one solution, no solution (inconsistent), or infinitely many solutions (identity).
• A system of linear equations involves two or more linear equations with the same variables.
• The solution to a system of linear equations is the set of values for the variables that satisfy all equations simultaneously.
• Systems can be solved graphically by finding the intersection point of the lines, algebraically using substitution, or elimination (addition) methods.
• The number of solutions depends on the relationship between the lines: intersecting (one solution), parallel (no solution), or coincident (infinite solutions).

Polynomials

• A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• A polynomial is typically written in the form anxn + an-1xn-1 + ... + a1x + a0, where an, an-1, ..., a1, a0 are coefficients and n is a non-negative integer representing the degree of the polynomial.
• Terms are separated by addition or subtraction.
• The degree of a polynomial is the highest power of the variable in the polynomial.
• Polynomials can be classified by their degree: constant (degree 0), linear (degree 1), quadratic (degree 2), cubic (degree 3), etc.
• Polynomials can be added, subtracted, multiplied, and divided.
• Addition and subtraction involve combining like terms (terms with the same variable and exponent).
• Multiplication involves using the distributive property to multiply each term in one polynomial by each term in the other polynomial.
• Division can be performed using long division or synthetic division.
• Special products include the square of a binomial (a + b)2 = a2 + 2ab + b2, the square of a binomial difference (a - b)2 = a2 - 2ab + b2, and the difference of squares (a + b)(a - b) = a2 - b2.

Factoring

• Factoring is the process of breaking down a polynomial into a product of simpler polynomials or factors and is the reverse process of multiplication.
• Common factoring techniques include:
  • Greatest Common Factor (GCF): Finding the largest factor that divides all terms in the polynomial.
  • Difference of Squares: Factoring expressions in the form a2 - b2 as (a + b)(a - b).
  • Perfect Square Trinomials: Recognizing and factoring expressions in the form a2 + 2ab + b2 as (a + b)2 or a2 - 2ab + b2 as (a - b)2.
  • Factoring quadratic trinomials of the form ax2 + bx + c: This often involves finding two numbers that multiply to ac and add to b.
  • Factoring by grouping: Used for polynomials with four or more terms, where terms are grouped and a GCF is factored from each group.
• Factoring is used to solve polynomial equations by setting each factor equal to zero and solving for the variable.
• The zero-product property states that if ab = 0, then either a = 0 or b = 0 (or both).

Graphing

• Graphing is the visual representation of mathematical equations on a coordinate plane.
• The coordinate plane consists of two perpendicular number lines, the x-axis (horizontal) and the y-axis (vertical).
• Points on the plane are identified by ordered pairs (x, y).
• Linear equations can be graphed by finding two or more points that satisfy the equation and connecting them with a straight line.
• The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
• The slope represents the rate of change of y with respect to x and can be calculated as rise over run.
• The y-intercept is the point where the line crosses the y-axis (where x = 0).
• Quadratic equations (polynomials of degree 2) graph as parabolas.
• Parabolas are U-shaped curves that open upwards or downwards depending on the sign of the leading coefficient.
• Key features of a parabola include the vertex (the minimum or maximum point), the axis of symmetry (a vertical line through the vertex), and the x-intercepts (where the parabola crosses the x-axis).
• The x-intercepts can be found by setting y = 0 and solving the quadratic equation which are also known as the roots or zeros of the quadratic function.
• Graphing inequalities involves shading the region of the coordinate plane that satisfies the inequality.
• For linear inequalities, the boundary line is graphed as either a solid line (for ≤ or ≥) or a dashed line (for < or >).
• The region to be shaded is determined by testing a point (e.g., (0, 0)) in the inequality and shading the region that contains the points that satisfy the inequality.