College Algebra: Review and Foundations Study Notes

Review of Fundamental Algebraic Concepts

Scientific Notation

Scientific notation is a method for expressing very large or very small numbers in the form a × 10n, where 1 ≤ |a| < 10 and n is an integer.

• Purpose: Simplifies calculations and representation of extreme values.

• Conversion: Move the decimal point to create a number between 1 and 10; count the moves to determine the exponent.

• Example: 0.00056 = 5.6 × 10-4

• Multiplication: Multiply the coefficients and add the exponents.

• Division: Divide the coefficients and subtract the exponents.

Addition, Subtraction, and Multiplication of Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication.

• Addition/Subtraction: Combine like terms (terms with the same variable and exponent).

• Multiplication: Use the distributive property or special products (e.g., FOIL for binomials).

• Example: (2x2 + 3x) + (x2 - 5x) = 3x2 - 2x

• Formula: $(a + b)(c + d) = ac + ad + bc + bd$

Factoring Polynomials

Factoring is the process of expressing a polynomial as a product of its factors.

• Common Methods: Factoring out the greatest common factor (GCF), factoring trinomials, difference of squares, and grouping.

• Example: x2 - 9 = (x + 3)(x - 3)

• Formula (Difference of Squares): $a^2 - b^2 = (a + b)(a - b)$

Multiplication and Division of Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials.

• Multiplication: Multiply numerators together and denominators together, then simplify.

• Division: Multiply by the reciprocal of the divisor.

• Example: $\frac{x}{x+2} \times \frac{x+2}{x-1} = \frac{x}{x-1}$

Addition and Subtraction of Rational Expressions

To add or subtract rational expressions, first find a common denominator.

• Steps:

  1. Find the least common denominator (LCD).

  2. Rewrite each expression with the LCD.

  3. Add or subtract the numerators.

  4. Simplify the result.

• Example: $\frac{1}{x} + \frac{2}{x^2} = \frac{x + 2}{x^2}$

Equations and Inequalities

Solving Linear Equations

Linear equations are equations of the first degree (the highest exponent of the variable is 1).

• General Form: $ax + b = 0$

• Solution: $x = -\frac{b}{a}$

• Example: 2x + 3 = 7 → x = 2

Solving Quadratic Equations by Factoring

Quadratic equations have the form ax2 + bx + c = 0.

• Factoring: Express as a product of two binomials and set each factor to zero.

• Example: x2 - 5x + 6 = 0 → (x - 2)(x - 3) = 0 → x = 2, 3

Using the Quadratic Formula

The quadratic formula provides solutions to any quadratic equation.

• Formula: $x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a}$

• Discriminant: $b^2 - 4ac$ determines the nature of the roots (real and distinct, real and equal, or complex).

• Example: Solve x2 + 2x - 3 = 0 using the formula.

Addition, Subtraction, and Multiplication of Complex Numbers

Complex numbers are numbers of the form a + bi, where i is the imaginary unit (i2 = -1).

• Addition/Subtraction: Combine real parts and imaginary parts separately.

• Multiplication: Use distributive property and i2 = -1.

• Example: (2 + 3i) + (1 - 4i) = 3 - i

• Example: (2 + i)(1 - 3i) = 2 - 6i + i - 3i2 = 2 - 5i + 3 = 5 - 5i

Solving Equations by Factoring

Factoring can be used to solve equations by setting each factor equal to zero.

• Example: x2 - 4x = 0 → x(x - 4) = 0 → x = 0, 4

Solving Inequalities

Inequalities involve expressions related by <, ≤, >, or ≥.

• Linear Inequalities: Solve similarly to equations, but reverse the inequality when multiplying/dividing by a negative.

• Quadratic Inequalities: Find zeros, test intervals.

• Example: 2x - 3 < 5 → x < 4

Graphs

Finding the Equation of a Line (Point-Slope Form)

The point-slope form is useful for writing the equation of a line given a point and the slope.

• Formula: $y - y_1 = m(x - x_1)$

• Example: Through (2, 3) with slope 4: $y - 3 = 4(x - 2)$

Finding Equations of Parallel and Perpendicular Lines

Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.

• Parallel: If a line has slope m, a parallel line also has slope m.

• Perpendicular: If a line has slope m, a perpendicular line has slope -1/m.

• Example: Line y = 2x + 1; parallel: y = 2x + b; perpendicular: y = -1/2 x + b