College Algebra: Review and Foundations Study Notes

Review of Algebraic Concepts

Scientific Notation

Scientific notation is a method of 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 to Scientific Notation: Move the decimal point so that only one nonzero digit remains to the left of the decimal. Count the number of places moved to determine the exponent.

• Operations:

  • Multiplication: Multiply the coefficients and add the exponents.

  • Division: Divide the coefficients and subtract the exponents.

• Example: $4,500,000 = 4.5 \times 10^6$

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., $(a+b)^2$).

• Example: $(2x^2 + 3x) + (x^2 - 5x + 4) = 3x^2 - 2x + 4$

Factoring Polynomials

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

• Common Methods:

  • Factoring out the greatest common factor (GCF)

  • Factoring trinomials

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

  • Sum/difference of cubes

• Example: $x^2 - 9 = (x-3)(x+3)$

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: $\frac{x}{x+2} \times \frac{x+2}{x-3} = \frac{x}{x-3}$

Addition and Subtraction of Rational Expressions

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

• Steps:

  1. Factor denominators if possible.

  2. Find the least common denominator (LCD).

  3. Rewrite each expression with the LCD.

  4. Add or subtract numerators, keep the common denominator.

  5. Simplify if possible.

• 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: $x = -\frac{b}{a}$

• Steps:

  1. Simplify both sides (expand, combine like terms).

  2. Isolate the variable on one side.

  3. Solve for the variable.

• Example: $2x + 3 = 7 \implies x = 2$

Solving Quadratic Equations by Factoring

Quadratic equations are equations of the form $ax^2 + bx + c = 0$.

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

• Example: $x^2 - 5x + 6 = 0 \implies (x-2)(x-3) = 0 \implies x = 2, 3$

Using the Quadratic Formula

The quadratic formula provides solutions to any quadratic equation.

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

• Discriminant: $D = b^2 - 4ac$ determines the nature of the roots.

• Example: $x^2 + 2x - 3 = 0 \implies x = \frac{ -2 \pm \sqrt{4 + 12} }{2} = \frac{ -2 \pm 4 }{2} \implies x = 1, -3$

Addition, Subtraction, and Multiplication of Complex Numbers

Complex numbers are numbers of the form $a + bi$, where $i = \sqrt{-1}$.

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

• Multiplication: Use distributive property and $i^2 = -1$.

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

Solving an Equation by Factoring

Factoring can be used to solve higher-degree polynomial equations by setting each factor equal to zero.

• Example: $x^3 - 4x^2 = x^2(x-4) = 0 \implies x = 0, 4$

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: $2x - 5 > 3 \implies x > 4$

Graphs and Equations of Lines

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

The point-slope form is useful when you know a point on the line and its slope.

• Formula: $y - y_1 = m(x - x_1)$

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

Finding the Equation of Parallel and Perpendicular Lines

Parallel lines have the same slope; 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 $-\frac{1}{m}$ (provided $m \neq 0$).

• Example: Line through (1,2) parallel to $y = 3x + 5$: $y - 2 = 3(x - 1)$

• Example: Line through (1,2) perpendicular to $y = 3x + 5$: $y - 2 = -\frac{1}{3}(x - 1)$