College Algebra: Review and Foundations Study Notes

Review of Fundamental 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 numbers in science and engineering.

• Standard Form: $a \times 10^{n}$

• Example: 0.00056 = $5.6 \times 10^{-4}$

• Operations:

  • Multiplication: Multiply the coefficients and add the exponents: $(a \times 10^{m}) \times (b \times 10^{n}) = (a \times b) \times 10^{m+n}$

  • Division: Divide the coefficients and subtract the exponents: $\frac{a \times 10^{m}}{b \times 10^{n}} = \left(\frac{a}{b}\right) \times 10^{m-n}$

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 to multiply each term in one polynomial by each term in the other.

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

• Example: $(x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6$

Factoring Polynomials

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

• Common Methods:

  • Greatest Common Factor (GCF): Factor out the largest common factor from all terms.

  • Factoring Trinomials: Express as $(x + a)(x + b)$ where $ab = c$ and $a + b = b$ for $ax^2 + bx + c$.

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

• Example: $x^2 - 9 = (x - 3)(x + 3)$

Multiplication and Division of Rational Expressions

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

• Multiplication: Multiply numerators and denominators, then simplify.

• Division: Multiply by the reciprocal of the divisor.

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

• Example: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

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).

• Standard Form: $ax + b = 0$

• Solution: $x = -\frac{b}{a}$

• Example: $2x - 5 = 7 \implies x = 6$

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 the solutions to any quadratic equation $ax^2 + bx + c = 0$.

• 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: $x^2 + 4x + 5 = 0 \implies x = \frac{-4 \pm \sqrt{16 - 20}}{2} = \frac{-4 \pm 2i}{2} = -2 \pm i$

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

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

Solving an Equation by Factoring

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

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

Solving Inequalities

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

• Linear Inequality: Solve as you would an equation, but reverse the inequality sign when multiplying or dividing by a negative number.

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

• Quadratic Inequality: Find zeros, test intervals, and determine where the inequality holds.

Graphs and Equations of Lines

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

The point-slope form is useful for writing the equation of a line when 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 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 $m$, any 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 $y = 3x + 1$: $y - 2 = 3(x - 1)$

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