Introduction to College Algebra

Jul 12, 2024

Introduction to College Algebra

Basics of Algebra

Multiplying and Dividing Exponents

  • Multiplication: $x^2 \cdot x^5 = x^{2+5} = x^7$
    • Explanation: When multiplying like bases, add the exponents.
  • Division: $x^5 / x^2 = x^{5-2} = x^3$
    • Explanation: When dividing like bases, subtract the exponents.
  • Negative Exponents: $x^4 / x^7 = x^{4-7} = x^{-3} = 1 / x^3$
    • Explanation: Negative exponents denote reciprocal values.

Raising Exponents to a Power

  • Example: $(x^3)^4 = x^{3 \cdot 4} = x^{12}$
    • Explanation: When an exponent is raised to another exponent, multiply them.
  • Zero Exponent Rule: Anything raised to the power of zero equals one.
    • Example: $x^0 = 1$

Simplifying Expressions and Combining Like Terms

Example Problems:

  1. Simplify: $5x + 3 + 7x - 4$
    • Combine like terms: $(5x + 7x) = 12x$ and $(3 - 4) = -1$
    • Result: $12x - 1$
  2. Simplify: $3x^2 + 6x + 8 + 9x^2 + 7x - 5$
    • Combine like terms: $(3x^2 + 9x^2) = 12x^2$, $(6x + 7x) = 13x$, $(8 - 5) = 3$
    • Result: $12x^2 + 13x + 3$
  3. Simplify: $5x^2 - 3x + 7 - 4x^2 - 8x - 11$
    • Distribute negative sign: $5x^2 - 3x + 7 - 4x^2 + 8x + 11$
    • Combine like terms: $(5x^2 - 4x^2) = x^2$, $(-3x + 8x) = 5x$, $(7 + 11) = 18$
    • Result: $x^2 + 5x + 18$

Multiplying Polynomials

FOIL Method

  • Multiply binomials: $(3x - 5)(2x - 6)$
    • First: $3x \cdot 2x = 6x^2$
    • Outer: $3x \cdot (-6) = -18x$
    • Inner: $-5 \cdot 2x = -10x$
    • Last: $-5 \cdot (-6) = 30$
    • Combine like terms: $6x^2 - 28x + 30$

Solving Linear Equations

Basic Steps

  1. Isolation: Move constants or variables to isolate $x$.
  • Example: $x + 6 = 11$, subtract 6 from both sides: $x = 5$
  1. Inverse Operations: Use inverse operations to solve for $x$.
  • Example: $4x = 8$, divide both sides by 4: $x = 2$
  1. Combined Operations: Combine above steps sequentially.
  • Example: $3x + 5 = 26$, subtract 5 then divide by 3: $x = 7$

Advanced Equations

  1. $4(2x - 7) + 8 = 20$
  • Distribute and isolate: $8x - 28 + 8 = 20$
  • Combine and solve: $8x - 20 = 20$ then $8x = 40$, thus $x = 5$

Inequalities and Absolute Values

Plotting Inequalities

  1. Solve and plot on the number line, using open/closed circles based on strict/weak inequalities.
  • Example: $2x + 5 > 11$, solve as $2x > 6$, thus $x > 3$

Absolute Value Equations and Inequalities

  1. $|2x + 3| = 11$
  • Create two equations: $2x + 3 = 11$ and $2x + 3 = -11$
  • Solve both: $x = 4$ and $x = -7$
  1. $|3x - 1| > 5$
  • Create two inequalities: $3x - 1 > 5$ and $3x - 1 < -5$
  • Solve both: $x > 2$ or $x < -4/3$

Graphing Functions

Slope-Intercept Form

  • Equation: $y = mx + b$, where $m$ is slope and $b$ is y-intercept.
  • Example: $y = 2x - 3$, plot and apply rise/run for slope.

Standard Form

  • Equation: $ax + by = c$, solve for intercepts by setting $x$ and $y$ to zero.
  • Example: $2x + 3y = 6$, x-intercept = $(3, 0)$, y-intercept = $(0, 2)$

Graphing Transformations

  1. Basic functions and transformations with vertical/horizontal shifts and reflections.
  • Example: $y = |x - 3| + 2$, shifts right 3 units and up 2 units.
  1. Reflection: Negative sign reflects the graph across the x-axis.
  • Example: $y = -|x|$ reflects downwards.

Solving Quadratic Equations

Factoring

  • Example: $x^2 - 25 = 0$
    • Factor: $(x - 5)(x + 5) = 0$
    • Solve: $x = -5, 5$
  • For non-trivial cases, use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Imaginary Numbers

  • Definition: $i = \sqrt{-1}$, important identities: $i^2 = -1$, $i^3 = -i$, $i^4 = 1$
  • Applying square roots to negatives yields imaginary results.

Systems of Equations

Solving by Elimination and Substitution

  1. Elimination: Align terms, multiply to cancel one variable, solve.
  • Example: Add equations $2x + y = 5$ and $3x - y = 0$ to eliminate $y$.
  1. Substitution: Solve one equation for one variable, substitute into another.
  • Example: $y = 2x + 5$, replace in other equation.

Functions and Transformations

Composite Functions

  • Combine functions: $f(g(x))$
    • Example: $f(x) = 3x + 5$, $g(x) = x^2 - 4$, $f(g(x)) = 3(x^2 - 4) + 5 = 3x^2 - 7$

Inverse Functions

  • Found by swapping $x$ and $y$ in $f(x)$ and solving for $y$.
    • Example: $f(x) = 7x - 5$, inverse $f^{-1}(x) = \frac{x + 5}{7}$
  • Verify by checking $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

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Ending Note: This guide covers basic concepts. For a comprehensive understanding, deeper exploration of each topic is advisable.