Learn & Review: College Algebra Introduction

This video provides a comprehensive overview of fundamental college algebra concepts, covering exponents, simplifying expressions, polynomial operations, solving equations and inequalities, absolute value expressions, graphing linear equations, function transformations, quadratic equations, and systems of equations.

I. Exponent Rules

• Multiplication: When multiplying terms with the same base, add the exponents.
  • Example: $x^2 \times x^5 = x^{2+5} = x^7$
• Division: When dividing terms with the same base, subtract the exponents.
  • Example: $x^5 \div x^2 = x^{5-2} = x^3$
• Negative Exponents: A negative exponent indicates a reciprocal. Moving a variable with a negative exponent from the numerator to the denominator (or vice versa) changes the sign of the exponent.
  • Example: $x^4 \div x^7 = x^{4-7} = x^{-3} = \frac{1}{x^3}$
• Power of a Power: When raising an exponent to another exponent, multiply the exponents.
  • Example: $(x^3)^4 = x^{3 \times 4} = x^{12}$
• Zero Exponent: Any non-zero base raised to the power of zero is equal to one.
  • Example: $x^0 = 1$

II. Simplifying Expressions and Polynomials

• Combining Like Terms: Combine terms that have the same variable and exponent.
  • Example: $5x + 3 + 7x - 4 = (5x + 7x) + (3 - 4) = 12x - 1$
• Distributing Negatives: When subtracting polynomials, distribute the negative sign to each term in the second polynomial.
  • Example: $(5x^2 - 3x + 7) - (4x^2 + 8x + 11) = 5x^2 - 3x + 7 - 4x^2 - 8x - 11 = x^2 - 11x - 4$
• Polynomial Terminology:
  • Monomial: One term (e.g., $5x^2$)
  • Binomial: Two terms (e.g., $2x - 5$)
  • Trinomial: Three terms (e.g., $x^2 + 3x + 2$)
  • Polynomial: An expression with one or more terms.
• Multiplying Polynomials (FOIL): Use the FOIL method (First, Outer, Inner, Last) for binomials.
  • Example: $(3x - 5)(2x - 6) = (3x \times 2x) + (3x \times -6) + (-5 \times 2x) + (-5 \times -6) = 6x^2 - 18x - 10x + 30 = 6x^2 - 28x + 30$
• Expanding Expressions: Rewrite expressions with exponents as repeated multiplication.
  • Example: $(2x - 5)^2 = (2x - 5)(2x - 5) = 4x^2 - 10x - 10x + 25 = 4x^2 - 20x + 25$

III. Solving Linear Equations

• Isolating the Variable: Use inverse operations to isolate the variable.
  • Addition/Subtraction: To undo addition, subtract; to undo subtraction, add.
    • Example: $x + 6 = 11 \implies x = 11 - 6 \implies x = 5$
  • Multiplication/Division: To undo multiplication, divide; to undo division, multiply.
    • Example: $4x = 8 \implies x = 8 \div 4 \implies x = 2$
• Multi-step Equations: Combine inverse operations to solve.
  • Example: $3x + 5 = 26 \implies 3x = 21 \implies x = 7$
• Equations with Parentheses: Distribute or divide first, then solve.
  • Example: $4(2x - 7) + 8 = 20 \implies 4(2x - 7) = 12 \implies 2x - 7 = 3 \implies 2x = 10 \implies x = 5$

IV. Solving and Graphing Inequalities

• Solving Inequalities: Follow the same rules as solving equations, but remember to flip the inequality sign when multiplying or dividing by a negative number.
  • Example: $2x + 5 > 11 \implies 2x > 6 \implies x > 3$
  • Example: $-3x \ge 9 \implies x \le -3$ (inequality sign flipped)
• Graphing Inequalities:
  • Use an open circle for $>$ or $<$.
  • Use a closed circle for $\ge$ or $\le$.
  • Shade to the right for $>$ or $\ge$.
  • Shade to the left for $<$ or $\le$.
• Interval Notation: Express solutions using parentheses (exclusive) and brackets (inclusive).
  • $x > 3$ is $(3, \infty)$
  • $x \ge 2$ is $[2, \infty)$
  • $x \le -3$ is $(-\infty, -3]$
• Compound Inequalities: Solve for all parts simultaneously.
  • Example: $1 \le 2x + 5 \le 13 \implies -4 \le 2x \le 8 \implies -2 \le x \le 4$, which is $[-2, 4]$ in interval notation.
  • OR Statements: Solutions are separate intervals.
    • Example: $x > 2$ or $x < -4/3$ is $(-\infty, -4/3) \cup (2, \infty)$

V. Absolute Value Expressions

• Definition: The absolute value of a number is its distance from zero, always resulting in a non-negative value.
  • $|x| = x$ if $x \ge 0$
  • $|x| = -x$ if $x < 0$
• Solving Absolute Value Equations: Set up two equations: one with the expression equal to the positive value, and one equal to the negative value.
  • Example: $|x| = 4 \implies x = 4$ or $x = -4$
  • Example: $|2x + 3| = 11 \implies 2x + 3 = 11$ (gives $x=4$) or $2x + 3 = -11$ (gives $x=-7$)
• Solving Absolute Value Inequalities: Set up two inequalities, one with the original inequality sign and one with the opposite inequality sign and a negative value.
  • Example: $|3x - 1| > 5 \implies 3x - 1 > 5$ (gives $x > 2$) or $3x - 1 < -5$ (gives $x < -4/3$)

VI. Graphing Linear Equations

• Slope-Intercept Form: $y = mx + b$
  • $m$: slope (rise over run)
  • $b$: y-intercept (where the line crosses the y-axis)
  • Graphing: Plot the y-intercept, then use the slope to find other points.
    • Example: $y = 2x - 3$. Plot $(0, -3)$, then rise 2, run 1 to get $(1, -1)$, etc.
    • Example: $y = -\frac{3}{4}x + 5$. Plot $(0, 5)$, then rise -3 (down 3), run 4 (right 4) to get $(4, 2)$.
• Standard Form: $Ax + By = C$
  • Intercept Method: Find the x-intercept (set $y=0$) and the y-intercept (set $x=0$), then plot these two points and draw a line through them.
    • Example: $2x + 3y = 6$.
      • x-intercept: $2x = 6 \implies x = 3$. Point: $(3, 0)$.
      • y-intercept: $3y = 6 \implies y = 2$. Point: $(0, 2)$.
    • Example: $3x - 4y = 12$.
      • x-intercept: $3x = 12 \implies x = 4$. Point: $(4, 0)$.
      • y-intercept: $-4y = 12 \implies y = -3$. Point: $(0, -3)$.

VII. Function Transformations

• Parent Functions: Basic shapes like $y = |x|$ (V-shape) and $y = x^2$ (U-shape/parabola).
• Transformations:
  • Vertical Shift: Add/subtract a constant outside the function (e.g., $y = |x| + 2$ shifts up 2).
  • Horizontal Shift: Add/subtract a constant inside the function (e.g., $y = |x - 3|$ shifts right 3; $y = |x + 2|$ shifts left 2).
  • Reflection: Multiply the function by -1 (e.g., $y = -|x|$ reflects across the x-axis).
  • Stretching/Compressing: Multiply the function or the variable by a factor (e.g., $y = 2|x|$ stretches vertically; $y = |2x|$ compresses horizontally).
• Combining Transformations: Apply shifts, reflections, and stretches/compressions in order.
  • Example: $y = -|x - 3| + 2$ shifts right 3, reflects across x-axis, and shifts up 2.

VIII. Solving Quadratic Equations

• Factoring:
  • Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$
    • Example: $x^2 - 25 = (x - 5)(x + 5) = 0 \implies x = 5$ or $x = -5$
    • Example: $4x^2 - 49 = (2x - 7)(2x + 7) = 0 \implies x = 7/2$ or $x = -7/2$
  • Trinomials (leading coefficient = 1): Find two numbers that multiply to the constant term and add to the middle coefficient.
    • Example: $x^2 + 10x + 24 = (x + 4)(x + 6) = 0 \implies x = -4$ or $x = -6$
  • Trinomials (leading coefficient ≠ 1): Multiply the leading coefficient by the constant term, find two numbers that multiply to this product and add to the middle coefficient, then use factoring by grouping.
    • Example: $2x^2 + 5x - 12 = 0$. Multiply $2 \times -12 = -24$. Numbers are 8 and -3. Rewrite as $2x^2 + 8x - 3x - 12 = 0$. Factor by grouping: $2x(x + 4) - 3(x + 4) = 0 \implies (2x - 3)(x + 4) = 0 \implies x = 3/2$ or $x = -4$.
• Quadratic Formula: For $ax^2 + bx + c = 0$, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • Used when factoring is difficult or impossible.
  • Example: $x^2 - 13x + 42 = 0$. $a=1, b=-13, c=42$. $x = \frac{13 \pm \sqrt{(-13)^2 - 4(1)(42)}}{2(1)} = \frac{13 \pm \sqrt{169 - 168}}{2} = \frac{13 \pm \sqrt{1}}{2} = \frac{13 \pm 1}{2}$. Solutions: $x = 14/2 = 7$ and $x = 12/2 = 6$.
• Working Backwards from Solutions: If solutions are $x=6$ and $x=11/12$, the factors are $(x-6)$ and $(x - 11/12)$. To remove the fraction, multiply the second factor by 12: $(x-6)(12x - 11)$.

IX. Complex Numbers

• Imaginary Unit (i): $i = \sqrt{-1}$
• Properties:
  • $i^2 = -1$
  • $i^3 = -i$
  • $i^4 = 1$
• Solving Equations with Negative Square Roots:
  • Example: $x^2 = -9 \implies x = \pm \sqrt{-9} = \pm \sqrt{9 \times -1} = \pm 3i$

X. Systems of Equations

• Solving for Two Variables: Requires two equations.
• Elimination Method: Align variables and add or subtract equations to eliminate one variable.
  • Example: $2x + y = 5$ $3x - y = 0$ Adding gives $5x = 5 \implies x = 1$. Substitute $x=1$ into the first equation: $2(1) + y = 5 \implies y = 3$. Solution: $(1, 3)$.
  • Multiplying Equations: If variables don't cancel directly, multiply one or both equations by constants to make coefficients opposites.
    • Example: $3x + 4y = 25$ $5x - 3y = 3$ Multiply first by 3, second by 4: $9x + 12y = 75$ $20x - 12y = 12$ Adding gives $29x = 87 \implies x = 3$. Substitute $x=3$ into $3x + 4y = 25$: $3(3) + 4y = 25 \implies 9 + 4y = 25 \implies 4y = 16 \implies y = 4$. Solution: $(3, 4)$.
• Substitution Method: Solve one equation for one variable, then substitute that expression into the other equation.
  • Example: $y = 2x + 5$ $5x - 4y = 10$ Substitute $(2x + 5)$ for $y$ in the second equation: $5x - 4(2x + 5) = 10 \implies 5x - 8x - 20 = 10 \implies -3x = 30 \implies x = -10$. Substitute $x=-10$ into $y = 2x + 5$: $y = 2(-10) + 5 = -20 + 5 = -15$. Solution: $(-10, -15)$.
• Graphical Method: The solution is the point of intersection of the graphed lines.

XI. Functions

• Evaluating Functions: Substitute a value for the input variable (e.g., $x$) into the function's expression.
  • Example: If $f(x) = 5x + 4$, then $f(3) = 5(3) + 4 = 15 + 4 = 19$.
• Finding Input from Output: Set the function equal to the given output value and solve for the input variable.
  • Example: If $f(x) = 5x + 4$ and $f(x) = 39$, then $5x + 4 = 39 \implies 5x = 35 \implies x = 7$.
• Synthetic Division: A shortcut for evaluating functions, especially polynomials, by using the coefficients and the input value. The remainder is the function's value.
• Composite Functions: Applying one function to the result of another.
  • $f(g(x))$: Substitute the entire function $g(x)$ into the $x$ of function $f(x)$.
  • $g(f(x))$: Substitute the entire function $f(x)$ into the $x$ of function $g(x)$.
  • Example: $f(x) = 3x + 5$, $g(x) = x^2 - 4$.
    • $f(g(x)) = 3(x^2 - 4) + 5 = 3x^2 - 12 + 5 = 3x^2 - 7$.
    • $g(f(x)) = (3x + 5)^2 - 4 = (9x^2 + 30x + 25) - 4 = 9x^2 + 30x + 21$.
• Evaluating Composite Functions: First evaluate the inner function, then use that result as the input for the outer function.
  • Example: $f(g(2))$. First find $g(2) = 2^2 - 4 = 0$. Then find $f(0) = 3(0) + 5 = 5$. So $f(g(2)) = 5$.

XII. Inverse Functions

• Definition: If $(a, b)$ is a point on the graph of $f(x)$, then $(b, a)$ is a point on the graph of its inverse, $f^{-1}(x)$. The x and y values are swapped.
• Graphical Property: Inverse functions are reflections of each other across the line $y = x$.
• Finding the Inverse:
  1. Replace $f(x)$ with $y$.
  2. Swap $x$ and $y$.
  3. Solve for $y$.
  4. Replace $y$ with $f^{-1}(x)$.
  • Example: $f(x) = 7x - 5$.
    1. $y = 7x - 5$
    2. $x = 7y - 5$
    3. $x + 5 = 7y \implies y = \frac{x+5}{7}$
    4. $f^{-1}(x) = \frac{x+5}{7}$
• Proving Inverses: Two functions $f(x)$ and $g(x)$ are inverses if $f(g(x)) = x$ AND $g(f(x)) = x$.