Concise Pre-Calculus & Algebra II Key Concepts

A. Order of Operations

• Follow PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right).

• Apply inside parentheses first, then exponents, then multiply/divide left-to-right, then add/subtract left-to-right.

• Use substitution after simplifying if expressions involve variables.

Example template: Evaluate expressions by substituting values after simplifying inside parentheses and applying order rules.

$ext{General rule: } a(b+c) = ab + ac <br>$

B. Solve Equations

• For linear equations: expand, collect like terms, then isolate the variable.

• Check solutions by substituting back.

• Common forms include explicit, factored, and isolated-variable steps.

C. Linear Equations - Graphs, Slope and Intercept

• Slope-intercept form: $y = mx + b$ where m is slope and b is y-intercept.

• Slope: $m = rac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$ (two-point form).

• x-intercept: set $y=0$ and solve for x; y-intercept: set $x=0$ and solve for y.

• Point-slope form: $y - y<em>1 = m(x - x</em>1)$.

• Given a point, you can form the equation using slope if known.

• Example: For line 2x - 3y = 6, solve for y: $-3y = -2x + 6 \ y = frac{2}{3}x - 2$.

D. Add, Subtract and Multiply Polynomials

• Operations:

  • Subtract polynomials: distribute the minus sign correctly.

  • Multiply polynomials using FOIL (binomials) or distribution.

  • Combine like terms.

• Quick results (examples):

  • $(2x^2+3x-4) - (x^2+x-1) = x^2+2x-3$

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

  • $2x(4x^2-3x+2) = 8x^3-6x^2+4x$

  • $(4x-7)(3x+2) = 12x^2-13x-14$

  • $(2x-5)^2 = 4x^2-20x+25$

  • $(x-3)(x^2+2x-3) = x^3 - x^2 -9x + 9$

E. Factoring - GCF, Trinomials, Difference of Squares

• Steps:

  • Always factor out a Greatest Common Factor first.

  • Look for special products: difference of squares, sum/difference of cubes, grouping when 4 terms.

  • For trinomials ax^2+bx+c, factor by trial or via ac-method (find two numbers with product ac and sum b).

• Key identities:

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

  • Sum/Difference of cubes: a^3 \n + b^3 = (a+b)(a^2 - ab + b^2) and $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$

• Examples:

  • $x^2 - 49 = (x-7)(x+7)$

  • $x^2 + 4x - 21 = (x+7)(x-3)$

  • $x^2 - 16x + 64 = (x-8)^2$

  • $2x^2 - 5x - 12 = (2x+3)(x-4)$

  • $x^3 + 5x^2 - 9x - 45 = (x+5)(x^2-9) = (x+5)(x-3)(x+3)$

F. Simplify Using Exponent Rules

• Core rules:

  • $a^m a^n = a^{m+n}$

  • $(ab)^m = a^m b^m$

  • $(a^m)^n = a^{mn}$

  • $rac{a^m}{a^n} = a^{m-n}$

  • $a^{-n} = rac{1}{a^n},\ a^0 = 1$

• Notes:

  • Handle negative bases and fractional exponents with care.

G. Quadratic Equations

• Solve by factoring or using the Quadratic Formula.

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

• Discriminant: $\Delta = b^2 - 4ac$ determines number/type of real solutions:

  • \Delta > 0 two real solutions

  • $\Delta = 0$ one real solution

  • \Delta < 0 no real solutions (two complex conjugates)

• Vertex coordinates for $y = ax^2 + bx + c$:

  • $x<em>v = -\frac{b}{2a}$, $y</em>v = f(x_v)$

• Isolate and square-root steps: if equation is $x^2 = k$ then $x = \pm \sqrt{k}$.

H. Complex (Imaginary) Numbers

• Complex form: $a + bi$ with $i^2 = -1$.

• Basic operations:

  • Addition/Subtraction: $(a+bi) \pm (c+di) = (a+c) + (b+d)i$

  • Multiplication: $(ac - bd) + (ad + bc)i$

• Simplify roots involving negatives:

  • $\sqrt{-k} = i\sqrt{k}$

• Examples: solving equations may yield real or complex solutions depending on discriminant.

I. Radicals and Rational Exponents

• Convert between radical and fractional exponent forms:

  • $x^{m/n} = \sqrt[n]{x^m}$

• Simplify radicals: prime-factor inside, pull out perfect squares, etc.

  • $\sqrt{18} = 3\sqrt{2},\ \sqrt{8} = 2\sqrt{2}$

• Domain considerations: even roots require nonnegative radicands in real numbers.

• Rational exponents with negative bases require care (when defined).

• Examples of simplification and combining like radicals.

J. Rational Expressions

• Key operations:

  • Factor numerators and denominators and cancel common factors (watch domain restrictions).

  • When dividing by a fraction, multiply by its reciprocal; simplify.

• Complex fractions:

  • Use LCD to clear complex fractions, or multiply numerator and denominator by the reciprocal of the denominator.

• Common steps:

  • Factor, cancel, and record domain restrictions where denominators are zero.

K. Common Errors in Algebra (Section 3)

• Errors with parentheses:

  • Distributing a negative sign: 3 - (x - 2) = 3 - x - 2, not 3x or similar.

  • Distributing left and right incorrectly across a product or sum.

  • Do not distribute exponents over addition/subtraction: (a + b)^2 ≠ a^2 + b^2.

  • Do not distribute when only multiplication is present: (ab)^2 = a^2 b^2, not ab^2 or similar.

• Fraction errors:

  • Incorrect combining of fractions; ensure common denominators when adding/subtracting.

  • Correct across complex fractions using reciprocal or LCD methods.

• Exponents:

  • Confusing nested exponents: (x^2)^3 = x^{6}, not x^{5} or similar.

  • Steps for calculator is to respect exponent precedence and parentheses.

• Radicals:

  • Misapplying √ rules: √(a^2 + b^2) ≠ √a^2 + √b^2; improper order of operations under radicals.

  • Correctly simplify nested radicals and avoid mixing radical signs improperly.

• Dividing out common factors:

  • Ensure factoring is done correctly before canceling; watch for restrictions on x in denominators.

• Formula sheet at a glance (highlights to memorize):

  • Slope-intercept: $y = mx + b$

  • Point-slope: $y - y<em>1 = m(x - x</em>1)$

  • Vertex: $x_v = -\frac{b}{2a}$

  • Discriminant: $\Delta = b^2 - 4ac$

  • Quadratic formula: $x = \frac{-b\pm \sqrt{\Delta}}{2a}$

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

  • Sum/Difference of cubes: $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$