methods / polynomials.mdon commit [spec] DEs of the dependent variable (a978feb)
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   2geometry: a4paper, margin=2cm
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   4author: Andrew Lorimer
   5header-includes:
   6- \usepackage{setspace}
   7- \usepackage{fancyhdr}
   8- \pagestyle{fancy}
   9- \fancyhead[LO,LE]{Year 12 Methods}
  10- \fancyhead[CO,CE]{Andrew Lorimer}
  11- \usepackage{graphicx}
  12- \usepackage{tabularx}
  13- \usepackage[dvipsnames]{xcolor}
  14---
  15
  16\setstretch{1.3}
  17\definecolor{cas}{HTML}{e6f0fe}
  18\pagenumbering{gobble}
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  20
  21# Polynomials
  22
  23## Quadratics
  24
  25\newcolumntype{R}{>{\raggedleft\arraybackslash}X}
  26\begin{tabularx}{\columnwidth}{Rl}
  27  General form& \parbox[t]{5cm}{$x^2 + bx + c = (x+m)(x+n)$\\ where $mn=c, \> m+n=b$} \\
  28  \hline
  29  Difference of squares & $a^2 - b^2 = (a - b)(a + b)$ \\
  30  \hline
  31  Perfect squares & \parbox[c]{5cm}{$a^2 \pm 2ab + b^2 = (a \pm b^2)$} \\
  32  \hline
  33  Completing the square & \parbox[t]{5cm}{$x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$ \\ $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$} \\
  34  \hline
  35  Quadratic formula & $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ \\
  36\end{tabularx}
  37
  38## Cubics
  39
  40**Difference of cubes:** $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$  
  41**Sum of cubes:** $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$  
  42**Perfect cubes:** $a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3$  
  43
  44$$y=a(bx-h)^3 + c$$
  45
  46- $m=0$ at *stationary point of inflection* (i.e. (${h \over b}, k)$)
  47- in form $y=(x-a)^2(x-b)$, local max at $x=a$, local min at $x=b$
  48- in form $y=a(x-b)(x-c)(x-d)$: $x$-intercepts at $b, c, d$
  49- in form $y=a(x-b)^2(x-c)$, touches $x$-axis at $b$, intercept at $c$
  50
  51## Linear and quadratic graphs
  52
  53### Forms of linear equations
  54
  55$y=mx+c$ where $m$ is gradient and $c$ is $y$-intercept  
  56${x \over a} + {y \over b}=1$ where $m$ is gradient and $(x_1, y_1)$ lies on the graph  
  57$y-y_1 = m(x-x_1)$ where $(a,0)$ and $(0,b)$ are $x$- and $y$-intercepts
  58
  59## Line properties
  60
  61Parallel lines: $m_1 = m_2$  
  62Perpendicular lines: $m_1 \times m_2 = -1$  
  63Distance: $|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
  64
  65## Quartic graphs
  66
  67### Forms of quadratic equations
  68$y=ax^4$  
  69$y=a(x-b)(x-c)(x-d)(x-e)$  
  70$y=ax^4+cd^2 (c \ge 0)$  
  71$y=ax^2(x-b)(x-c)$  
  72$y=a(x-b)^2(x-c)^2$  
  73$y=a(x-b)(x-c)^3$
  74
  75## Simultaneous equations (linear)
  76
  77- **Unique solution** - lines intersect at point
  78- **Infinitely many solutions** - lines are equal
  79- **No solution** - lines are parallel
  80
  81### Solving $\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>$ for $\{0,1,\infty\}$ solutions
  82
  83where all coefficients are known except for one, and $a, b$ are known
  84
  851. Write as matrices: $\begin{bmatrix}p & q \\ r & s \end{bmatrix}
  86  \begin{bmatrix} x \\ y \end{bmatrix}
  87  =
  88  \begin{bmatrix} a \\ b \end{bmatrix}$
  892. Find determinant of first matrix: $\Delta = ps-qr$
  903. Let $\Delta = 0$ for number of solutions $\ne 1$  
  91   or let $\Delta \ne 0$ for one unique solution.
  924. Solve determinant equation to find variable  
  93   - *--- for infinite/no solutions: ---*
  945. Substitute variable into both original equations
  956. Rearrange equations so that LHS of each is the same
  967. $\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x$ ($\infty$ solns)  
  97   $\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x$ (0 solns)
  98
  99\colorbox{cas}{On CAS:} Matrix $\rightarrow$ `det`
 100
 101### Solving $\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\
 102a_2 x + b_2 y + c_2 z = d_2 \\
 103a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}$
 104
 105- Use elimination
 106- Generate two new equations with only two variables
 107- Rearrange & solve
 108- Substitute one variable into another equation to find another variable
 109- etc.