c826306003a45111c78051b0c8446e844a2e2727
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  10
  11# Polynomials
  12
  13## Quadratics
  14
  15\newcolumntype{R}{>{\raggedleft\arraybackslash}X}
  16\begin{tabularx}{\columnwidth}{|R|l|}
  17  Quadratics & $x^2 + bx + c = (x+m)(x+n)$ \\
  18  & where $mn=c, \> m+n=b$ \\
  19  Difference of squares & $a^2 - b^2 = (a - b)(a + b)$ \\
  20  Perfect squares & $a^2 \pm 2ab + b^2 = (a \pm b^2)$ \\
  21  Completing the square & \parbox[t]{5cm}{$x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$ \\ $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$} \\
  22  Quadratic formula & $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ \\
  23\end{tabularx}
  24
  25## Cubics
  26
  27**Difference of cubes:** $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$  
  28**Sum of cubes:** $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$  
  29**Perfect cubes:** $a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3$  
  30
  31## Linear and quadratic graphs
  32
  33### Forms of linear equations
  34
  35$y=mx+c$ where $m$ is gradient and $c$ is $y$-intercept  
  36${x \over a} + {y \over b}=1$ where $m$ is gradient and $(x_1, y_1)$ lies on the graph  
  37$y-y_1 = m(x-x_1)$ where $(a,0)$ and $(0,b)$ are $x$- and $y$-intercepts
  38
  39## Line properties
  40
  41Parallel lines: $m_1 = m_2$  
  42Perpendicular lines: $m_1 \times m_2 = -1$  
  43Distance: $\vec{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
  44
  45
  46## Cubic graphs
  47
  48$$y=a(bx-h)^3 + c$$
  49
  50- $m=0$ at *stationary point of inflection* (i.e. (${h \over b}, k)$)
  51- in form $y=(x-a)^2(x-b)$, local max at $x=a$, local min at $x=b$
  52- in form $y=a(x-b)(x-c)(x-d)$: $x$-intercepts at $b, c, d$
  53- in form $y=a(x-b)^2(x-c)$, touches $x$-axis at $b$, intercept at $c$
  54
  55## Quartic graphs
  56
  57### Forms of quadratic equations
  58$y=ax^4$  
  59$y=a(x-b)(x-c)(x-d)(x-e)$  
  60$y=ax^4+cd^2 (c \ge 0)$  
  61$y=ax^2(x-b)(x-c)$  
  62$y=a(x-b)^2(x-c)^2$  
  63$y=a(x-b)(x-c)^3$
  64
  65## Literal equations
  66
  67Equations with multiple pronumerals. Solutions are expressed in terms of pronumerals (parameters)
  68
  69## Simultaneous equations (linear)
  70
  71- **Unique solution** - lines intersect at point
  72- **Infinitely many solutions** - lines are equal
  73- **No solution** - lines are parallel
  74
  75### Solving $\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases}$ for one, infinite and no solutions
  76
  77where all coefficients are known except for one, and $a, b$ are known
  78
  791. Write as matrices: $\begin{bmatrix}p & q \\ r & s \end{bmatrix}
  80  \begin{bmatrix} x \\ y \end{bmatrix}
  81  =
  82  \begin{bmatrix} a \\ b \end{bmatrix}$
  832. Find determinant of first matrix: $\Delta = ps-qr$
  843. Let $\Delta = 0$ for number of solutions $\ne 1$  
  85   or let $\Delta \ne 0$ for one unique solution.
  864. Solve determinant equation to find variable  
  87   - *--- for infinite/no solutions: ---*
  885. Substitute variable into both original equations
  896. Rearrange equations so that LHS of each is the same
  907. If $\text{RHS}(1) = \text{RHS}(2)$, lines are coincident (infinite solutions)  
  91   If $\text{RHS}(1) \ne \text{RHS}(2)$, lines are parallel (no solutions)
  92
  93Or use Matrix -> `det` on CAS.
  94
  95### Solving $\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\
  96a_2 x + b_2 y + c_2 z = d_2 \\
  97a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}$
  98
  99- Use elimination
 100- Generate two new equations with only two variables
 101- Rearrange & solve
 102- Substitute one variable into another equation to find another variable
 103- etc.