methods / polynomials.mdon commit [methods] clarify cubic points of inflection (d2489a6)
   1---
   2geometry: margin=1.5cm
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   4
   5# Polynomials
   6
   7## Factorising
   8
   9#### Quadratics
  10**Quadratics:** $x^2 + bx + c = (x+m)(x+n)$ where $mn=c$, $m+n=b$  
  11**Difference of squares:** $a^2 - b^2 = (a - b)(a + b)$  
  12**Perfect squares:** $a^2 \pm 2ab + b^2 = (a \pm b^2)$  
  13**Completing the square (monic):** $x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$  
  14**Completing the square (non-monic):** $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$  
  15**Quadratic formula:** $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ (if $\Delta$ is a perfect square, rational roots)
  16
  17#### Cubics
  18**Difference of cubes:** $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$  
  19**Sum of cubes:** $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$  
  20**Perfect cubes:** $a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3$  
  21
  22## Linear and quadratic graphs
  23
  24### Forms of linear equations
  25
  26$y=mx+c$ where $m$ is gradient and $c$ is $y$-intercept  
  27${x \over a} + {y \over b}=1$ where $m$ is gradient and $(x_1, y_1)$ lies on the graph  
  28$y-y_1 = m(x-x_1)$ where $(a,0)$ and $(0,b)$ are $x$- and $y$-intercepts
  29
  30## Line properties
  31
  32Parallel lines: $m_1 = m_2$  
  33Perpendicular lines: $m_1 \times m_2 = -1$  
  34Distance: $\vec{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
  35
  36
  37## Cubic graphs
  38
  39$$y=a(bx-h)^3 + c$$
  40
  41- $m=0$ at *stationary point of inflection* (i.e. ({h \over b}, k)$)
  42- in form $y=(x-a)^2(x-b)$, local max at $x=a$, local min at $x=b$
  43- in form $y=a(x-b)(x-c)(x-d)$: $x$-intercepts at $b, c, d$
  44- in form $y=a(x-b)^2(x-c)$, touches $x$-axis at $b$, intercept at $c$
  45
  46## Quartic graphs
  47
  48### Forms of quadratic equations
  49$y=ax^4$  
  50$y=a(x-b)(x-c)(x-d)(x-e)$  
  51$y=ax^4+cd^2 (c \ge 0)$  
  52$y=ax^2(x-b)(x-c)$  
  53$y=a(x-b)^2(x-c)^2$  
  54$y=a(x-b)(x-c)^3$
  55
  56## Literal equations
  57
  58Equations with multiple pronumerals. Solutions are expressed in terms of pronumerals (parameters)
  59
  60## Simultaneous equations (linear)
  61
  62- **Unique solution** - lines intersect at point
  63- **Infinitely many solutions** - lines are equal
  64- **No solution** - lines are parallel
  65
  66### Solving $\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases}$ for one, infinite and no solutions
  67
  68where all coefficients are known except for one, and $a, b$ are known
  69
  701. Write as matrices: $\begin{bmatrix}p & q \\ r & s \end{bmatrix}
  71  \begin{bmatrix} x \\ y \end{bmatrix}
  72  =
  73  \begin{bmatrix} a \\ b \end{bmatrix}$
  742. Find determinant of first matrix: $\Delta = ps-qr$
  753. Let $\Delta = 0$ for number of solutions $\ne 1$  
  76   or let $\Delta \ne 0$ for one unique solution.
  774. Solve determinant equation to find variable  
  78   - *--- for infinite/no solutions: ---*
  795. Substitute variable into both original equations
  806. Rearrange equations so that LHS of each is the same
  817. If $\text{RHS}(1) = \text{RHS}(2)$, lines are coincident (infinite solutions)  
  82   If $\text{RHS}(1) \ne \text{RHS}(2)$, lines are parallel (no solutions)
  83
  84Or use Matrix -> `det` on CAS.
  85
  86### Solving $\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\
  87a_2 x + b_2 y + c_2 z = d_2 \\
  88a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}$
  89
  90- Use elimination
  91- Generate two new equations with only two variables
  92- Rearrange & solve
  93- Substitute one variable into another equation to find another variable
  94- etc.