Andrew's git - notes.git/blob - methods/polynomials.md

   1# Polynomials
   2
   3## Factorising
   4
   5#### Quadratics
   6**Quadratics:** $x^2 + bx + c = (x+m)(x+n)$ where $mn=c$, $m+n=b$  
   7**Difference of squares:** $a^2 - b^2 = (a - b)(a + b)$  
   8**Perfect squares:** $a^2 \pm 2ab + b^2 = (a \pm b^2)$  
   9**Completing the square (monic):** $x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$  
  10**Completing the square (non-monic):** $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$  
  11**Quadratic formula:** $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ (if $\Delta$ is a perfect square, rational roots)
  12
  13#### Cubics
  14**Difference of cubes:** $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$  
  15**Sum of cubes:** $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$  
  16**Perfect cubes:** $a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3$  
  17
  18## Linear and quadratic graphs
  19
  20$$y=mx+c, \quad {x \over a} + {y \over b}=1$$
  21
  22Parallel lines - $m_1 = m_2$  
  23Perpendicular lines - $m_1 \times m_2 = -1$
  24
  25
  26## Cubic graphs
  27
  28$$y=a(x-b)^3 + c$$
  29
  30- $m=0$ at *stationary point of inflection*
  31- in form $y=(x-a)^2(x-b)$, local max at $x=a$, local min at $x=b$
  32- in form $y=a(x-b)(x-c)(x-d)$: $x$-intercepts at $b, c, d$
  33
  34
  35## Quartic graphs
  36
  37$$y=ax^4$$
  38
  39$$=a(x-b)(x-c)(x-d)(x-e)$$
  40
  41$$=ax^4+cd^2 (c \ge 0)$$
  42
  43$$=ax^2(x-b)(x-c)$$
  44
  45$$=a(x-b)^2(x-c)^2$$
  46
  47$$=a(x-b)(x-c)^3$$
  48
  49where
  50- $x$-intercepts at $x=b,c,d,e$
  51
  52## Literal equations
  53
  54Equations with multiple pronumerals. Solutions are expressed in terms of pronumerals (parameters))
  55
  56## Simultaneous equations (linear)
  57
  58- **Unique solution** - lines intersect at point
  59- **Infinitely many solutions** - lines are equal
  60- **No solution** - lines are parallel
  61
  62Solving in matrix form - use inverse $A^{-1}= {1 \over {ad-bc}}\begin{bmatrix}d & -b \\ -c & a\end{bmatrix}$. $A^{-1}$ exists for infinite solutions or no solutions ($ad-bc=0$), does not exist for unique solutions ($ad-bc \ne 0$).  
  63Or use Matrix -> `det` on CAS.