c3d38f37b609cecf6e5d3d8f42a086762757ca04
   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### Forms of linear equations
  21
  22$y=mx+c$ where $m$ is gradient and $c$ is $y$-intercept  
  23${x \over a} + {y \over b}=1$ where $m$ is gradient and $(x_1, y_1)$ lies on the graph  
  24$y-y_1 = m(x-x_1)$ where $(a,0)$ and $(0,b)$ are $x$- and $y$-intercepts
  25
  26## Line properties
  27
  28Parallel lines: $m_1 = m_2$  
  29Perpendicular lines: $m_1 \times m_2 = -1$  
  30Distance: $\vec{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
  31
  32
  33## Cubic graphs
  34
  35$$y=a(x-b)^3 + c$$
  36
  37- $m=0$ at *stationary point of inflection*
  38- in form $y=(x-a)^2(x-b)$, local max at $x=a$, local min at $x=b$
  39- in form $y=a(x-b)(x-c)(x-d)$: $x$-intercepts at $b, c, d$
  40
  41
  42## Quartic graphs
  43
  44### Forms of quadratic equations
  45$y=ax^4$  
  46$y=a(x-b)(x-c)(x-d)(x-e)$  
  47$y=ax^4+cd^2 (c \ge 0)$  
  48$y=ax^2(x-b)(x-c)$  
  49$y=a(x-b)^2(x-c)^2$  
  50$y=a(x-b)(x-c)^3$
  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
  62
  63
  64### Solving $\begin{cases}px + qy = a \\ rx + sy = b\end{cases}$ for one, infinite and no solutions
  65
  66where all coefficients are known except for one, and $a, b$ are known
  67
  681. Write as matrices: $\begin{bmatrix}p & q \\ r & s \end{bmatrix}
  69  \begin{bmatrix} x \\ y \end{bmatrix}
  70  =
  71  \begin{bmatrix} a \\ b \end{bmatrix}$
  722. Find determinant of first matrix: $\Delta = ps-qr$
  733. Let $\Delta = 0$ for number of solutions $\ne 1$  
  74   or let $\Delta \ne 0$ for one unique solution.
  754. Solve determinant equation to find variable  
  76   - *--- for infinite/no solutions: ---*
  775. Substitute variable into both original equations
  786. Rearrange equations so that LHS of each is the same
  797. If $\text{RHS}(1) = \text{RHS}(2)$, lines are coincident (infinite solutions)  
  80   If $\text{RHS}(1) \ne \text{RHS}(2)$, lines are parallel (no solutions)
  81
  82Or use Matrix -> `det` on CAS.
  83
  84### Solving $\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\
  85a_2 x + b_2 y + c_2 z = d_2 \\
  86a_3 x + b_3 y + c_3 z = d_3\end{cases}$
  87
  88- Use elimination
  89- Generate two new equations with only two variables
  90- Rearrange & solve
  91- Substitute one variable into another equation to find another variable
  92- etc.