1# Polynomials
2## Factorising
4#### 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#### 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## Linear and quadratic graphs
19$$y=mx+c, \quad {x \over a} + {y \over b}=1$$
21Parallel lines - $m_1 = m_2$
23Perpendicular lines - $m_1 \times m_2 = -1$
24## Cubic graphs
27$$y=a(x-b)^3 + c$$
29- $m=0$ 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## Quartic graphs
36$$y=ax^4$$
38$$=a(x-b)(x-c)(x-d)(x-e)$$
40$$=ax^4+cd^2 (c \ge 0)$$
42$$=ax^2(x-b)(x-c)$$
44$$=a(x-b)^2(x-c)^2$$
46$$=a(x-b)(x-c)^3$$
48where
50- $x$-intercepts at $x=b,c,d,e$