+---
+geometry: margin=1.5cm
+columns: 2
+header-includes:
+- \usepackage{tabularx}
+---
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+\renewcommand{\arraystretch}{1.4}
+
# Polynomials
-## Factorising
+## Quadratics
+
+\newcolumntype{R}{>{\raggedleft\arraybackslash}X}
+\begin{tabularx}{\columnwidth}{|R|l|}
+ Quadratics & $x^2 + bx + c = (x+m)(x+n)$ \\
+ & where $mn=c, \> m+n=b$ \\
+ Difference of squares & $a^2 - b^2 = (a - b)(a + b)$ \\
+ Perfect squares & $a^2 \pm 2ab + b^2 = (a \pm b^2)$ \\
+ Completing the square & \parbox[t]{5cm}{$x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$ \\ $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$} \\
+ Quadratic formula & $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ \\
+\end{tabularx}
-#### Quadratics
-**Quadratics:** $x^2 + bx + c = (x+m)(x+n)$ where $mn=c$, $m+n=b$
-**Difference of squares:** $a^2 - b^2 = (a - b)(a + b)$
-**Perfect squares:** $a^2 \pm 2ab + b^2 = (a \pm b^2)$
-**Completing the square (monic):** $x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$
-**Completing the square (non-monic):** $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$
-**Quadratic formula:** $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ (if $\Delta$ is a perfect square, rational roots)
+## Cubics
-#### Cubics
**Difference of cubes:** $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$
**Sum of cubes:** $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$
**Perfect cubes:** $a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3$
## Cubic graphs
-$$y=a(x-b)^3 + c$$
+$$y=a(bx-h)^3 + c$$
-- $m=0$ at *stationary point of inflection*
+- $m=0$ at *stationary point of inflection* (i.e. (${h \over b}, k)$)
- in form $y=(x-a)^2(x-b)$, local max at $x=a$, local min at $x=b$
- in form $y=a(x-b)(x-c)(x-d)$: $x$-intercepts at $b, c, d$
-
+- in form $y=a(x-b)^2(x-c)$, touches $x$-axis at $b$, intercept at $c$
## Quartic graphs
- **Infinitely many solutions** - lines are equal
- **No solution** - lines are parallel
-
-
-### Solving $\begin{cases}px + qy = a \\ rx + sy = b\end{cases}$ for one, infinite and no solutions
+### Solving $\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases}$ for one, infinite and no solutions
where all coefficients are known except for one, and $a, b$ are known
Or use Matrix -> `det` on CAS.
-### Solving $\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\
+### Solving $\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\
a_2 x + b_2 y + c_2 z = d_2 \\
-a_3 x + b_3 y + c_3 z = d_3\end{cases}$
+a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}$
- Use elimination
- Generate two new equations with only two variables
- Rearrange & solve
- Substitute one variable into another equation to find another variable
-- etc.
\ No newline at end of file
+- etc.