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# Polynomials
## Factorising
**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 $\Delta=b^2-4ac$
+**Quadratic formula:** $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ (if $\Delta$ is a perfect square, rational roots)
#### 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:** $a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3$
+
+## Linear and quadratic graphs
+
+### Forms of linear equations
+
+$y=mx+c$ where $m$ is gradient and $c$ is $y$-intercept
+${x \over a} + {y \over b}=1$ where $m$ is gradient and $(x_1, y_1)$ lies on the graph
+$y-y_1 = m(x-x_1)$ where $(a,0)$ and $(0,b)$ are $x$- and $y$-intercepts
+
+## Line properties
+
+Parallel lines: $m_1 = m_2$
+Perpendicular lines: $m_1 \times m_2 = -1$
+Distance: $\vec{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
+
+
+## Cubic graphs
+
+$$y=a(x-b)^3 + c$$
+
+- $m=0$ at *stationary point of inflection*
+- 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 $b, c, d$
+
+
+## Quartic graphs
+
+### Forms of quadratic equations
+$y=ax^4$
+$y=a(x-b)(x-c)(x-d)(x-e)$
+$y=ax^4+cd^2 (c \ge 0)$
+$y=ax^2(x-b)(x-c)$
+$y=a(x-b)^2(x-c)^2$
+$y=a(x-b)(x-c)^3$
+
+## Literal equations
+
+Equations with multiple pronumerals. Solutions are expressed in terms of pronumerals (parameters)
+
+## Simultaneous equations (linear)
+
+- **Unique solution** - lines intersect at point
+- **Infinitely many solutions** - lines are equal
+- **No solution** - lines are parallel
+
+### 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
+
+1. Write as matrices: $\begin{bmatrix}p & q \\ r & s \end{bmatrix}
+ \begin{bmatrix} x \\ y \end{bmatrix}
+ =
+ \begin{bmatrix} a \\ b \end{bmatrix}$
+2. Find determinant of first matrix: $\Delta = ps-qr$
+3. Let $\Delta = 0$ for number of solutions $\ne 1$
+ or let $\Delta \ne 0$ for one unique solution.
+4. Solve determinant equation to find variable
+ - *--- for infinite/no solutions: ---*
+5. Substitute variable into both original equations
+6. Rearrange equations so that LHS of each is the same
+7. If $\text{RHS}(1) = \text{RHS}(2)$, lines are coincident (infinite solutions)
+ If $\text{RHS}(1) \ne \text{RHS}(2)$, lines are parallel (no solutions)
+
+Or use Matrix -> `det` on CAS.
+
+### Solving $\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\
+a_2 x + b_2 y + c_2 z = d_2 \\
+a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}$
+
+- Use elimination
+- Generate two new equations with only two variables
+- Rearrange & solve
+- Substitute one variable into another equation to find another variable
+- etc.
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