1--- 2geometry: margin=1.5cm 3--- 4 5# Polynomials 6 7## Factorising 8 9#### Quadratics 10**Quadratics:** $x^2 + bx + c = (x+m)(x+n)$ where $mn=c$, $m+n=b$ 11**Difference of squares:** $a^2 - b^2 = (a - b)(a + b)$ 12**Perfect squares:** $a^2 \pm 2ab + b^2 = (a \pm b^2)$ 13**Completing the square (monic):** $x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$ 14**Completing the square (non-monic):** $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$ 15**Quadratic formula:** $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ (if $\Delta$ is a perfect square, rational roots) 16 17#### Cubics 18**Difference of cubes:** $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$ 19**Sum of cubes:** $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$ 20**Perfect cubes:** $a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3$ 21 22## Linear and quadratic graphs 23 24### Forms of linear equations 25 26$y=mx+c$ where $m$ is gradient and $c$ is $y$-intercept 27${x \over a} + {y \over b}=1$ where $m$ is gradient and $(x_1, y_1)$ lies on the graph 28$y-y_1 = m(x-x_1)$ where $(a,0)$ and $(0,b)$ are $x$- and $y$-intercepts 29 30## Line properties 31 32Parallel lines: $m_1 = m_2$ 33Perpendicular lines: $m_1 \times m_2 = -1$ 34Distance: $\vec{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ 35 36 37## Cubic graphs 38 39$$y=a(x-b)^3 + c$$ 40 41- $m=0$ at *stationary point of inflection* 42- in form $y=(x-a)^2(x-b)$, local max at $x=a$, local min at $x=b$ 43- in form $y=a(x-b)(x-c)(x-d)$: $x$-intercepts at $b, c, d$ 44 45 46## Quartic graphs 47 48### Forms of quadratic equations 49$y=ax^4$ 50$y=a(x-b)(x-c)(x-d)(x-e)$ 51$y=ax^4+cd^2 (c \ge 0)$ 52$y=ax^2(x-b)(x-c)$ 53$y=a(x-b)^2(x-c)^2$ 54$y=a(x-b)(x-c)^3$ 55 56## Literal equations 57 58Equations with multiple pronumerals. Solutions are expressed in terms of pronumerals (parameters) 59 60## Simultaneous equations (linear) 61 62- **Unique solution** - lines intersect at point 63- **Infinitely many solutions** - lines are equal 64- **No solution** - lines are parallel 65 66### Solving $\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases}$ for one, infinite and no solutions 67 68where all coefficients are known except for one, and $a, b$ are known 69 701. Write as matrices: $\begin{bmatrix}p & q \\ r & s \end{bmatrix} 71\begin{bmatrix} x \\ y \end{bmatrix} 72 = 73\begin{bmatrix} a \\ b \end{bmatrix}$ 742. Find determinant of first matrix: $\Delta = ps-qr$ 753. Let $\Delta = 0$ for number of solutions $\ne 1$ 76 or let $\Delta \ne 0$ for one unique solution. 774. Solve determinant equation to find variable 78- *--- for infinite/no solutions: ---* 795. Substitute variable into both original equations 806. Rearrange equations so that LHS of each is the same 817. If $\text{RHS}(1) = \text{RHS}(2)$, lines are coincident (infinite solutions) 82 If $\text{RHS}(1) \ne \text{RHS}(2)$, lines are parallel (no solutions) 83 84Or use Matrix -> `det` on CAS. 85 86### Solving $\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ 87a_2 x + b_2 y + c_2 z = d_2 \\ 88a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}$ 89 90- Use elimination 91- Generate two new equations with only two variables 92- Rearrange & solve 93- Substitute one variable into another equation to find another variable 94- etc.