1. Ex: (3+2i)/(4+5i)× (4- 5i)/(4- 5i)=21/24
2. Multiplying
2.1. Ex: (3+2i)(4+5i) 12+15i+8i+10i^2 12+23i- 10 2+23i
3. Adding and Subtracting
3.1. Ex: (3+2i)-(4+5i) - 1- 3i
4. Patterns in factoring
4.1. Difference of squares Ex: x^2=9
4.2. Perfect square trinomials Ex: x^2- 10x+25
4.3. Sum of cubes Ex: x^3+125 (a+b)(a^2- ab+b^2) Same Opposite Always Positive!
4.3.1. Ex: x^3+4^3 (x+4)(x^2- 4x+16) Check work: x^3- 4x^2+16x+4x^2- 16x+64=x^3+64
4.4. Difference of cubes Ex: x^3- 125 (a- b)(a^2+ab+b^2) Same Opposite Always Positive!
4.4.1. Ex: x^3- 3^3 (x- 3)(x^2+3x+9) Check work: x^3+3x^2+9x- 3x^2- 9x- 27=x^3- 27
5. Solving Quadratic Equations
5.1. With leading coefficient= 1
5.1.1. 1) Variables on the left, constants on the right. 2) Take 1/2 of the coefficient of the "x" variable term. 3) Square that value and add to both sides of the equation. 4) This makes the left side a Perfect Square Binomial.
5.1.1.1. Ex: x^2+8x- 43=0 x^2+8x=- 43 8/2=4 x^2+8x+16=43+16 x^2+8x+16=59 (x+4)^2=59
5.2. With leading coefficient= not 1
5.2.1. 1) Variables on the left, constants on the right. 2) Make leading coefficient =1 by dividing every term by the leading coefficient. 3) Take 1/2 of the coefficient of the "x" variable term. 4) Square that value and add to both sides of the equation. 5) This makes the left side a Perfect Square Binomial.
5.2.1.1. Ex: 3x^2- 6x=31 x^2- 2x=31/3 - 2/2=- 1 x^2- 2x+1=31/3+3/3 (x- 1)^2=34/3
6. Imaginary number patterns
6.1. i^1=i
6.2. i^2= - 1
6.3. i^3= - i
6.4. i^4= 1
7. Dividing imaginary numbers
8. Factor by grouping (four terms)
8.1. 1) split the four terms into two groups 2) find the greatest common factor or shared factor of each group
8.2. Ex: 8r^3- 64r^2+r- 8 (8r^3- 64r^2)+(r- 8) 8r^2 (r- 8)+1 (r- 8) (8r^2+1)(r- 8)
9. Factoring trinomials
9.1. With leading coefficient= 1
9.1.1. Ex: x^2- 3x- 28 - 7×4=- 28 - 7+4=- 3 (x- 7)(x+4)
9.2. With leading coefficient = not 1
9.2.1. Ex: 6x^2- 11x- 2 x^2- 11x- 12 (6x^2- 12x)+(x- 2) 6x (x- 2)+1 (x- 2) (6x+1)(x- 2)
10. Discriminants
10.1. (b)^2- 4 (a)(c)
10.2. D>0: 2 x- intercepts 2 solutions
10.3. D=0 1 x- intercept 1 solution
10.3.1. D<0 no x- intercepts 2 solutions