Complex NumbersThe imaginary unit “i”has the following properties:Any negative square root can be defined as :if a > 0 thenFor example Imaginary numbers can be added or multiplied in much the same way as polynomials. Simply combine like termse.g. (2 + 3i) + (4 + 5i) = 6 + 8ior “distribute” (2 + 3i)(4 + 5i) = 8 + 10i+ 12i+ 15i2= 8 + 22i+ (15)(-1)= -7 + 22iii2= -1 =-1i-a =aii-25 =25 = 5

Practice1. (2 + 6i) + (5 – 4i)2. (3 + i)(3 – i)3. Find real numbers such that 4 - 5i= 2x + yi4. Find the value of i, i2, i3, 1.Determine a general pattern for i1+4ni2+4n, i3+4n, i4+4nwhere n is any integer(x = 2, y = -5)(i, -1, -i, 1)i4, (7 + 2i)(10)

Factoring ReviewWhenever you are asked to factor an expression start with the following steps:1.Find the prime factors of each term (e.g. 42a2)2.Find the GCF of all terms3.If no GCF for every term try grouping terms that have a GCFe.g. ab-ac+zb-zc4.If a quadratic function (highest degree term is 2) then:a)Identify the number of termsi.If binomial use difference of perfect squares rulesii.If trinomial then use factoring trinomial rules

Factoring/Solving Quadratic Trinomial EquationsThe standard form of any quadratic equation can be written as ax2+ bx + c = yQuadratic trinomials can be factored in three ways1.Using factoring rules2.Completing the square3.Using the quadratic formula

FactoringIf a = 1 then:1.Find factors of “c” that add up to “b” (a “t-chart” might be helpful)2.Express the equation in the form (x+d)(x+e) where dand e are the factors“c” that add up to “b”of .

FactoringIf a 1 then1.Multiply a x c2.Find factors of a x c that add up to b3.Replace b with two linear terms whose coefficients add up to b.4.Factor by grouping

FactoringIf a 1 thenAlternatively1.Multiply a x c2.Set a = 1 and replace “c” with “a x c”3.Factor as if a = 14.