POLYNOMIALS

POLYNOMIALS

1. Factoring: Breaking up an expression into components that can be (x) together to get the original equation E.X.: a(x-h)(x-k) Final Goal: Factored Form

   • Trinomial: An expression with three mathematical terms seprated by addition or subtraction E.G.: x^2+3x+1 / 8x+4d-6
     • Simple Trinomials a=1:
       • Steps: 1) Identify a,b,c 2) List all facor pairs of "c" 3) Identify praticular factor forms that add to "b" 4) Place factor pairs into 2 adjacent binomials
       • Example: x^2-2x-3 1) Factors: -3,1 2) -3+1= -2 3) (x-3)(x+1)
     • Complex Trinomials a=1:
       • Steps: 1)Calculate product of (a*c) 2)Determine which factors multiply to (a*c) and add to "b" 3) Decompose middle term to become sum of factors in step 2 4)Group the pairs
       • Example: 3x^2+7x+4 3x4=12 3+4=7 3x^2+3x+4x+4 =3x(x+1)+4(x+1) = (3x+4)(x+1)
     • Squares:
       • Difference of Squares (b=0) Every difference of squares has a general forma^2-c^2 this form can then be factored into (a-c)(a+c) in order to factor a^2-c^2 you must determine if the factor can be factored any further
         • Example: 36x^2-100 =(6x-10)(6x+10) =(2x-5)(2x+5)
       • Perfect Square: General form (m+n)^2 or (m-n)^2 can be expanded into m^2+2mn+n^2 or m^2-2mn+n^2
         • Example: x^2+12x+36 =(x+6)(x+6) =(x+6)^2
     • Factor by Grouping: The greates common factor (GCF) for 2 or more powers with the same base is the power with the smallest exponent
       • GCF for 2 or more numbers is the largest term that is divisible by all number
       • When having a negative first term you will have a negative GCF
       • Example: as+ag+sd+gc =a(s+g)+g(s+g) =(a+g)+(s+g)
   • Common Factoring: 1) Find greatest common factor (GCF) 2) Write GCF in front of 1st brackets each successive term with the common factor divides out of each term
     • Example: -144p^5q-12p^3-6p =-6p(24p^4q+6p^2
     • Factor Common Binomials: A binomial that is shared by all terms of a polynomial e.g.d(x+2)-2b(x+2)
       • Example: 2x^2+5x-3 =2x-x+6x-3 =3x^2-x+6x-3 =x(2x-1)+3(2x-1) =(x+3)(2x-1) All these expressions have the same binomial factor (x+3)

2. Expanding: To distribute terms and collect like terms by adding or subtracting F.O.I.L ( First, Outer, Inner, Last) or shower powerExample (x-2)(x+1) =x^2+1x-2x-2 =x^2-x-2

   • Multiplying Binomials: Using F.O.I.L multiply brackets and add like terms Example: (v+2)(2-v) =2v-v^2+4-2v =v^2+4