Factoring Polynomials *GCF *Factor by grouping *Factor a trinomial *Special Cases: -Difference of Squares -Perfect Square Trinomials -Sum or Difference of Cubes WARM UP - Your HW ?s HERE: Factoring
Factoring is the process of writing a polynomial as the product of two or more polynomials. Ex: 6x2 x 2 = (2x + 1)(3x 2) Polynomials that cannot be factored using integer coefficients are called irreducible over the integers or prime. Ex: Not prime: 6x+3 = 3(2x+1) Prime: 2x+1 The goal in factoring a polynomial is to use one or more factoring techniques until each of the polynomials factors is prime or irreducible. In this situation, the polynomial is said to be factored completely. In any factoring problem, the first step is to look for the greatest common factor.
Factor by Grouping x3 + 4x2 + 3x + 12 (x3 + 4x2) (+3x+12) 2. Pull out a GCF from each grouping. ___(x3 + 4x2) ___ (+3x+12) x2(x +4) + 3 (x+4) 1. Group first two and last two terms. 3. Pull out the GCF of (x+4) for final answer (x+4) (x2 +3)
Factor by Grouping x3 + 5x2 2x - 10 (x3 + 5x2) (-2x+10) 2. Pull out a GCF from each grouping. ___(x3 + 5x2) ___ (-2x+10) x2 (x+5) -2 (x+5) 1. Group first two and last two terms. 3. Pull out the GCF of (x+4) for final answer (x+5) (x2 -2) Factoring Methods
Guess & Check BIG X - By Grouping Box Method A Strategy for Factoring ax2 + bx + c GUESS & CHECK (Assume, for the moment, that there is no greatest common factor.) 1. Find two First terms whose product is ax2: ( x + )( x + ) = ax2 + bx + c 2. Find two Last terms whose product is c: (x + )(x + ) = ax2 + bx + c 3. By trial and error, perform steps 1 and 2 until the sum of the Outside product and Inside product is bx: ( x + )( x + ) = ax2 + bx + c I
O (sum of O + I) If no such combinations exist, the polynomial is prime. Factoring Methods GUESS & CHECK Guess & Check (Easy with prime #s or when a =1) 3x2 + 7x + 2
x2+6x + 8 Possible with others too, but harder. 6x2 + 19x + 10 Factoring Methods Grouping- Big X 6x2 + 19x + 10 6x2 + 15x + 4x + 10
You need 4 terms to factor by grouping, so were going to split the 19x: What multiplies to A*C (6x2 *10) and adds to 19? The X can help organize your thoughts. Now, follow steps to factor by grouping. (6x2 +15x)(+ 4x +10) 3x(2x+5) +2 (2x+5) (3x+2)(2x+5) Factoring Methods
BOX METHOD 6x2 + 19x + 10 BEFORE YOU BEGIN, YOU MUST PULL OUT ANY GCFs! 6x2 10 A*C 60x2 Find the factors that add to 19x 1*60
FILL IN THE BOX: A 2*30 3*20 4*15 5*12 6*10 6x2 15x 4x 10 2x
+ 5 C 3x +2 Fill in box with the factors: What multiplies to A*C (6x2 *10) and adds to 19? Now, factor rows and columns. A
FINAL ANSWER: (3x+2)(2x+5) C Special Cases Difference of Squares (a2 b2)=(a+b)(a-b) Ex: 36x2 64y2 25x2 - 121 x4 81 2
Special Cases Perfect Square Trinomials a2 2ab + b2 = (ab)2 Ex: 25x2 60x + 36 Sum or Difference of Cubes Special Cases Sum or Difference of Cubes a3 + b3 = (a+b)(a2 ab + b2) a3 - b3 = (a-b)(a2 + ab + b2) Notice, each only has one negative sign and
the first grouping matches the original sign. Ex: 27x3 + 81 Sum or Difference of Cubes Factoring Polynomials Common Factors In any factoring problem, the first step is to look for the greatest common factor. The greatest common factor is a n expression of the highest degree that divides each term of the polynomial. The distributive property in the reverse
direction can be used to factor out the greatest common factor. ab + ac = a(b + c) a. 18x3 + 27x2 b. x2(x + 3) + 5(x + 3) Text Example Factor: a. 18x3 + 27x2 b. x2(x + 3) + 5(x + 3) Solution a. We begin by determining the greatest common factor. 9 is the greatest integer that divides 18 and 27. Furthermore, x 2 is the greatest expression that divides x3 and x2. Thus, the greatest common factor of the two terms in the polynomial is 9x2. 18x3 + 27x2
= 9x2(2x) + 9x2(3) Express each term with the greatest common factor as a factor. = 9x2(2 x + 3) Factor out the greatest common factor. b. In this situation, the greatest common factor is the common binomial factor (x + 3). We factor out this common factor as follows. x2(x + 3) + 5(x + 3) = (x + 3)(x2 + 5) Factor out the common binomial factor.
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