What is Factoring? Breaking apart a polynomial into the expressions that were MULTIPLIED to create it. If a Polynomial can not be factored, it is called PRIME

Greatest Common Factor 1. LARGEST common number 2. SMALLEST exponent on EACH letter. 3. *DIVIDE the coefficients and SUBTRACT exponents.* Example: 15x4 + 18x3 + 30x2 Largest number: 3 Smallest exponent:2 Example: 12x4y2 + 4x3y + 20x2y5 Largest number: 4 Smallest exponent on x :2 Smallest exponent on y: 1 Example: 7xy2 + y + 14x2 GCF = 1 PRIME Example: 15x4 + 18x3 + 30x2 Largest number: 3 Smallest exponent:2 GCF=3x2 (3x2)(5x2+6x+10) Now divide by 3x2: 5x2 +6x1 +10 Example: 15ax5 + 25a3x3 + 10a2x2 Largest number: 5 Smallest exponent: a1x2 GCF=5a1x2 Now divide by 5a1x2: 3x3 +5a2x +2a1 (5ax2) (3x3 + 5a2x + 2a) Example: 16f4g9 + 24g3 + 4fg2 Largest number: 4 Smallest exponent: g2 GCF=4g2 Now divide by 4g2: 4f4g7 +6g +1f (4g2) (4f4g7 + 6g + 1fg) Largest number: 5 Example: 5h4 + 30h2 - 15h Smallest exponent: 1 GCF = 5h Now divide by 5h1: 1h3 + 6h - 3 (5h) (h3 + 6h – 3)

A value made of a number times ITSELF or x2 or x4 Difference of 2 squares What do I look for? Must be a BINOMIAL Both terms are perfect squares SUBTRACTION in the middle A value made of a number times ITSELF or x2 or x4 a2 – b2 = (a+b)(a-b) Square root the first term to get the first part of each answer Square root the second term to get the 2nd part of each answer One + and one – in the middle 16x2 - 81 ( )( ) 4x 4x + - 9 9 X2 - 25 ( ) ( ) X X + - 5 5 You might need to find a GCF First! 3x2 - 75 GCF 3 (x2 – 25) 3( )( ) X X + - 5 5 6x3 – 24x 6x ( ) X2 - 4 6x( )( ) X X + - 2 2 4x3 + 8x2 – 60x 4x (x2 + 2x – 15) (4x)(x+5)(x-3)

Factoring with Boxes (x + 5) (3X2 -2) (x + 7) (X2 + 2) 3X3 15x2 X3 7x2 P 46 Use this for: 4 – term polynomials Quadratic Trinomials with a > 1 4 Term Polynomials (known as grouping) Place the terms into the boxes in Z Pattern. Find the GCF of each row and column *** Keep the sign of the closest box *** Answer factors on top and right side Example: x3 + 7x2 + 2x + 14 Example 2: 3x3 + 15x2- 2x - 10 x 5 X 7 3X2 3X3 15x2 X2 X3 7x2 -2x -10 -2 2x 14 2 (x + 5) (3X2 -2) (x + 7) (X2 + 2)

Factor by Grouping Works for 4 term polynomials!! 1. Group the first 2 terms together and the last 2 terms together. 2.Factor out a GCF for each group 3.The GCF’s combine to form one factor and the matching binomials will be the other factor. Example: x3 + 7x2 + 2x + 14 Practice 1: 3x3 + 15x2- 2x - 10 Step 1: (x3 + 7x2 ) +(2x + 14) (3x3 + 15x2) (- 2x – 10) Step 2: X2( ) + 2( ) X+7 X+7 3x2 (x + 5) -2 (x + 5) (3x2 -2) (x + 5) Step 3: (X2+ 2) ( X+7 ) x 7 x2 x3 7x2 2 2x 14

C C Factors factors added added b b one Positive negative one 2 3 positive 6 5 -10 3 1 6 7 2 3 5

C C Factors factors added added b b one Negative negative one positive

Factoring Trinomials (a=1) Standard Trinomial: aX2 + bX + c a, b, c are numbers Factor using 2 binomials ( )( ) First term in each binomial is x (x )(x ) Number game  Find 2 numbers that multiply to c and add to b. Example: x2 + 5x + 4 a= b= c= 1 5 4 (x )(x ) +4 +1 Find 2 numbers that multiply to 4 and add to 5 Example: x2 + 7x + 12 (x )(x ) +3 +4 Positive C means factors have same sign Find 2 numbers that multiply to 12 and add to 7 Example: x2 -9x + 18 (x )(x ) -6 -3 Find 2 numbers that multiply to 18 and add to -9 Example: x2 + 9x +5 (x )(x ) PRIME Find 2 numbers that multiply to 5 and add to 9 Example: x2 + 2x - 15 (x )(x ) +5 -3 Negative C means factors have different sign Find 2 numbers that multiply to -15 and add to 2

Factoring Trinomials Pg 48 Turn a TRI-nomial into 4 terms EXAMPLE: 3x2 + 11x + 6 x 3 Fill the first and last terms into The top left and bottom right. 3x 3x2 9x 2 2x 6 (3x+2)(x+3) Split up the middle term into two boxes by finding the numbers that multiply to a*c 3*6 = 18 how can we split up 11x so that it multiplies to 18? 9 and 2! Write the GCF in every row and column. EX. 1 2x2 + 5x + 3 Ex. 2 5m2 – 17m + 6 EX. 3 6y2 – 5y - 4 EX. 4 12c2 + 11c - 5