Monday June 30 Factoring Factoring out the GCF

Monday, June 30 Factoring

Factoring out the GCF

Greatest Common Factor The greatest common factor (GCF) is the product of what both items have in common. Example: 18 xy , 36 y 2 18 xy = 2 · 3 · x · y 2 36 y = 2 · 3 · y · y GCF = 2 · 3 · y = 18 y

Now you try! Find the greatest common factor of the following: Example 1: 2 2 2 12 a b , 90 a b c 2 GCF = 6 a b Example 2: 2 15 r , 35 s 2 , 70 rs GCF = 5

Factoring - Opposite of distributing - Breaking down a polynomial to what multiplies together to form the polynomial

Example: Factor: 2 12 a + 16 a 1. Factor each term. = 2· 2· 3·a·a + 2· 2·a 2. Pull out the GCF. = 2 · a (3·a + 2· 2) 3. Multiply. = 4 a (3 a + 4) You can check by distributing.

Example: Factor: 2 2 18 cd + 12 c d + 9 cd = 2· 3· 3·c·d·d + 2· 2· 3·c·c·d + 3· 3·c·d = 3 · c · d (2· 3·d + 2· 2·c + 3) = 3 cd (6 d + 4 c + 3)

Now you try! Example 1: 2 15 x + 25 x = 5 x(3 + 5 x) Example 2: 2 2 4 12 xy + 24 xy – 30 x y 3 = 6 xy(2 + 4 y – 5 xy )

Factoring by Grouping

1. Group Factor: terms with ( ). Example: 5 xy – 35 x + 3 y – 21 (5 xy – 35 x) + (3 y – 21) = 5 x (y – 7) + 3 (y – = (5 x + 3) (y – 7) 2. Pull out 7) GCF from each group. 3. Split into factors.

Notes - What is in parentheses MUST be the same!! - Grouping only works if there are 4 terms!!

Now you try! Factor. Example 1: 2 5 y – 15 y + 4 y - 12 = (5 y + 4)(y – 3) Example 2: 2 5 c – 10 c + 2 d – 4 cd = (5 c + 2 d)(1 – 2 c)

2 more important examples: Example 1: 2 xy + 7 x + 2 y + 7 (2 xy + 7 x) + (2 y + 7) = x (2 y + 7) + (2 y 1(2 y++7)7) = (x + 1) (2 y + 7)

Example 2: 15 a – 3 ab – 20 + 4 b (15 a – 3 ab) – (20 + – 4 b) = 3 a (5 – b) – 4 (5 – b) = (3 a – 4) (5 – b) If there is a negative in the middle, you MUST change the sign after it.

Factoring Trinomials

Example 1: 2 Factor: x + 5 x + 6 Look for factors of 6 that ADD to positive 5 (x + 2)(x + 3) 6 1· 6 2· 3

Example 2: 2 Factor: x + 7 x + 12 Look for factors of 12 that ADD to positive 7 (x + 3)(x + 4) 12 1 · 12 2· 6 3· 4

Now you try! 2 Example: x + 6 x + 8 (x + 2)(x + 4) 2 Example: x + 11 x + 10 (x + 1)(x + 10)

To determine the signs: Last sign Positive Middle sign Positive ( + ) Negative ( – ) Negative ( + )( – )

Example 3: 2 Factor: x – 12 x + 27 Look for factors of 27 that ADD to negative 12 (x – 3) (x – 9) 27 1 · 27 3· 9

Example 4: 2 Factor: x + 3 x – 18 Look for factors of 18 that SUBTRACT to positive 3 (x + 6)(x – 3) 18 1 · 18 2· 9 3· 6

Now you try! 2 Example: x – 20 (x + 4)(x – 5) 2 Example: x – 7 x – 18 (x + 2)(x – 9)

Please note! 2 Example: x – 5 x – 6 (x + 1)(x – 6) 2 Example: x – 5 x + 6 (x – 2)(x – 3)

More Factoring Trinomials

Example 1: 2 Factor: 6 x + 17 x + 5 30 1 · 30 22 6 x ++2 x) 2 x + 15 x (6 x (15 x++55) 2 · 15 2 x(3 x + 1) + 5(3 x + 1) 3 · 10 5 · 6 (2 x + 5)(3 x + 1)

Example 2: 2 Factor: 4 x + 24 x + 32 Always check your factors to see if there is anything more that can be factored out.

OR Example 2: 2 Factor: 4 x + 24 x + 32 It is usually faster if you factor out the GCF first. Always check to see if there is anything you can factor out first.

Now you try! 2 Example: 5 x + 27 x + 10 (5 x + 2)(x + 5) 2 Example: 24 x – 22 x + 3 (4 x – 3)(6 x – 1)