Factoring – GCF, Grouping, & a = 1 FRIDAY – January 31st, 2014

Warm Up Find the greatest common factor: 72 and 96 Simplify

Greatest Common Factor (GCF) The Greatest Common Factor (GCF) of two or more numbers is the largest number that can divide into all of the numbers. To find the GCF, start by writing out the prime factorization. What factors do the numbers have in common? Circle these then multiply the common factors to get your answer.

We can also find the GCF of monomials… Example 1: Find the GCF of 40a2b and 48ab4

Examples 2. Find the GCF of 12a3b4 and 3a5b. 3. Find the GCF of 7x2y2 and 10xy3.

Undoing?!?!? What operation undoes addition for numbers? What operation undoes division for numbers? How can we undo multiplication for polynomials?

Remember this warm up problem? Multiply: 2x(5x + 3) We can work backwards. What if I gave you the answer 10x2 + 6x, and asked you for the original problem?

We can also “undistribute” our warm up problem Find the GCF then use the distributive property to factor out the GCF. What are we really doing? Take out the GCF and then write your “leftovers” on the inside. Example 13: 10x2 + 6x

We can also find the GCF of a polynomial… Factor out the GCF of all of the terms. Example 9: 3x3y – 9x2y2 Example 10: 12x3 – 8x2 + 16x

Examples 11 & 12 18z3 + 9z2 – 6z 12a2b + 24a – 48b

Examples Factor out the GCF from each of the following: 14. 3x3y – 9x2y2 15. 12a2b + 24a – 48b

Practice 24c5 – 16c2d 19x2y + 9xy2 32x3 – 4x – 16 Factor out the GCF of each of the following: 24c5 – 16c2d 19x2y + 9xy2 32x3 – 4x – 16

Practice Find the GCF of 120x2y5 and 60x4y5 60x4y5 60x2y5 2x4y5 2x2y2

Practice What is the GCF of the terms of A. 2 B. 4 C. 2c D. 4c

Don’t forget to check your answer!! Practice Don’t forget to check your answer!!

Practice Factor 15x2 -12x + 5 5(3x2 – 2x + 1) x(15x2 – 12x + 5) Prime

Practice Factor 12x3y2 – 9x2y4 + 6x2y. 3x2y(4xy – 3y3 + 2)

***Your leftovers must match*** Factoring by Grouping You can try to factor by grouping when your polynomial has FOUR terms. Always look for a GCF first! Then group your terms in pairs using parenthesis. Find the GCF for each binomial. Your answer will be (GCF’s)(leftovers). ***Your leftovers must match***

Example #1 Factor: 12x3 + 3x2 + 20x + 5 Always look for a GCF first! Then group your terms in pairs using parenthesis. Find the GCF for each binomial. Your answer will be (GCF’s)(leftovers). Your leftovers must match Factor: 12x3 + 3x2 + 20x + 5

How can we check our answers?

Don’t forget to look for a GCF first! Example #2 Factor: 8x2 + 8xy + 2y2 + 2xy Don’t forget to look for a GCF first!

Examples 3 & 4: 8j3 + 4j2 + 10j + 5 2m3 + 8m2 + 9m + 36

THINK/PAIR/SHARE Where did the teacher make an error? Factor 20p3 + 40p2 + 15p + 30 (20p3 + 40p2) + (15p + 30) 20p2(p + 2) + 15(p + 2) (p + 2)(20p2+15)

Factoring trinomial with a = 1 Solve the x-puzzles!

Factoring with a = 1 Start with x2 + bx + c FIND the 2 factors! Example 1: x2 + 11x + 18 We need 2 #’s whose product is 18 and sum is 11

Examples 2. Factor. a2 – 9a + 20 3. Factor. c2 – c – 30

Patterns in Factoring If c is positive (x2 + bx + c), then both binomials will have the sign of b. If c is negative (x2 + bx – c), then you will have one binomial with addition, and one with subtraction.

Examples 4. Factor x2 - 4x – 21 5. Factor x2 - 8x + 7

How do I know what to do when? Always check for a GCF! And if there is one, factor (un-distribute) it FIRST. After it’s factored out or if there isn’t one, see how many terms it has. 4 terms  factor by grouping 3 terms  factor the trinomial (like we did today)

Homework Worksheet