Factoring – GCF and Grouping

Warm Up Multiply: 2x(5x + 3) What are the factors of 8? What does it mean to be a factor of a number? Which of the following numbers are prime? 3, 4, 5, 6, 7, 8, 9, 10, 11

Objectives: Students will be able to determine whether a number is prime or composite and give its prime factorization. Students will be able to give the prime factorization of a monomial. Students will be able to find the greatest common factor of a polynomial. Students will be able to use the distributive property to factor out the GCF from a polynomial.

Prime vs. Composite Numbers A prime number is a number that can only be divided by 1 and itself. A composite number is a number that is not prime. Prime or composite? 15 19 23 27

Sticky Notes! I will give you a sticky note with a number on it. Decide whether the number is prime or composite then place it in the correct category on the board. Prime Composite

Prime Factorization Prime factorization means finding all of the prime factors of a number. To find the prime factorization, make a factor tree! Example: 84

You try! Give the prime factorization of: 1. 40 2. 210

We can also factor monomials… Remember how we broke down monomials when we wanted to multiply/divide them? Example: 9a4b3 To give the prime factorization of a monomial, use the same process. Just remember to ALSO factor the coefficient.

You try! Give the prime factorization: 1. 16xy2 2. 24m4n2

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.

GCF 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. Example: find the GCF of 42 and 60.

You Try! Find the GCF of 36 and 60. Find the GCF of 27 and 81.

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

You try! Find the GCF of 12a3b4 and 3a5b. Find the GCF of 7x2y2 and 10xy3.

Hands up, pair up Pick a partner! Move your desks together. Take your notes and calculator with you!

Partner race! Race your partner to complete the problems. Whoever finishes all of the problems first (they must be done correctly) will win a prize! Let me know when you are done so that I can check your work!

Partner race solutions I. 1. composite: (7)(3) 2. composite: (3)(3)(2)(2) 3. prime 4. composite: (17)(2)(2) II. 1. (3)(2)(2)(2)(a)(a)(b)(b)(b)(b)(b)(b) 2. (5)(5)(2)(x)(x)(x)(x)(y) III. 1. 5 2. 3r2s2 3. 13xy

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

You try! 18z3 + 9z2 – 6z 12a2b + 24a – 48b

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 need to “undistribute” We did this with basic problems in August and September. How can we “undistribute” (10x – 5)?

We can also “undistribute” our warm up problem Fancy math language: 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: 10x2 + 6x

Examples Factor out the GCF from each of the following: 1. 3x3y – 9x2y2 2. 12x3 – 8x2 + 16x

You try! Factor out the GCF from each of the following: 18z3 + 9z2 – 6z 12a2b + 24a – 48b

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

Ticket Out – on a separate sheet of paper Factor out the GCF of each of the following: 1. 9x2 – 12x + 3 2. 12x3 + 6x2 + 36 3. What does it mean to FACTOR a number, monomial, or polynomial? Write 2-3 sentences. When you finish, write your name on it and turn it into Mrs. S!

Warm Up 1. 200 2. 34a4b3c Find the GCF of each of the following: Give the prime factorization of each of the following: 1. 200 2. 34a4b3c Find the GCF of each of the following: 3. 144, 36 4. 144x2y4, 36x5y3 Factor out the GCF: 5. 144x2y4 + 36x5y3 6. 14x2 – 7x + 21