1. Multiplying and Factoring Module VII, Lesson 2 Online Algebra VHS@PWCS

2. Multiplication Review 1. -2(4x) 2. 3a(5b) 3. 4x(-3x) 4. -4(2x – 3) 1. -8x 2. 15ab 3. -12x 2 4. -8x + 12 Try these problems. Before going on with the lesson make sure that you can get these right.

3. Multiplying Problems like -4(2x – 3) use the distributive property to solve. The distributive property says that -4(2x – 3) = -4(2x) - -4(3) = -8x + 12 We can use this property to multiply any polynomial by a monomial. 5y(8y 3 + 7y 2 – 3y) 5y(8y 3 ) + 5y(7y 2 ) – 5y(3y) 40y 4 + 35y 3 – 15y 2

4. Try these. 1. 2a(5a 3 -7a 2 + 2) 2a(5a 3 ) – 2a(7a 2 ) + 2a(2) 10a 4 – 14a 3 + 4a 2. -2x(5x + 11) -2x(5x) + -2x(11) -10x 2 – 22x 3. 5m 2 (m + 7) – 2m(5m 2 – 3m + 7) 5m 2 (m) + 5m 2 (7) + (-2m)(5m 2 ) + (-2m)(-3m) + (-2m)(7) 5m 3 + 35m 2 +-10m 3 + 6m 2 + -14m -5m 3 + 41m 2 – 14m Now combine all like terms!

5. Factoring When two or more numbers are multiplied, each number is a factor of the product. 18(2) = 36 So 18 and 2 are factors of 36 Other factors of 36 are: 1 and 36, 3 and 12, 4 and 9, 6 and 6

6. Greatest Common Factor The Greatest Common Factor of two or more numbers is the largest factor that the numbers have in common. For example: The factors of 54 are: 1, 2, 3, 6, 9, 18, 27, 54 The factors of 63 are: 1, 3, 7, 9, 21, 63 The largest factor they have in common is 9, so 9 is the GCF of 54 and 63.

7. Find the GCF of each pair of numbers. 15 and 50 The factors of 15 are: 1, 3, 5, 15 The factors of 50 are: 1, 2, 5, 10, 25, 50 The largest factor that 15 and 50 have in common is 5, so 5 is their GCF. 88 and 40 The factors of 88 are: 1, 2, 4, 8, 11, 22, 44, 88 The factors of 40 are: 1, 2, 4, 5, 8, 10, 20, 40 The largest factor that 88 and 40 have in common is 8, so 8 is their GCF.

8. Find the GCF of: 12ab and 4a 2 b 2 When variables are involved the GCF will be the GCF of the coefficients and the lowest power of each variable. The factors of 12 are: 1, 2, 3, 4, 6, 12 The factors of 4 are: 1, 2, 4 The lowest power of a is 1 The lowest power of b is 1 Putting all that together the GCF of 12ab and 4a 2 b 2 is 4ab

9. Find the GCF of the following. 1. 50n 4 and 40n 3 2. 56x 2 y and 49xy 3. 12mn, 10mn and 15mn 1. 10n 3 2. 7xy 3. mn

10. Factoring Polynomials Factoring a polynomial can be done several ways. The first step is always to factor out the GCF. To do this we must 1. Find the GCF of the terms of the polynomial. 2. Divide each term of the polynomial by the GCF. 3. Write as multiplication.

11. Factor: 11x + 44x 2 y 1. Find the GCF of the terms of the polynomial. The GCF of the terms is 11x 2. Divide each term of the polynomial by the GCF. 3. Write as multiplication. 11x(1 + 4xy)

12. Try to factor these. 16xy 2 – 24y 2 z + 40y 2 1. Find the GCF. 8y 2 2. Divide each term by the GCF. 3. Write as multiplication. 8y 2 (2x – 3z + 5) 28a 2 + 21a – 35 1. Find the GCF. 7 2. Divide each term by the GCF. 3. Write as multiplication. 7(4a 2 + 3a – 5)

13. Wrap-Up  We can use the distributive property to multiply a polynomial by a monomial.  Numbers that are multiplied together are called factors.  Greatest Common Factors are the largest factor that 2 or more numbers have in common.  To factor a polynomial, find the GCF of its terms and divide each term by that GCF. Finally write as multiplication.