1.  Polynomials Lesson 3 Multiplying and Factoring Polynomials using the Distributive Method

2. Todays Objectives  Students will be able to demonstrate an understanding of the multiplication of polynomial expressions, including:  Generalize and explain a strategy for multiplication of polynomials  Identify and explain errors in a solution for a polynomial expression  Generalize and explain strategies used to factor a trinomial  Express a polynomial as a product of its factors

3. Multiplying Polynomials  Today we will look at a strategy for multiplying and factoring polynomials:  Distributive Method  When binomials have negative terms, it is difficult to show their product using algebra tiles, but we can use a rectangle diagram or the distributive property.  Algebra tiles are limited to multiplying 2 binomials. The distributive property can be used to multiply any polynomials

4. Distributive Property

5. Example (You do)

6. Example  Factor the trinomial x 2 – 4x – 21 into two binomials.  Answer will be in the form (x + a)(x + b), where we know the following:   a + b = -4   (a)(b) = -21  So we need to find two numbers that ADD to -4, and MULTIPLY to -21

7. Example  Find all factors of -21  (1)(-21), (-1)(21), (3)(-7), (-3)(7)  Find one set of factors that add to -4  (3)(-7)  So, a = 3, b = -7  (x + 3)(x – 7)

8. Example (You do)  Factor the trinomial z 2 – 12z + 35  Solution: Factors of 35 are (1)(35), (-1)(-35), (5)(7), (-5)(-7)  Two factors that add to -12 are (-5)(-7)  So, a = -5, b = -7  (z – 5)(z – 7)

9. Example  List all the factors for the polynomial 2x 2 – 10x + 12  Solution: First, remove the GCF  2(x 2 – 5x + 6)  *you must include the factored out 2 as a factor in your final answer!  Now, factor the trinomial:  (x + a)(x + b), a + b = -5, (a)(b) = +6  Find all the factors of +6  (1)(6), (2)(3), (-1)(-6), (-2)(-3)  Find one set of factors that add to – 5  (-2)(-3)  So, a = -2, b = -3  Factors of the polynomial 2x 2 – 10x + 12 are:  (2)(x - 2)(x – 3)

10. Example (You do)

11. Distributive Method  The distributive property can be used to perform any polynomial multiplication. Each term of one polynomial must be multiplied by each term of the other polynomial.

12. Using the distributive property to multiply two polynomials

13. Example  -3f 2 (4f 2 – f – 6): -12f 4 + 3f 3 + 18f 2  3f(4f 2 – f – 6): 12f 3 – 3f 2 – 18f  -2(4f 2 – f – 6): -8f 2 + 2f + 12  Add: -12f 4 +15f 3 + 7f 2 –16f + 12 (-3f 2 + 3f – 2)(4f 2 – f – 6) Use the distributive property. Multiply each term in the 1 st trinomial by each term in the 2 nd trinomial. Align like terms.

14. Check your answers!  One way to check that your product is correct is to substitute a number for the variable(s) in both your answer and the original question. If both sides are equal, then your answer is correct. For this example, let’s let f = 1.  = (-3f 2 + 3f – 2)(4f 2 – f – 6) = -12f 4 + 15f 3 + 7f 2 – 16f + 12  = [-3(1) 2 + 3(1) – 2][4(1) 2 – 1 – 6] = -12(1) 4 + 15(1) 3 + 7(1) 2 – 16(1) + 12  = (-2)(-3) = 6  = 6 = 6  Since the left side equals the right side, the product is likely correct.

15. Example (You do)  Use the distributive property to multiply (3x – 2y)(4x – 3y + 5)  Solution:  = 3x(4x – 3y – 5) – 2y(4x – 3y + 5)  = 3x(4x) + 3x(-3y) + 3x(-5) – 2y(4x) – 2y(-3y) – 2y(5)  = 12x 2 – 9xy – 15x – 8xy + 6y 2 – 10y  = 12x 2 – 9xy – 8xy – 15x + 6y 2 – 10y  collect like terms  = 12x 2 – 17xy – 15x + 6y 2 – 10y

16. Example  Simpify (2c – 3)(c + 5) + 3(c – 3)(-3c + 1)  Solution:  Use the order of operations. Multiply before adding and subtracting. Then, combine like terms.  = 2c(c + 5) – 3(c + 5) + 3[c(-3c + 1) – 3(-3c + 1)]  = 2c 2 + 10c – 3c – 15 + 3[-3c 2 + c + 9c – 3]  = 2c 2 + 7c – 15 + 3[-3c 2 + 10c – 3]  = 2c 2 + 7c – 15 – 9c 2 + 30c – 9  = -7c 2 + 37c – 24

17. Example (You do)

18. Homework  pg. 186-187, # 5,7,9,11,15-17, 21