1. Topic 2 Unit 7 Topic 2

2. Information To multiply two binomials you need to apply the distributive property twice. For example, to multiply you need to multiply a by. Then, you need to multiply b by. In total, you need to perform four multiplications. You can use the acronym FOIL or use a multiplication box to keep track of the four products. Both approaches are shown.

3. Information

4. Multiplication Box Draw a two-by-two box. Along one side, write the terms of the first binomial. Along another side, write the terms of the second binomial. In the cells of the box write the product of each pair of terms.

5. Explore Investigating Multiplying Binomials and Factoring Trinomials 1.Multiply the following factors so that the expression is in the form of. a) b) c) Try this on your own first!!!!

6. Explore Investigating Multiplying Binomials and Factoring Trinomials 1.Multiply the following factors so that the expression is in the form of. a) b) Try this on your own first!!!! Collect the like terms

7. Explore Investigating Multiplying Binomials and Factoring Trinomials c) Try this on your own first!!!!

8. Explore 2.Use inductive reasoning to create a conjecture about how to determine b, when rewriting in the form. (Hint: Look at the original factors and the mathematical operations applied) 3.Use inductive reasoning to create a conjecture about how to determine c, when rewriting in the form. (Hint: Look at the original factors and the mathematical operations applied) Collect the like terms by determining the sum of the two numbers = b Multiply the two numbers to obtain a product = c

9. Example 1 Solving Using the Zero Product Rule Find the roots to the following quadratic equations. a) b) c) d) Try this on your own first!!!!

10. Example 1: Solutions Solving Using the Zero Product Rule Find the roots to the following quadratic equations. a)b)

11. Example 1: Solutions c) d)

12. More Information A polynomial with three terms is called a trinomial. For example, is a trinomial. To factor a trinomial of the form, find two numbers, r and s, with a product of the constant term, c, and a sum of the coefficient of the x term, b. If rs = c and r + s = b, then Below is the factorization of.

13. Example 2 Factoring Trinomials of the Form Identify two integers with the given product and sum. a) product = 18; sum =11 b) product = 12; sum = -7 Try this on your own first!!!!

14. Example 2:Solutions a)product p= 18b) product p= 12 sum s= 11 sum s=  7 Positive product and a negative sum indicates that both integers are negative

15. Example 3 Factoring Trinomials of the Form Factor and determine the roots. a) b) c) d) e) Try this on your own first!!!!

16. Example 3a: Solution Since the c-value is +, and the b-value is +, we know that both signs will be positive. sum product Then, set each factor equal to zero and solve for the variable.

17. Example 3a: Solution To check substitute each of our solutions into the equation to make sure they work.

18. Example 3b: Solution Since the c-value is +, and the b-value is -, we know that both signs will be negative. sum product Then, set each factor equal to zero and solve for the variable.

19. Example 3c: Solution Since the c-value is -, and the b-value is +, we know that one sign will be positive and one sign will be negative. Because the sum is positive the larger integer will have the positive sign. sum product Then, set each factor equal to zero and solve for the variable.

20. Example 3d: Solution Since the c-value is -, and the b-value is -, we know that one sign will be positive and one sign will be negative. Because the sum is negative the larger integer will have the negative sign. sum product Then, set each factor equal to zero and solve for the variable.

21. Example 3e: Solution Since the c-value is +, and the b-value is +, we know that both signs will be positive. sum product Then, set each factor equal to zero and solve for the variable.

22. Example 4 Solving Quadratic Equations by Factoring Factor and then solve each of the following quadratic equations. a) b) c) d) Try this on your own first!!!! Helpful Hint When factoring, check for common factors first.

23. Example 4a: Solution sum product Then, set each factor equal to zero and solve for the variable. Since the c-value is -, and the b-value is +, we know that one sign will be positive and one sign will be negative. Because the sum is positive the larger integer will have the positive sign. When trinomials look like they have an a - value that is not 1, look for a common factor that can be factored out.

24. Example 4a: Solution To check substitute each of our solutions into the equation to make sure they work.

25. Example 4b: Solution sum product Then, set each factor equal to zero and solve for the variable. When trinomials look like they have an a - value that is not 1, look for a common factor that can be factored out. Since the c-value is +, and the b-value is -, we know that both signs will be negative.

26. Example 4c: Solution sum product Then, set each factor equal to zero and solve for the variable. When trinomials look like they have an a - value that is not 1, look for a common factor that can be factored out. Since the c-value is -, and the b-value is +, we know that one sign will be positive and one sign will be negative. Because the sum is positive, the larger integer will have the positive sign.

27. Example 4d: Solution sum product Then, set each factor equal to zero and solve for the variable. When trinomials look like they have an a - value that is not 1, look for a common factor that can be factored out. Since the c-value is +, and the b-value is -, we know that both signs will be negative.

28. Example 5 Determining the Dimensions of a Ping Pong Table The area of a rectangular Ping Pong table is 45 ft 2. The length is 4 feet longer than the width. a)If w represents the width of the table, then write an expression for the length. b)Substitute into the area formula A = lw and solve for the width. Start by rewriting the equation with the left side equal to zero. Try this on your own first!!!!

29. Example 5 Determining the Dimensions of a Ping Pong Table The area of a rectangular Ping Pong table is 45 ft 2. The length is 4 feet longer than the width. a)If w represents the width of the table, then write an expression for the length.

30. Example 5 The area of a rectangular Ping Pong table is 45 ft 2. The length is 4 feet longer than the width. b)Substitute into the area formula A = lw and solve for the width. Start by rewriting the equation with the left side equal to zero.

31. Example 5 Determining the Dimensions of a Ping Pong Table c) What are the dimensions of a Ping Pong Table? w is the width of the ping pong table so only positive values make sense in this context The dimensions of the Ping Pong table are a width of 5 feet and a length of 9 feet.

32. Need to Know: To factor a trinomial of the form, find two numbers, r and s, with a product of the constant term, c, and a sum of the coefficient of the x term, b. If rs = c and r + s = b, then When trinomials look like they have an a -value that is not 1, determine whether or not a common factor can be factored out. You can solve some quadratic equations by factoring. First write the equation in the form, with one side of the equation equal to zero. Then factor the other side. Next, set each factor to zero, and solve for the unknown. You’re ready! Try the homework from this section.