AIP Factoring Practice/Help

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1 The following pages include many problems to practice factoring skills. There are also several activities with examples to help you with factoring if you feel like you are not proficient with it. There are also several different methods of factoring represented here. Please do not feel like you are expected to complete this entire packet if you are proficient with factoring polynomials. It is being provided for you just for your use. 0

2 Factor each polynomial using GCF: Part 1 1) x 5x + ) m + 45 ) 15y + 0y 10 x( + ) 5( + ) 5( + - ) 4) 10 y 9 y + 1 5) 1t 5 + 6t 6) 4 9 6x + 15x + x Factor each polynomial using grouping 7) 4 6 y y y + 8) 4m 1m m ( 4y ) + ( + 6) y ( ) ( ) (y )( y ) 9) + 5x + 6 x 10) x + 5x ) x + 9x + 0 Challenge: 1) x + x 1) x x 8 1

3 Factoring ax + bx + c when a = 1 Part 1) 5x + 6 x ) x + x 6 ) 4x + 4 x 4) x + 8x ) + 1x + 6 x 6) x 5x + 4 7) + x 40 x 8) x 11x + 8 9) + 8x + 15 x 10) x + 11x 1 11) 16x + 64 x 1) x + 1x + 6

4 Factoring Using the Box Method Part EXPLORE 1 Factor x + 16x + 5 using the box method Place the first and last terms in the box STEP 1 Use the box model to factor x + 16x + 5. Place the x term in the upper left square of the box. Place the constant term in the lower right square of the box. List factors STEP Find the product of the terms in the box. Then list the factors of the product. Be sure to list the factors as the product of a number and x. Choose factors STEP Find the sum of the factors you found in Step. Circle the factors that add up to the middle term of x + 16x + 5. Place the factors in the box STEP 4 Place one of the factors you circled in Step in one of the empty squares. Place the other factor in the remaining empty square. Find the greatest common factor STEP 5 Find the GCF of the 1st column. Put this value in box (a). Use multiplication STEP 6 The product of boxes (a) and (c) must equal the value in the upper left-hand square. To find the value of (c) ask, what do you multiply the value in box (a) by to get x? Put your answer in box (c). Fill in remaining boxes STEP 7 Repeat the procedure in Step 6 to find the values for boxes (b) and (d). Write the factors STEP 8 The sum of boxes (a) and (b) form one of the factors. The sum of boxes (c) and (d) form the other. Write the factors of the quadratic on your worksheet. Your final answer is

5 Use your observations to complete these exercises DRAW CONCLUSIONS 1. Use the box model to factor x + 1x + 1. You may want to refer to the steps in Explore 1. In Exercises 4, use the box method to find the factors of the quadratic. ) x + 11x + 10 ) 4x + 15x + 9 4) x + 11x

6 EXPLORE x + 5x 6 using the box method Use your observations to complete these exercises. In Exercises 5 8, use the box method to find the factors of the quadratic. 5) x 19x ) 5x 8x 4 7) 6x + 5x 4 8) x x + 1 5

7 Activity Worksheet EXPLORE 1: x + 16x + 5 Product: Factors of the product: x + 16x + 5 = ( )( ) EXPLORE : 4x + 5x 6 Product: Factors of the product: 4x + 5x 6 = ( )( ) 6

8 Factoring ax + bx + c when a > 1 Part 4 When the coefficient of x is greater than 1, factoring can be a challenge. Factor by grouping is one method. The box method that you previously looked at is another method. Example: x + 15x + 7 Step 1: Find the product of A and C. (7) = 14 Step : Write down all the factors of the above product: 1 and 14, and 7 Step : Then pick the set of factors that add up to the B term of 15: 1 and 14 Step 4: Replace the middle term with the set of factors. Be sure to include the variable. x + 1x + 14x + 7 Step 5: Separate the first two terms from the second two terms and factor using GCF (x + 1x) + (14 x + 7) x ( x + 1) + 7(x + 1) Note that what is in parentheses is the same. If it is not the same, you need to retry the problem. Step 6: Use the distributive property backwards. What is not in parentheses will be grouped together and the matching pair of parentheses will be written once. ( x + 7)(x + 1) Your Final Answer! If you aren t sure, use the distributive property and you should get back to your original trinomial. Try some on your own! 1) + 9x + 7 x ) x + 8x + 5 ) x x 4) x + x 10 7

9 Factoring with Special Cases Part 5 Section 1: Factor the following trinomials using any method you prefer. Pay close attention to each answer. If you can spot the pattern, you can complete these problems very quickly. 1) 4 + 1x + 9 x ) x + 8x + 16 ) 10x + 5 x 4) 49y + 14y + 1 5) 9 + 4s + 16 s 6) r 18r ) 4 0x + 5 x 8) 16x + 7x ) 6 + 1d + 1 d 10) y + 4y ) These types of problems are called perfect square trinomials. Why do you think they are called this? What is the trick to complete these problems quickly? 8

10 Special Cases Continued Section : Factor the following binomials using any method you prefer. To make them easier to understand, you might want to insert a middle term of 0x. Pay close attention to each answer. If you can spot the pattern, you can complete these problems very quickly. 1) x 4 ) 4a 5 ) 9x 16 4) 16t ) 5w 81 6) 81 4m 7) 4t 1 8) p q 9) 4 5s k 10) 100 6q 11) This type of problem is called the difference of two squares. Why do you think they are called this? What is the trick to complete these problems quickly? 9

11 Factoring (Sums and Differences) Part 6 The Sum and Difference of Two Cubes Practice We have established the following two identities: ( a b ) = ( a b)( a + ab + b ( a + b ) = ( a + b)( a ab + b x + 7 y x + ( y) Treat x like your a and treat y like your b. Example 1: Find the factors of ( x + (y))( x x(y) + (y) ( x + y)( x xy + 9y ) Example : Find the factors of ) 8a 15b ( a) (5b) (a 5b)((a) + (a)(5b) + (5b) (a 5b)(4a + 10ab + 5b ) Find the factors of: 1) 1 x ) 8c 1 ) ) ) ) 1 c + 4) y 8 5) 64 h 6) a + 15b 7) 7a ) 8a + 4b 10

12 Factoring by Completing the Square Part 7 For an expression of the form x + bx, you can add a constant c to the expression so that the expression x + bx + c is a perfect square trinomial. This process is called completing the square. In this activity, you will use algebra tiles to complete the square. As you will see, this method can be use to solve any quadratic equation. EXPLORE Complete the square Find the value of c that makes x + 4x + c a perfect square trinomial. STEP 1 Model expression Use algebra tiles to model the expression x + 4x. How many x -tiles and x-tiles do you need? STEP Rearrange tiles Arrange the tiles to form a square. The arrangement will be incomplete in one of the corners. Draw your arrangement. STEP Complete the square Add 1-tiles to your model to complete the square. Draw the perfect square model. STEP 4 Find the value of c The number of 1-tiles is the value of c. The perfect square trinomial is x + 4x + or (x + ). 11

13 DRAW CONCLUSIONS Use your observations to complete these exercises 1. Complete the table using algebra tiles. Expression Number of 1-tiles needed to complete the square Expression written as a square x + 4x 4 x + 4x + 4 = (x + ) x + 6x x + 8x x + 10x. In the statement x + bx + c = (x + d), how are b and d related? How are c and d related?. Use your answer to Exercise to predict the number of 1-tiles you would need to add to complete the square for the expression x + 18x. Practice completing the square: 4) x x 5) x 8x 6) x 7x + 7) x x 1 8) x ) x + x 1

14 Completing the Square continued. Any Equation of the form ax + bx + c = 0 where a 0 can be written equivalently as a ( x h) + k = 0 for some real numbers h and k. For the equations below, write in the form a ( x h) + k = 0. Example: x + 6x 7 = 0 Step 1: Move the constant term x + 6x = 7 Step : Take half of the middle coefficient and square it. Add this number to both sides. x + 6 x + () = 7 + (9) Step : Convert ( x + ) = 16 x + Step 4: Move constant term back: ( ) 16 0 Practice 10) + 5x + 6 = 0 x 11) x 7x + 10 = 0 = 1) + 11x + 0 = 0 x 1) x + 7x 15 = 0 1

15 Solving equations by Factoring Part 8 Example: x + 5x + 6 = 0 Step 1: Factor ( x + )( x + ) = 0 Step : Separate the problem: Either x+=0 or x+=0 or both. Step : Solve both halves. x = - or x = -. Practice: 1) + 5x 4 = 0 x ) 4x + x 5 = 0 ) 6x + 9 = 0 x 4) x + 7x + 10 = 0 5) 15x + 6 = 0 x 6) x x + 1 = 0 7) x = 1 x 8) x 16x = 6 14

16 Factoring with the Calculator Part 9 15

17 Factoring a Sum or Difference of Cubes. When given a binomial to factor where both of the terms are perfect cubes, you may use the following method. a b = ( a) ( b) ( ) ( a b) ( a) + ab + ( b) S O AP OR a + b = ( a) + ( b) ( ) ( a + b) ( a) ab + ( b) S O AP S = Same sign as the binomial O = Opposite of the sign in the original binomial AP = Always a Positive Sign Examples: ( ) x x x x x x x + 8 = ( ) + () = ( + ) ( ) ( )() + () = ( )( ) x y x y xy x y x y x y xy 64 = ( ) ( ) (4) = 4 ( ) ( ) + ( )( )(4) + (4) = Practice: (Don t forget to use S-O-AP.) y. a a b 1000c x y

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