2.1 Intro to Algebra

Transcription

1 MFM1P Unit 2: Algebra Lesson 1 Date: Learning goal: I understand polynomial terminology and can create algebraic expressions. 2.1 Intro to Algebra What is algebra? Learning algebra is like learning another language. By learning algebra, mathematical models of realworld situations can be created and solved! In algebra, letters are often used to represent numbers. KEY TERMS à variable à coefficient à expression à constant Brainstorm words that represent Addition Subtraction Multiplication Division 1

2 Choose a variable to represent the number and write an algebraic expression for the following phrases. a) 19 decreased by a number b) 30 more than a number c) 12 more than 3 times a number d) A number divided by seven e) double the amount of money f) 5 years younger than Rebecca Write an English statement that could represent each of the following: a) g + 5 b) 4d c) 2a 1 2

3 Homework 2.1 Variables and Expressions 1. Indicate with math symbols what operations are being described by the given word(s). Use +, -, x, or symbols. a) sum b) product c) decreased by d) times e) increased by f) difference g) more than h) less than i) twice something 2. Write a verbal expression for the algebraic expression. a) x + 7 b) 2x c) x 6 d) y 2 e) 3x 4 f) ab 3. Write an algebraic expression to the given verbal expression. a) eight less than a number b) a number increased by seven c) a number squared d) nine times a number e) a number decreased by three f) two less than five times a number g) twice a number increased by three times the number 4. The junior girls volleyball team has just started tryouts. Write an expression for the number of girls who made the team. a) 25 girls tried out for the team and the coach cut 13 girls b) A lot of girls tried out for the team and the coach cut 21 girls n 21 c) 16 girls tried out for the team and the coach cut some d) Some girls tried out for the team and the coach only cut 2 3

4 5. For each of the following, identify the variable. Then write an expression. a) triple the width of a rectangle b) 8 years younger than Vijay c) the area increased by 15 cm 2 d) some pencils shared equally among 4 students e) double the length decreased by 6 cm f) Sarah bought some coffee for $1.90 each 6. What is the expression for the number of cats in each example: a) if there are 12 dogs in a shelter of 30 animals b) if there are x dogs in a shelter with 30 animals c) if there are 11 cats in a shelter of n animals d) if there are x cats in a class of p animals? 7. Salma gets $12 per hour to baby-sit. She gets a bonus if she has to baby-sit past 10 p.m. The expression 12h + 25 represents what Salma was paid last night. a) What is the variable in the expression? Explain what it represents in real life. b) How much did she earn last night? 9. You have to pay a one-time fee of $65 to join the gym plus $2 every class that you take. What is an expression for the cost for x classes?? 2.1 Answers 1. a) + b) x c) - d) x e) + f) - g) + h) - i) x 2. a) 7 more than a number b) double of a number c) 6 less than a number d) a number squared e) 4 less than three times a number f) the product of two numbers 3. a) n-8 b)n+7 c) n 2 d) 9n e) n-3 f) 5n-2 g) 2n+3n 4. c) 16-x d) x-2 5. a) Width of a rectangle b) Vijay s age c)the area d) Pencils e)length f)coffee 6. a) 30-12n b)30-x c)n-11 d)p-x 7. a) Yes. $25 b) h. It represents the number of hours she baby-sits. c) $ x 4

5 MFM1P Unit 2: Algebra Lesson 2 Learning goal: I can substitute a value into a math expression then evaluate. Date: 2.2 Substitution WARM UP 1. Evaluate each of the following. Remember to use BEDMAS! a) (4-7) b) (-4) c) (-5) (-7) d) 6 2 4(2 + 7) 2. A builder rents a digger. He pays a fixed charge of $30 plus $10 per hour to rent the digger. Work out how much he pays to rent the digger for: a) 4 hours b) 8 hours c) n hours A formula is an expression that uses variables to express a relationship between two or more quantities. List some formulas that you already know New Vocab Substitution 5

6 3. Evaluate the following -replace the variable with a numeric value. -use order of operation rules to simplify the expression. a) 3x + 4y, if x = 7, y = 2. b) xy, if x = 6, y = 3. c) 8 y + z, if x = 3, y = 2, z = 9. d) 3x2 6x, if x = 4 4. A basketball court is 20m long and 15m wide. The perimeter of the court is given by P = 2l +2w where l represents the length, and w represents the width. Evaluate (use substitution) to find the perimeter. 5. You are saving for a skateboard. Your aunt gives you $45 to start and you save $3 each week. The expression w gives the amount of money you save after w weeks. Complete the table of values to show much money you will have over time. w 3w

7 6. The formula to calculate the power for an electrical circuit is modeled by the equation P = i 2 r, where i represents the current in amperes and r represent the resistance in ohms. Determine the power if the current is 12 amperes and the resistance is 2 ohms. Homework 2.2 Substitution 1. Evaluate using the correct order of operations a) (-4) - 8 (-2) - 15 b) (-3) + (-18) 2 (-3) c) (4-7) d) (-4) d) 3 (14 18) 8 (-4) e) (3 + 1) 2. Evaluate each expression a) t + 5 when t = 3 d) 3 + 2y when y=4 b) d - 4 when d = 7 e) m 10 when m = -30 c) 4r - 3 when r = -5 f) 3x+11 when x = Complete the table of values for the following expressions 3x + 4. Show your work. x 3x (-1) + 4 = If p = 4, q = 5, and r = -2, what is the value of each expression? a) 3p + 5 b) 2q 3 c) 4q + r d) pq 5. Describe and correct the error in evaluating the expression when m = 8. 7

8 6. Evaluate each expression for the given values of the variables. a) 3x 2 when x = 2 b) 2x 2 + 5x + 1 when x = 4 7. The formula B = m is used to compute the monthly bill for the use of a cellular phone. Compute the value of B when m = The cost of a school banquet is $ n, where n is the number of people attending. What is the cost for 62 people? 9. An employee who receives a weekly salary of $250 and a 5% commission is paid according to the formula p = 0.05s + 250, where p represents the total amount earned weekly and s represents the total weekly sales. Find the earnings for a week with $2529 total sales. 10. Imagine that you own your own T-shirt business. The cost of making the designs and buying the T-shirts is $475. In addition to these one time charges, the cost of printing each T-shirt is $1.75. The average cost 1.75x per T-shirt for the business to manufacture x T-shirts is modeled by A =. Find the average cost x per T-shirt when x = The formula C = 23.95d (m 780) is used to compute the cost of renting a car. Calculate the value of C when d = 7 and m = Answers 1. a) -3 b) 0 c) 6 d) -14 e) 10 f) a) 8 b) 11 c) 3 d) -3 e) -23 f) ,7,10,13 4. a)17 b)7 c)18 d) =43 6. a) 12 b)53 7. $ $ $ $ $

9 MFM1P Unit 2: Algebra Lesson 3 Date: Learning goal: I understand polynomial terminology and can create algebraic expressions. WARM UP Draw a model to represent x, x 2, and x Algebraic Expressions Start to fill out the FISHBONE diagram (next page) with the following definitions, using your notes: ü Variable ü Coefficient ü Constant ü Like Terms 9

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11 Representing Polynomials Algebra Tiles can help you visualize algebraic expressions. *shape tells us the type of tile **Colour tells us the sign (positive/negative) Represent the following polynomials with algebra tiles a) 3x + 1 b) 4x 2 3x c) 2x 2 2x + 4 d) x 2 5 New Vocabulary: **ADD TO YOUR FISHBONE! A term is simply a part of the expression. 11

12 A polynomial is an algebraic expression made of 1 or more terms connected by addition or subtraction. Number of Terms Type of Polynomial Examples Other examples 1 term Monomial 5 7x -3a y 2 2 terms Binomial 5 + x 3x 2 2x 7a + b 3 terms Trinomial 2x 2 3x+5 a + b +c 2x + 3y 6z Complete the chart. Expression 4x + 3 7a 2 2a + 5 5x +3y +2z a 2 + 4a - 2 6c 2 4 Number of Terms Name (monomial, binomial, trinomial or polynomial?) 12

13 Homework 2.3 Algebraic Expressions *separate sheet of paper! 1. Model each polynomial. a) 3x + 2 b)-x 2-2 c) 2x x 2. What expression does the model show? 3. Sonja and Myron are discussing this algebra tile model. Sonja says, This model shows the expression 3x 2 + x + 2. Myron says, It shows 3x 2 x 2. a) Who is correct? Circle SONJA or MYRON. b) Give 1 reason for your answer 4. Complete the table 5. Draw out each of the following expressions. 2 a) 2x 5x+ 3 2 b) 5+ 3p+ 2p c) 5m+ 3+ 2m 13 2

14 6. Which of the polynomials in question 5 can be represented by the same algebra tiles? Explain why. 7. Write an expression for each polynomial. 8. Explain the meaning of 2 tiles with the same shape but different colours. 9. Use algebra tiles to model each polynomial. Is the polynomial a monomial, binomial, or trinomial? Explain. 2 a) 5x 2 b) 3b 5b+ 6 c) 4 2x d) 5 + y 2 e) 3a 2 2a + 1 f) v 2 4v 10. For the polynomial 6x 5, state the following: a) number of terms b) coefficient of the first term c) constant term 11. Explain what each of the following words mean using examples to help with your explanation. a) Binomial b) Coefficient c) Constant d) Term 2.3 Answers 1. a) 3 Shaded x plus 2 shaded 1 b) one unshaded x2 plus two unshaded 1 c) two shaded x2 plus one unshaded x plus three shaded x2-3x+4 3. a) MYRON. b) Both shaded and unshaded tiles are in the expression. 4. a) 3. Trinomial b) 1 Monomial c)4 Polynomial d) 1 Monomial a and c have the same algebra tiles. 7. a) 2x2-3 b) x2-2x+1 c)-x2+3x-2 d)4 8. They are the opposite terms. 9. a) Monomial b) Trinomial c)binomial d)binomial e) Trinomial f) Binomial 10. a) 2 b)6 c) a)x+y two terms connected by subtraction or addition. b) 3x the number in front of a variable c) 6 number d) 6x a part of the expression 14

15 MFM1P Unit 2: Algebra Lesson 4 Learning goal: I can simplify algebraic expressions by collecting like terms. Date: 2.4 Like Terms Term Coefficient Variable Exponent 6b -3a 2 7x c 3 WARM UP What are like terms? Like terms have the same variable(s) and exponents; only the coefficients can be different. Examples: Example 1: From the list, circle the terms that are like 2w 2. Draw the algebra tiles if needed 5w, 6w 2, 2, 4w, 3x 2, w 2, 7w, 2 Example 2: From the list, circle the terms that are like -5x. Draw the algebra tiles if needed 4w, 5x 2, 6, 5x, 3w 2, x 2, 3x, 7 COLLECTING LIKE TERMS METHOD 1: Using Algebra Tiles 4x 2x x - 2 Step 1) Draw algebra tiles to show each term. Step 2) Group the tiles to form zero pairs. Step 3) Remove the zero pairs. The remaining tiles is your expression. 15

16 Simplify each of the following. (Collect like terms!) a) 2 4m m m 2 b) 2x 2 + 3x 1 + x 2 4x 2 METHOD 2: Simplify by Grouping Example a) 4 + x x + 1 b) 2x x 2 + 5x 2 Simplify each of the following a) 7d 2d b) 4 + 2a + 7 4a c) 3a 2 2a 4 + 2a 3a d) 6x x x 7x 2 16

17 Homework 2.4 Like Terms 1. a) Circle the terms that are like 3x: -5x, 3x 2, 3, 4x, -11, 9x 2, -3x, 7x, x 3 b) Identify terms that are like -2x 2 : 2x, -3x 2, 4, -2x, x 2, -2, 5, 3x 2 2. Fill in the blanks. a) The opposite of (+1). b) The opposite of is. c) Adding and produces a result of. 3. Combine like terms to write the expression in its simplest form. The first one is done for you. a) = x 2 -x + 3 b) = c) = d) = 4. Combine like terms. First, rewrite the like terms together. Then combine the like terms. Tiles can help! a) -3x x 4 b) 6x - 3 4x + 5 = -3x + 1x = -2x 2 c) 3x x x 5 d) 6 3x 2 + 7x

18 5. Simplify each polynomial using a method of your choice. a) 4 + x x + 1 b) 3y 2 + 3y 2 c) 2x x 2 + 5x 2 d) 3y + 7y y 2y 3y 2 e) 3a 2 2a 4 + 2a 3a f) 7z z z Which of the following can not be simplified? Simplify only the expression that can be. What do you notice? a) 5y 2 3y 4 b) 10x 1 c) 1 + x x 2 d) 2y y 2 3y + 16 e) 7 + 5x 7x x f) 5x x 6x 2 6 x 2x 7. Write an expression with 5 terms that has only 2 terms when it is simplified. 8. The following are some of Terry s homework answers that he did incorrectly. Circle where he made mistakes and explain WHY it is wrong. Question: 5x 2 + 6x x - 3x Step 1 = 5x 2-3x 2 + 6x + 4x Step 2 = 2x x - 4 Step 3 = 12x Determine the perimeter of the following figures. Remember that perimeter is the distance around a figure. Show your work clearly and completely. Express your answer as a polynomial in simplified form. 2.4 Answers 1. a) -5x, 4x, -3x,7x b) -3x 2, x 2, 3x 2 2. a)-1 b) shaded x c) 0 3. b) x 2 +3x-3 c)x 2 +x d)-x b) 2x+2 c)12x-4 d)4x a) 6x+6 b) -3y 2 +3y-2 c)3x 2-3 d)4y 2 +1 e)1 f)7z-4 6. a) No b) No c) No d)-5y 2-3y-4 e) 10x-1 f) x 2 +x x-3 8. a) Step 3 b) Can only add like terms together. 9. a) 5x+9 b) 9x+8 18

19 MFM1P Unit 2: Algebra Lesson 5 Learning goal: I can add and subtract polynomials. Date: WARM UP ACTIVITY: 2.5 Adding and Subtracting Polynomials Use tiles and two different colours to record your solution. Create zero pairs if you are adding positive & negative tiles. Draw the tiles under each polynomial & write answer in chart. Compare with neighbor. **compare with a neighbor. Adding Polynomials 1. To separate one polynomial from another, often brackets are used: 2 2 Evaluate: (3x + 5x 1) + (4x 2 x) USING TILES: WITHOUT TILES: 19

20 2. Write a polynomial for the perimeter of this rectangle. Simplify the polynomial. 3x + 7 2x + 1 Subtracting Polynomials Review: Subtracting is the same as the. This same idea can be used to subtract polynomials. Add the opposite Step 1: Change the sign and write the opposite of the second polynomial. Step 2: Rewrite without the brackets, group like terms, combine like terms. Ex. (2x 2 + 3x + 5) - (x 2 + 2x + 4) 2 2 Ex. Try without tiles: (7b 2b+ 13) (4b 5b 6). 20

21 3. Practice Problems: Subtract (choose your method). a) (4x + 2) (2x + 1) b) (4x + 2) (2x 1) c) (4x + 2) ( 2x 1) Bring it all together! 4. SIMPLIFY. (what does this mean?) a) (5a 2 + 2a) + (6a 2 a) b) (3y 2 2y + 5) ( 4y 2 + 6y + 5) c) (x 2 + 2x 4) + (4x 2 2x 5) d) ( 9z 2 z 2) (3z 2 z 3) 21

22 Homework 2.5 Adding and Subtracting Polynomials 1. Add each polynomial. Use algebra tiles if it helps. a) ( 4h + 1) + (6h + 3) b) (3y 2 2y + 5) + ( y 2 + 6y + 3) c) (x 5) + (2x + 2) d) (y 2 + 6y) + ( 7y 2 + 2y) e) (x 5) + (2x + 2) f) (b 2 + 3b) + (b 2 3b) g) (2a 2 + a) + ( 5a 2 + 3a) h) (3y 2 2y + 5) + ( y 2 + 6y + 3) i) (3 2y + y 2 ) + ( 1 + y 3y 2 ) j) (5n 2 + 5) + ( 1 3n 2 ) 2. For each shape below, write the perimeter as a sum of polynomials and in simplest form. i) ii) iii) iv) 3. The sum of two polynomials is 3r 2 4r + 5. One polynomial is 2r 2 + 2r 8 ; what is the other polynomial? Explain how you found your answer. ( ) + ( 2r 2 + 2r 8 ) = 3r 2 4r Subtract the following polynomials: a) (2x + 3) (5x + 4) b) (4 8w) (7w + 1) c) (4x + 2) ( 2x 1) d) (x 2 + 2x 4) (4x 2 + 2x 2) e) ( 9z 2 z 2) (3z 2 z 3) f) (2s 2 3s + 6) (s 2 s + 2) 5. A student subtracted a) Explain why the student s solution is incorrect. b) What is the correct answer? Show your work. 6. Molly has (4x + 10) dollars and Ron has (-5x + 20) dollars. a) How much money do they have altogether? b) How much more money does Molly have than Ron? 22

23 7. The perimeter of each polygon is given. Determine each unknown length. 2.5 Answers 1. a) 2h+4 b) 2y 2 +4y+8 c) 3x-3 d) -6y 2 +8y e) 3x-3 f)2b 2 g) -3a2+4a h) 2y 2 +4y +8 i) -2y 2 -y+2 j) 2n i) 6n + 6 ii) 9p + 12 iii) 16y + 4 iv) 2a r 2-6r a) -3x-1 b) 3-15w c) 6x+3 d) -3x 2-2 e) -6z 2 +1 f) s 2-2s+4 5. a) -5x 2 since -2-3 = -5. b) -5x 2 -x a) x+ 30 b) 9x a) w+4 b) s+3 23

24 MFM1P Unit 2: Algebra Lesson 6 Learning goal: I can use the distributive property to multiply a number and a polynomial. Date: 2.6 The Distributive Property - Part 1 Think When you distribute something you give that thing to each person in a group. The Distributive Law If you multiply a number by a bracket, then multiply each term in the bracket by that number. We call this EXPANDING New Vocab: Expand à Example 1: Expand the following. a) 3(x + 2) b) 2(4r 4) = c) (3n + 6)(4) d) 2(x + 3) e) (2 n)(8) f) 4(y + 2) DISTRIBUTING A NEGATIVE can be TRICKY! BE CAREFUL WITH YOUR NEGATIVES! Examples 2: Expand the following. a) -5(n + 4) b) (4 y) c) (7m 5)(-3) 24

25 Sometimes you need to distribute then collect like terms. Examples 3: Fully simplify each of the following. a) 2(6x 4) + x b) 4 2(m + 5) Distribute Distribute Simplify. Simplify. c) 15t (t-4) d) -6(v + 1) + v APPLICATIONS Example 4: Write a simplified expression for the area of the following rectangles 25

26 1. Fully simplify each of the following. Homework 2.6 The Distributive Property Part 1 a) 4(3a + 2) b) (d 2 + 2d)( 3) c) 2(4c 2 2c + 3) d) ( 2n 2 + n 1)(6) e) 3( 5m 2 + 6m + 7) f) (5t 2 2t)( 9) g) 3(x 2 + x 4) h) 2(m 2 3m + 5) i) 4(b 2 2b 3) j) 5(c 2 6c 1) k) 3(4 h 2 ) l) (n 2 + 4n + 3)( 2) g) (5t 2 2t)( 3) h) (w 2 + 2w 5)(4) h) (4x 2 3x 5) 2. Here is a student s solution for this question: a) Explain why the student s solution is incorrect. b) What is the correct answer? Show your work. (The student made 2 errors!!) 3. Use the Distributive Property to write and simplify an expression for the area of the rectangle. a) b) c) 4. Expand and Simplify x + 5 a) ( x ) ( ) b) 3 ( k 4) 2( k + 1) ( ) 3( j 3) d) 3( y 2) + 2( 4 2y) + ( 6 7y) c) 5 j e) 4( 3) 2( k 3k + 4) ( k 5) k f) 4 ( 3a 2b) 2( 5a + b) 5. A computer repair technician charges $50 per visit plus $30/hour for house calls. a) Write an algebraic expression that describes the service charge for one household visit. b) Use your expression to find the total service charge for a 2.5 hour repair job. c) Suppose all charges are doubled for holidays. Write a simplified expression for these service charges on a holiday. d) Use your simplified expression from part c) to calculate the cost for a 2.5 hour repair job on a holiday. 26

27 2.6 Answers 1. a) 12a + 8 b) -3d2-6d c) 8c2-4c+6 d)-12n2+6n-6 e) 15m2-18m-21 f)-45t2+18t g) 3x2+3x-12 h)2m2-6m+10 i) -4b2+8b+12 j)5c2-30c-5 k)-12 +3h2 l)-2n2-8n-6 m)-15t2+6t n)4w2+8w-20 o)-4x2+3x+5 2. a) 2r and -14 b) -8r2+2r a) 91-13x b) 14x+35 c) 8x a) 5x+9 b) k-14 c) 2j-6 d) -8y+8 e) -3k2+10k-15 f) 2a-10b 5. a) C= 50+30h b)$125 c) h d)$250 27

28 MFM1P Unit 2: Algebra Lesson 7 Learning goal: I can use the distributive property to multiply two polynomials. Date: 2.7 The Distributive Property - Part 2 WARM UP. How would we model MULTIPLYING with tiles? Sign rules for multiplying: Steps 1. Use tiles to model each expression on the sides of a rectangle. 2. Fill in the area of the rectangle using tiles. Remember your sign rules for multiplication.see chart à 3. Count the tiles inside the area. Write this by collecting like terms. 1. Examples to get us warmed up a) b) 2(4) -3(4) Answer: = Answer: = Now try some examples using variables (letters) c) d) 3(2x) 3x(4) Answer: = Answer: = 28

29 e) (3x)(2x) f) (-4x)(2x) Answer: = Answer: = What about letters with exponents? Like (x 2 )(x 3 ) or (x 2 )(x)? We cannot use area models for these. Let s learn how to do this without tiles.. 2. Practice: Multiply each of the following monomials. *there are rules for exponents!! a) (-3x 2 )(7x) b) (2x 2 )(-3x) c) (-3x)(-12x) When multiplying the same variables (letters), you the. 3. What if there are 2 terms in bracket? Simplify: 3x(2x + 4). Method 1 (tiles): Method 2 (no tiles): 29

30 Simplify a)2a(5a+3) b) 4b(3b 2) c) 3c( 5c-1) 4. Jacob is designing a pool and a deck that will sound the pool. He has sketched the following. a) Determine the area of the pool b) Determine the area of the deck (without the pool) 30

31 1. Simplify Assignment 2.7 Distribution Part 2 a) (x)(3x) b) (x)(5x) c) (3x)(5x) d) (7x)(2x) e) (x 2 )(x) f) (3x)(3x 2 ) g) (12x)(3x 2 ) h) (6x)(6x 2 ) 2. Multiply. a) (-3)(2x) b) (2x)(-9) c) (4x)(-4) d) (5)(-2x) e) (8)(-x) f) (5x)(-7) g) (-9x)(9) h) (-3x)(-6) i) (2x 2 )(-3x) j) (4x 2 )(-8x) k) (-3x)(4x 2 ) l) (2x)(-2x 2 ) 3. This diagram shows one rectangle inside another. a) Determine the area of the small and big rectangle. b) Determine the area of the shaded region. 4. A student thinks that the product 2x(x +1) is 2x Choose a model. Use the model to explain how to get the correct answer. 5. Determine each product (multiply) a) 2x(x 6) b) 3t(5t + 2) c) 2w(3w 5) d) - 3g(5 g) 31

32 6. Simplify a) 3(x 2 +x 4) b) 4(b 2 2b 3) c) 3h(4 h 2 ) d) (5t 2 2t)( t) e) -x(2 + 8x) f) (4 +3y)(-2y) g) 2(m 2 3m+5) h) 5c(2c 2 6c 1) 7. An L-shaped patio is built from two rectangular areas A and B. a) Write a simplified expression for the total area of the patio. b) Find the total area of the patio when x is 3 m. 2.7 Answers 1. a) 3x 2 b) 5x 2 c) 15x 2 d) 14x 2 e) x 3 f) 9x 3 g) 36x 3 h) 36x 3 2. a) -6x b) -18x c) -16x d) -10x e) -8x f) -35x g) -81x h) 18x i) -6x 3 j) -32x 3 k) -12x 3 l) -4x 3 3. a) Large = 18x 2, Small = 8x 2 b) 10x 2, Subtract the area of the small rectangle from the area of the large rectangle. 4. 2x*x = 2x 2 2x*1 = 2x Correct answer : 2x2+2x 5. a) 2x 2-12x b) 15t 2 +6t c) 6w 2-10w d) 3g 2-15g 6. a) 3x 2 +3x-12 b) -4b 2 +8b+12 c) 3h 3-12h d) -5t 3 +2t 2 e) -8x 2-2x f) -6y 2-8y g) 2m 2-6m+10 h) 10c 3-30c 2-5c 7. a) 4x 2 +15x-5 b) 76m 2 32