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1 Unit #: Powers and Polynomials Unit Outline: Date Lesson Title Assignment Completed.1 Introduction to Algebra. Discovering the Exponent Laws Part 1. Discovering the Exponent Laws Part. Multiplying and Dividing Polynomials Mid Unit Test àlesson.1 to..5 Collecting Like Terms.6 Adding and Subtracting Polynomials.7 Multiplying a Polynomial by a Monomial Mid Unit Test à Lessons.5 to.7.8 Simplifying Algebraic Expressions Part 1.9 Simplifying Algebraic Expressions Part *** Final Algebra Test Mathwithsheppard.weebly.com 1

2 .1 Introduction to Algebra Learning Goals: By the end of this lesson, you will be able to: 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. Visualizing Algebra: x x x x+x (x)(x) Vocabulary à Term: à Variable: à Coefficient: à Constant: Draw the following algebraic terms: x x Vocabulary à Like Terms: Come up with some examples!

3 Terms can be added and/or subtracted together to form polynomials. Important: Polynomials are in simplest form when they contain no like terms. Vocabulary à Polynomial: à Trinomial: à Binomial: à Monomial: ***Polynomials are in standard form when they are written with exponents highest à lowest. Practice Problems: 1. State the coefficient and variable(s) in each term a) 15x b) -y c) -w Coefficient: Coefficient: Coefficient: Variable: Variable: Variable:. Circle the constant term in each polynomial. a) x + x + b) y 5 c) a + 5a

4 . Draw a model of each of the following. a) x b) -x c) x + 1 d) x - e)x f) x Writing Algebraic Expression Writing algebraic expressions from word problems is a very important skill in mathematics. 1. Seven times a number plus three. Seven times the square of a number plus three. Three quarters of a number. A gym membership costs $5 upfront, then $10 per month 5. The sum of a number and seven 6. The difference between seven and a number 8. times a number plus seven

5 Assignment.1: Introduction to Algebra 1. Sketch models to represent each of the following algebraic expressions. a) x b) x c) y d) 5x. For each term, identify the coefficient and the variable. a) x b) 5p c) m n d) g h e) y 5 f) p q 5 g) ab h) 0.6r s. Classify each polynomial by type. a) x + 1 b) p p + c) b d d) 6 + gh 5 e) y 5 y + y f) x y + g) ab b h) 6p q. We read as squared and as cubed. Sketch models that show why this makes sense. 5. Write an algebraic expression for each phrase. a) double a number b) triple a number c) quadruple a number d) one half of a number e) one third of a number f) one quarter of a number g) 6 more than a number h) a number increased by h) increased by a number j) 5 decreased by a number k) 7 less than a number l) a number decreased by 6 6. Write an algebraic expression for each phrase. a) more than triple a number b) half a number, less 5 c) quadruple a number decreased by 1 d) less than double a number 7. Sate the problems that are in standard form. If it is not in standard form, re-write in standard form. a. x 11x b. + x + x + x c. x + 17x + x d. 1+ x + x 8. Given: x 5x x + 1 a) How many terms are there? b) What is the coefficient of the rd term? c) What is the constant? 5

6 . Discovering the Exponent Laws Part 1 Learning Goals: By the end of this lesson, you will be able to: Investigation #1: Multiplication of Powers Product Expanded Form Single Power x y m a x = ( x x x x) ( x x x) y = ( y y y y y) ( y y y y y y) 5 6 m = = 5 6 t w h p a = = t = = w = = 7 h = = 8 5 x p = = a b x = = 7 = x 11 = y Complete this statement: When multiplying powers with the same base. Challenge: cd cd = ( c c d d d) ( c c c c d d d d d) = cd k j k j = = 5 pq pq = = vz vz = = 5 gt gt = = 5 abc abc = = 5 6

7 Investigation #: Division of Powers Product Expanded Form Single Power x y m x 7 y 5 x x x x x x x = x x x y y y y y = y y m = = 6 = x = y a a = = 6 t t = = x a b x = Complete this statement: When DIVIDING powers with the same base Challenge: cd cd c c c c c c c d d d = c c c c d d 7 = cd k j k j = = 8 5 pq pq = = 6 vz vz = = 5 8 gt gt = = 6 5 abc abc = = 7 6 7

8 Let s Review: a 1 = a 0 = x y = x y = a 5 a = a 5 a = a 5 b 6 a b = b 5 b 5 = Consolidation: a) ( x 7 )( x ) b) ( a 5 b )( a b ) b) x 6 ( x y )( x y 5 ) x 5 y 5 Confirm your answers by writing the expressions in standard form. (When it doubt, write it out) 8

9 Assignment. Discovering the Exponent Laws Part 1 1. Write each expression as a single power. a) 7 7 b) 5 c) 5 5 d) e) ( ) ( ) f) ( 1) ( 1) ( 1) g) h). Evaluate each expression in question 1.. Write each expression as a single power. a) b) c) d) 8 5 e) ( 9) 7 ( 9) 6 f) g)( 0.) ( 0.) h). Evaluate each expression in question. 5. Simplify. a) b 5 b b) p p c) w 5 w d) x 8 x e) a b 5 ab f) m n m n g) p 6 q 5 p q h) xy y 6. Simplify each of the following, write as a single power if possible. a) b) a 7 a 1 c) c c 8 d) d 7 d 9 e) x x 8 f) w 10 w 100 g) a 6 b 5 h) 10 a 10 b i) g 1 g 19 g 11 j) k) l) Explain why it is necessary for the bases to be the same in order to apply the multiplication and division principles for exponents, use examples in your explanations. Enrichment: Write as a single power 9

10 Warm Up:. Discovering the Exponent Laws Part Evaluate by writing the expanding form Simplify using the exponent principles. Investigation : Power of a Power Product ( ) Expanded Form x = ( x x x x) ( x x x x) ( ) y = ( y y y) ( y y y) ( y y y) ( y y y) ( m ) = = ( g ) 5 = = ( t ) = = a ( x ) b = = 8 = x Single Power 1 = y Do these using your SHORTCUT! ( m ) 5 = When simplifying POWERS OF POWERS. 8 ( q ) 6 = ( ) 5 ab = ( x y z ) 6! # " a b $ & % 10

11 '! )# ( )" x $ & %! # " *, +, x 5 $ & % Let s Review: There are still two more: Example: (xy) = Example a b = Practice Problems: Simplify a) (x y ) b) (5xy) c) (x y ) d)! " # $ % &!# $ " e) %' ( & % + ) 5 * &, f)! # " x 5 $ & % Exponent Rules can only be used when. 11

12 Practice: Simplify ( x 5 y ) a) (n ) (n ) b) xy c) ( x y ) xy Extension: Determine the value that makes each statement true a) =? b) 7 =? c) 5? = 6 0 1

13 Assignment. Discovering the Exponent Laws Part 1. Simplify each expression using exponent laws. 5 9 a) ( ) b) ( 5 ) 5 c) ( ) 7 d) ( 7 ) 5 e) f) ( ) 7. Simplify each expression using exponent laws. a) ( x ) 5 b) ( 5 ) 5 c) ( ) 7 d) ( 7 ) 5 7 e) ( a ) 5 f) ( x ) 5 g) ( x ) h) ( ) y i) y ( ) ( x ) 5 j) ( x x x ) 5 k) ( s t ) 5 l) ( x y 5 ) 6 m) a b n) x y. Simplify each expression using exponent laws. 1 6 y x 8 5 o) 5 a b a b a) (m 5 ) b) (k ) k c) g 5 g 5 g 7 d) (a 6 ) (a 5 ) e) (gh ) f) k m (k ) g) (g 5 h ) gh 6 h*). Show that 10 is the same as 9 5 using your understanding of exponent. 5. Determine the value of the exponent that makes each expression true. a) 7 =? b) (-15) 7 = (-5)? c) 6? = bd bd ( bd ) 6. Evaluate for a = 5 and b =. Are the expressions equal? If not, which expression has the lesser value? a) a + b or a + b b) a b or (ab) 1

14 . Multiplying and Dividing Monomials Warm Up: After marking a quiz, a teacher recorded the most common errors made by the students. In each case, identify the error made by the student and provide the correct simplification. a) 1 5 b) ( ) = = c) d) ( 5a b) = 5a b = Multiplying Monomials x y xy When in doubt, write it out. MULTIPLYING MONOMIALS (6x y 6 z)(xy ) 1

15 Practice Problems: Multiply the following monomials. a) ( x)( x) b) ( 5m n) m ( ) c) ( 6z)( x y) d) ( 7mn )( mn ) e) 7 x ( 15 x ) f) ( mnp )( 8n) DIVIDING MONOMIALS 81a b 5 c 18a b 5 c Practice Problems: Divide the following monomials. a) 10x b) 5x 16m n mn c) 8x y 16xy 5 15

16 e) x yz 1 xyz 5 5 d) ( 15mp) ( 5p) f) x x Challenge Questions: 5 6x y a) 8y ( )( 5 6x y 9x y ) b) x y 16

17 Assignment. Multiplying and Dividing Monomials 1. Multiplying Monomials: Simplify each expression using exponent laws. a) ( b)( b) b) ( 6b) d) 5 x( 5x) e) ( b ) ( b ) 5 5 g) ( b )( b ) h) 5 ( y )( 6x )( y ) 6 j) 7 ( a ) 5 c) ( b)( b ) f) ( x)( x ) x i) 5x ( ) a k) ( 5 x)( x)( x) l) ( x )( x)( x) m) ( a b )( ab ) n) ( xy)( 5y ) o) a ( a)( 5a ) 5 p) 7m n ( m )( 6n) q) x ( x y )( 8y) r) 8x y ( x y )( x y ). Dividing Monomials: Simplify each expression using exponent laws. Fractions must be in lowest terms no decimals! a) 9x b) 8y 5 c) y 1n 18n 6 5 d) g) j) 5z 5z 5x 7 0x y 7 8 y 1 7x 6 y 6x y ( )( x y) e) h) 7t 9t 9a a b b 5 7m ( 5n ) k) 9 5m n 8 f) x 6x 7 5 9x ( y ) i) 11 81( x ) y ( 5x) ( 5y ) l) ( 5x y ). Write an expression to represent the area of the following shapes. a) xy b) x 1 m 17

18 . Determine the volume of the cube shown. 5. Determine the volume of the following shapes. 6. Write an expression to represent the volume of the following shape. 7. The volume of a box is 6a 5 b. The length of the box is a and the width of the box is 8b. Determine the height of the box. 18

19 . 5 Collecting Like Terms Warm Up: Draw a model to represent x, x, and x Illustrate and explain the difference between a) x and x. b) x and x Start to fill out the FISHBONE diagram (next page) with the following definitions: ü Variable ü Coefficient ü Constant ü Like Terms ü Polynomial 19

20 0 Powers and Polynomials

21 Representing Polynomials Algebra Tiles are often used to represent polynomials. Algebra tiles can help you visualize equivalent algebraic expressions and/or equations. Represent the following polynomials with algebra tiles a) x + 1 b) x x c) x x + d) x 5 Like and Unlike Terms Like terms have the same set of variable bases and corresponding exponents. Examples: Represent each of the following sets with algebra tiles a) x and -5x b) -6x and 9x c) xy and 5xy Did you notice how algebra tiles with the same size and shape represent like terms? Define and give an example of Unlike Terms - 1

22 Collecting Like Terms Algebraic expressions that contain like terms can be simplified by combining each group of like terms. Examples: x + x 9x 6x 1x y - 5x y Why can t you simplify? x + x x 7 6x y + 5xy Practice Problems: Simplify 1) 7x + 5 x ) 6w + 11w + 8w 15w ) 6x + 5 (-7x) ) (-1x) 5 7x 11 5) x (-x) + 7 (-x ) + x 7 6) 11a b 1ab

23 Assignment. 5: Collecting Like Terms 1. Are the terms in each pair like or unlike? a) 5a and a b) x and x c) p and p d) ab and ab e) b and b f) 6a b and a b g) 9pq and p q h) x y and x y. Simplify. a) + v + 5v 10 b) 7a b a b c) 8k k 5k + + k d) x x + 8x + 5x e) 1 m 8 m + m f) 6y + y + 10 y 6 y. Simplify. a) a + 6b + b + a b) x + xy + y + 5x xy y c) m m m + m d) x + xy + y x + xy y. Simplify: a) w w + 1 w w b) a 1 5 b a b 5. Find the perimeter of each figure below. a) b) n

24 .6 Adding and Subtracting Polynomials Warm Up: Determine a simplified expression the perimeter of the following rectangle. Give an example of A monomial A binomial A trinomial A constant term ACTIVITY: Adding 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.

25 To separate one polynomial from another, often brackets are used: Evaluate: (x + 5x 1) + (x x) USING TILES: WITHOUT TILES: Practice Problems: Simplify. Choose your method! a. (y 7) + ( y+ ) b. ( a + a) + (6a + 10 a) + (5a+ 1) PROBLEM SOLVING: The measures of two sides of a triangle are given. P is the perimeter. Find the measure of the third side. P = x + 5x + 5 x + x 5 x + x + 6 Enrichment: Determine the polynomial that must be added to x xy + 6y to get: 6x 6xy + 8y Create two three-term polynomials that when they are combined you get a -term polynomial 5

26 Subtracting Polynomials Review: Subtracting is the same as the. This same idea can be used to subtract polynomials. Ex. (x + x + 5) - (x + x + ) Ex. Try without tiles: (7b b 1) (b 5b 6) +. Practice Problems: Simplify (choose your method). a. (x ) (x+ 9) b. (5z 1 z) ( z 8) c. 1 (x 5x 1) + d. (8x + 7 x) (x ) ( 5x+ 9) 6

27 Assignment.6 Adding and Subtracting Polynomials 1. Add. a) (y + 6y 5) + ( 7y + y ) b) ( n + n + ) + ( 1 7n + n) c) (m + m) + ( 10m m ) d) ( d + ) + ( 7d + d). For each shape below, write the perimeter as a sum of polynomials and in simplest form. i) ii) iii) iv). Subtract. a) (x + ) (5x + ) b) ( 8w) (7w + 1) c) (x + x ) (x + x ) d) ( 9z z ) (z z ). A student subtracted (y + 5y + ) (y + y + ) like this: = y 5y y y = y y 5y y = y 8y a) Explain why the student s solution is incorrect. b) What is the correct answer? Show your work. 5. The difference between two polynomials is (5x + ). One of the two polynomials is (x + 1 x ). What is the other polynomial? Explain how you found your answer. 6. Subtract. a) (mn 5m 7) ( 6n + m + 1) b) (a + b a + b ) ( a + 8b + a b) c) (xy x 5y + y ) (6y + 9y xy) 7. The sum of the perimeters of two shapes is represented by 1x + y. The perimeter of one shape is represented by x y. Determine an expression for the perimeter of the other shape. Show your work. 8. A rectangular field has a perimeter of 10a 6 meters. The width is a meters. Determine an expression for the length of this field. 7

28 .7 Multiplying a Polynomial by a Monomial Warm Up: Simplify a) (b) b) (6h) c) (b ) d) (x ) e) ( y ) f) ( f) Simplify. a) (6k ) + (k + ) Simplify 5 a) a a b) (a + 1) (a + ) b) 6xy x y 7 c) (b 6) ( 5b) + (b + ) c) ( xy 5 )(x y ) d) 5 (9 y ) d) (m + m + 1) (m + m 6) e) 8bd bd ( bd ) 8

29 The Distributive Property The property known as the distributive property is also known as expanding. We are making sure EVERY TERM in bracket gets multiplied. Distributive Property: ab ( + c) = ab+ ac With an integer. a) (g + ) b) 7(q + ) c) (t 1) d) ( w 5) With a variable.. Example 1) Simplify: x(x+ 5) Practice Problems: Expand. a) b(b + 1) b) p(p + ) c) r( 5r + ) d) w(w 1) 9

30 With a variable and coefficient. Example ) Expand (9 x x ) x. Practice Problems: Simplify. 6 a) y( y + y ) b) 5 z (z 1) c) x ( x x+ 9) d) ( a 1) 11 a Example ) Simplify (b b + ) + 5b(b + ) Practice Problems: Expand and simplify. a) -(p ) + 6(p + 1) b) [ (6 t) + 5t] c) 5m(m + 5) (m m 7) 0

31 Assignment.7 Multiplying a Polynomial by a monomial 1. Determine each product. a) (a + ) b) (d + d)( ) c) (c c + ) d) ( n + n 1)(6) e) ( 5m + 6m + 7). Here is a student s solution for a multiplication question. ( 5k k )( ) = (5k ) (k) () = 10k k 6 a) Explain why the student s solution is incorrect. b) What is the correct answer? Show your work.. Expand. a) (x + x ) b) (m m + 5) c) (b b ) d) 5c(c 6c 1) e) h( h ) f) (n + n + )( ) g) (5t t)( t) h) (w + w 5)(w). Write a simplified expression for the area of the following rectangles a) b) 5. Expand a) x (x + ) b) n(n ) c) 5m(m + m) d) (y y )y e) pq(p + q) f) d(d d + 1) g) (t + t )(-t) h) a b(a b ) 6. Simplify a) (x + ) (x + ) b) a(a + b ) b(a + ab) c) p(p ) + 6(p + p ) d) x(x y ) + y(6x + y) Enrichment: Do some research to learn how to multiply a binomial by a binomial (or a polynomial) a) (x + 1)(x ) b) (x 5)(x ) c) (x y)(5x + y) 1

32 .8 Simplifying Algebraic Expressions Solve the following word problems. Include your steps and a clear and thorough solution. 1. The area of a rectangle is represented by the other side? 6 6x y. One side is represented by 6x y. What is the length of..

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34 Assignment.8 - Simplifying Algebraic Expressions Simplify 1) p q + p + + 5q 1 ) 5x y + 1xy 8x 1xy ) 5 y + y 8y 9y 1 ) x + x 7x (-5x ) 5) 5(a b) + (-b a) 6) (x + y) 5(-y + x) 7) -b(5a b) + (-ab 5b ) 8) x(x - y) (-x - xy) 9) -(x + y) + 5(-6y x ) 10) -(7xy - 11y ) y(-x + y) 11) ( x) - (-5 1x) 1) 7( - x) 6(8-1x) 1) 15) x y y + x 1) 1 x y x + y ) s t s t a b a b 18) 6 [ ( x y) ] + 19) (x y) [ (x y)] 0) 7 { x [ x (x 1) ]} ( ) 1) 5x (x + 6) " # x 1 + x $ % { } ) a a ( b + a ) ) [ x ( y w) ] [ x (y w) ] ) 6b { 5a [ a + ( b a) ]} + + 5) x x + x! + 5 6) A triangle has sides of length a centimeters, 7b centimeters, and 5a + centimeters. What is the perimeter of the triangle? 7) A rectangle has sides of length 7x meters and x + meters. What is the perimeter of the rectangle? 8) A square has a side of length 9x inches. Each side is shortened by inches. What is the perimeter of the new smaller square? 9) A triangle has sides of length a 5 feet, a + 8 feet, and 9a + feet. Each side is doubled in length. What is the perimeter of the new enlarged triangle? 0) Subtract four times the quantity x + 5 from three times the quantity 7x 8.

35 Possible Test Questions 1. If x =, what is the value of x + 5x? a 1 b 7 c d 51. Simplify the following expression: x(x + ) 5x a 6x 5x + b 6x 6x c 15x 5x d 6x + x. Simplify the following algebraic expression: a a b b a b 6 a b a b c a b d a b. Tim shows the steps he took in simplifying the following algebraic expression: In which step did Tim make an error? a Step 1 b Step c Step d Step = Step 1 = Step = Step 5. Sabeeta expands and simplifies the expression below. (x 5x) + x(7 + x) Which expression is equivalent to the one above? a 6x + x b 10x + 18x c 10x 8x d 8x = 1 Step 1) x + ( 1) (+5x) ( 6) ) ( 5a + 6) + (8a 7) ) (b ) (b 8) ) y(y 5) 5) (x + 6) (x + ) 6) 1 x ( ) + 9x +1 ( ) 7) m(m 5) (m m 1) 8) [(p + ) p] 6 ( x y) x ( y + w) x + (y w) 9) [ + ] 10) [ ] [ ] 5