Section 6: Polynomials

Foundations of Math 9 Updated September 2018 Section 6: Polynomials This book belongs to: Block: Section Due Date Questions I Find Difficult Marked Corrections Made and Understood Self-Assessment Rubric Category Sub-Category Description Expert 6 Work meets the objectives; is clear, error free, and demonstrates a mastery of the Learning Targets 5 Work meets the objectives; is clear, with some minor errors, and demonstrates a clear understanding of the Learning Targets Apprentice 4 Work almost meets the objectives; contains errors, and demonstrates sound reasoning and thought concerning the Learning Targets 3 Work is in progress; contains errors, and demonstrates a partial understanding of the Learning Targets Novice 2 Work does not meet the objectives; frequent errors, and minimal understanding of the Learning Targets is demonstrated 1 Work does not meet the objectives; there is no or minimal effort, and no understanding of the Learning Targets You could teach this! Almost Perfect, one little error. Good understanding with a few errors. You are on the right track, but key concepts are missing. You have achieved the bare minimum to meet the learning outcome. Learning Outcomes not met at this time. Learning Targets and Self-Evaluation Learning Target Description Mark 6 1 Understanding terms, degree, coefficients, and constants Grouping like terms 6 2 Addition and Subtraction of degree-2 polynomials 6 3 Multiplication and Division of degree-2 polynomials Basic operations with algebra tiles Combined Operations with degree-2 polynomials

Competency Self-Evaluation A valuable aspect to the learning process involves self-reflection and efficacy. Research has shown that authentic self-reflection helps improve performance and effort, and can have a direct impact on the growth mindset of the individual. In order to grow and be a life-long learner we need to develop the capacity to monitor, evaluate, and know what and where we need to focus on improvement. Read the following list of Core Competency Outcomes and reflect on your behaviour, attitude, effort, and actions throughout this unit. Rank yourself with a check mark: E (Excellent), G (Good), S (Satisfactory), N (Needs Improvement) Personal Responsibility I listen during instruction period and come to class ready to ask questions I am fully prepared for Unit Quizzes I am fully prepared to re-quizzes I follow instructions and assist peers I am on task during work blocks I complete assignments on time E G S N Self-Regulation I keep track of my Learning Targets I take ownership over my goals, learning, and behaviour I can solve problems myself and know when to ask for help I can persevere in challenging tasks I take responsibility to be actively engaged in the lesson and discussions I only use my phone for school tasks Classroom Responsibility and Communication I am focused on the discussion and lessons I ask questions during the lesson and class I give my best effort and encourage others to work well I am polite and communicate questions and concerns with my peers and teacher Collaborative Actions I can work with others to achieve a common goal I make contributions to my group I am kind to others, can work collaboratively and build relationships with my peers I can identify when others need support and provide it I present informative clearly, in an organized way I ask and respond to simple direct questions Communication I am an active listener, I support and encourage the speaker Skills I recognize that there are different points of view and can disagree respectfully Overall Goal for next Unit refer to the above criteria. Please select (underline/highlight) two areas you want to focus on 1

Section 6.1 Polynomials Vocabulary Term: There will be a lot vocabulary necessary to accurately understand Polynomials Any variable, constant, or product of the two 3, 4x, t, 2r 2, xyz Like Terms: Terms that have the same variable(s) to the same exponents x 2 and 4x 2, 7t and 3t, 4 and 9 Degree of a Term: The exponent on the variable or sum of exponents on different variables of one term 3x is Degree 1, 4x 2 is Degree 2, 5xyz is Degree 3 Polynomial: Any term or terms separated by addition or subtraction where all exponents on the variables are whole numbers 5t 2 + 2t 7 Leading Term: The term in a Polynomial with the highest degree From above: 5t 2 is the leading term, it has the highest degree Descending Order: Writing terms from highest to lowest degree From Above: Is in descending order, degree goes 2, 1, 0 Polynomial Degree: The highest degree on a term, becomes the degree of the polynomial From Above: 5t 2 is the leading term with degree 2, so the Polynomial is of degree 2 2

Combining like Terms Combining like terms is doing exactly that When we have a long list of terms written as a Polynomial, we can combine any that are Like Terms: same variables, same exponent 3q 2 4 + 2q 2q 2 + 4q 8 Like Terms are: 3q 2 and 2q 2, 4 and 8, 2q and 4q So, 3q 2 2q 2 = q 2 2q + 4q = 6q 4 8 = 12 And Descending Order: q 2 + 6q 12 Degree 2 Degree 0 Degree 1 Combine the Like Terms and leave the simplified expression in Descending Order Like Terms are: 5xy + 5x 2 + 2x 6 4yx + 2x + 6 3x 2 +5x 2 and 3x 2 so 5x 2 3x 2 = 2x 2 Degree of 2 5xy and 4yx so 5xy 4xy = xy Degree of 2 +2x and + 2x so 2x + 2x = 4x Degree of 1 xy and yx are the same, in multiplication order doesn t matter, xy = yx 6 and + 6 so 6 + 6 = 0 Degree of 0 Since x 2 and xy are both degree 2, which one goes first? We list them ALPHABETICALLY, x 2 = xx and xx comes before xy 2x 2 + xy + 4x 3

Section 6.1 Practice Questions Identify the number of terms, what are they, and their degrees? 1. 3x 4x 2 5 2. 4xyz 3. 2xyz 5xy + 4 4. 5x 3 y + 4xy 3 6xyz 5. 5 6. 3x + 4y + 5z x 2 Put the following Polynomials in DESCENDING ORDER 7. 3 + 4x 2 5x 8. 2t + 4t 3 2t 3t 2 9. 2 x + 5x 2 10. z 2 4z + 5 11. x + xy + xz y 12. 5xy y + 2x + x 2 4

Simplify the following, put your answer in DESCENDING ORDER 13. 3t + 4 6t + 2t 2 + 4 3t 2 14. 7z 3 + 2z 4z 2 + 1 4z 2 5 + 3z 2 + 3z 15. 5xy + 3 5yx + 2 16. 4q + 5q 2 7 5q 2 + 4q + 7 5

17. 1 3 i2 + 2i 1 6 i2 + 4 9 18. 4.9x 3.2y 1.3x + 4.2y + 1 19. 11 x + 2 y 3 x 1 y + 10 5 3 5 3 20. 1 4 j2 j 1 2 j + 3 8 j2 + 5 16 j2 6

Section 6.2 Addition and Subtraction of Polynomials All we are doing here is grouping like terms, however it involves a couple extra steps Let s start with Addition Addition of Polynomials (x 2 + 4x 7) + (2x 2 3x + 4) We have 2 Polynomials, shown in brackets, and we are adding the second Polynomial to the first. Step 1: In addition just drop the Brackets, keep the sign on the first term of the second Polynomial, since it s positive, nothing changes x 2 + 4x 7 + 2x 2 3x + 4 Step 2: Group the Like Terms x 2 + 4x 7 + 2x 2 3x + 4 = 3x 2 + x 3 Make sure your answer is in DESCENDING ORDER! We can t SOLVE for the unknown yet, this is as far as we will go in this class. If you make the Polynomial equal to something, then we can solve: We do this in Grade 10. Rearrange the terms so like terms are together, descending order right away is a bonus ( 4p 2 + 3 2p) + (2 3p 2 + 7p) 4p 2 + 3 2p + 2 3p 2 + 7p 4p 2 3p 2 2p + 7p + 3 + 2 Drop the brackets, leave the sign on the 1 st term of the second Polynomial 7p 2 + 5p + 5 Leave the solution in Descending Order 7

Rearrange the terms so like terms are together, descending order right away is a bonus (5xy + 3 2x) + ( 3 + 2x 5xy) 5xy + 3 2x 3 + 2x 5xy 5xy 5xy 2x + 2x + 3 3 0 Drop the brackets, leave the sign on the 1 st term of the second Polynomial Leave the solution in Descending Order, if everything cancels out, zero is a valid answer! (4 + 3t 2 7x) + (6x 2t 2 12) 4 + 3t 2 7x + 6x 2t 2 12 3t 2 2t 2 7x + 6x + 4 12 Subtraction of Polynomials t 2 x 8 There is 1 very important concept to understand with subtraction Consider this: (x 2 + 5x 4) (2x 2 5x 4) We are subtracting this from the 1 st one. The subtraction symbol MUST affect each term. Think about WATERBOMBING in the negative symbol The signs change (x 2 + 5x 4) (2x 2 5x 4) After you WATERBOMB in the negative you can change the signs and DROP the BRACKETS Remember: o negative negative = positive o negative positive = negative x 2 + 5x 4 2x 2 + 5x + 4 8

Now we GROUP the LIKE TERMS and we re done So, in Descending Order, x 2 2x 2 = x 2 5x + 5x = 10x 4 + 4 = 0 x 2 + 10x (3x 2 4x + 2) (6x 2 + 5x 12) Step 1: Drop the Brackets 3x 2 4x + 2 (6x 2 + 5x 12) of 1 st Polynomial Step 2: Waterbomb in the 3x 2 4x + 2 6x 2 5x + 12 ( ) to the second one and Drop the Brackets Step 3: Group the LIKE TERMS 3x 2 9x + 14 and put the result in DESCENDING ORDER (9r 2 + 4r + 5) ( 3r 2 4r + 5) 9r 2 + 4r + 5 + 3r 2 + 4r 5 9r 2 + 3r 2 + 4r + 4r + 5 5 Waterbomb in the negative sign Group the LIKE TERMS 12r 2 + 8r (2t 2 + 4t 6) (8t 2 5t + 2) Waterbomb in the negative sign 2t 2 4t + 6 8t 2 + 5t 2 2t 2 8t 2 4t + 5t + 6 2 10t 2 + t + 4 Waterbomb in the negative sign Group the LIKE TERMS 9

Section 6.2 Practice Questions Add the following Polynomials, leave answer in DESCENDING order. 1. (x + 4) + (x 7) 2. (2x 2 4x 7) + (3x 2 7 + 4x) 3. (3xy + 4x 3 + 4) + (2xy 4x 3 4) 4. (10 + 4t 2 + 4t) + (2t 7t 2 8) 5. (j 3 + 2j 2 + j + 4) + (3j 3 2j 2 7j + 15) 6. (4 + 6x 2x 2 ) + ( x 2 2x) 7. (t 2 + 4) + ( t 2 4) 10

8. ( x 2 + 2 3x) + ( 4x 2 + x 5) 9. ( 2x + x 2 2y 2 ) + ( y 2 x + 2x 2 ) 10. (4x 2x 2 ) + ( 5 + x 2 ) 11. ( 3 + 4x 2 + 4x) + (5x 2x 2 + 4) 12. (3x 2xy + 2y) + (xy 3y) + ( 3y x) 13. ( 2y + 3x + xy) + (2xy x y) + ( x 4xy) 11

Subtract the Polynomials, leave answer in DESCENDING order. 14. (3x 2 + 4x 7) ( 2x 2 + 4x + 9) 15. (t 3 5t + 4t 2 ) (t 2 7t 2t 2 ) 16. (z 4) (3z 7) 17. (w 7) (2w + 4) 18. (r + 6) ( 2r 2) 19. (j + 14) ( 5j + 7) 20. (2k 2 + k 7k) (3k 2 k 4k) (6k 2 8k + 7k) 21. (5 t) ( 7 + t) (12 + t 2 ) 12

22. ( 2x 3y) (4x + 2y) (x 3y) 23. ( 5x 2y + 3z) ( 2x + 9y) ( x + y 2z) Perform the Combined Operations 24. (2st s t) ( 3st + t) + ( s + 2t) 25. ( 3x + 4y) + (6x 5y) (2x + 11y 5z) 26. ( 2xy + 9z) + (4x 2 11z) (6x 2 + 8xy 11z) 13

Section 6.3 Multiply, Divide, Combined Operations, and Tiles Multiplication of Polynomials Multiplication is awesome, all we do is WATERBOMB (DISTRIBUTIVITY) and use our Exponent Laws for the Variables Remember: When we multiply a COMMON BASE we ADD the exponents! Also, the Order of Multiplication does not matter! 2 3 is the same as 3 2 xyz is the same as yxz, zyx, or zxy It makes no difference, keep this in mind 3(x + 4) 3(x + 4) Waterbomb in the 3 So, 3 times x is 3 3 times 4 is 12 3(x + 4) = 3x + 12 4x(x + 6) x(x + y) 4x x + 4x 6 x x + x y 4x 2 24x x 2 + xy 4k(3km 2m) t 2 (6p 4t) 4k 3km + 4k 2m t 2 6p + t 2 4t 12k 2 m 8km 6t 2 p 4t 3 4t 3 + 6t 2 p 14

14q(2q 2 4q) 14q 2q 2 + 14q 4q 28q 3 56q 2 Division of Polynomials Division of Polynomials is just fractions and exponent laws Consider this: 2 + 3 7 = 2 7 + 3 7 Remember this? We can break Polynomials down the same way. So, 4r + 2 2 = 4r 2 + 2 2 2r + 1 t 2 + 7t t = t2 t + 7t t t + 7 t 2 t = t 7t t = 7 Exponent Laws Anything divided by itself is 1 4z 3 2z 2 + 12z 2z 4z 3 2z + 2z2 2z + 12z 2z 2z 2 z + 6 15

8xyz + 4yz + 2z 2z 8xyz 2z + 4yz 2z + 2z 2z 4xy + 2y + 1 Combined Operations It is very rare that you only have to add, subtract, multiply, or divide only. More often than not it involves a combination of steps 3r(r + 4) 2r(4r + 6) 3r 2 + 12r 8r 2 12r Waterbomb to remove the BRACKETS 5r 2 Combine LIKE TERMS and SIMPLIFY So it ll take a few steps, multiply first, add/subtract, then combine the terms and leave your answer in Descending Order 4t(t 2 + 5) t 4t 3 + 20t t 4t 3 t + 20t t = 4t 2 + 20 16

3x(x + 4) + x 5x(3x 12) 3x 3x 2 + 12x x + 15x2 60x 3x Waterbomb to remove brackets 3x 2 x + 12x x + 15x2 3x + 60x 3x Divide each term by the denominator 3x + 12 + 5x 20 Group LIKE TERMS and SIMPLIFY 8x 8 2r 2 (r 4) r 6(r2 + 2r) 2 Waterbomb to remove brackets 2r 3 8r 2 r ( 6r2 + 12r ) 2 Since you re subtracting, put brackets around the second Polynomial so you don t forget to subtract each term 2r 3 r 8r2 r (6r2 2 + 12r 2 ) Divide each term by its denominator 2r 2 8r (3r 2 + 6r) Waterbomb in the NEGATIVE symbol 2r 2 8r 3r 2 6r Group LIKE TERMS 2r 2 3r 2 8r 6r Simplify the final solution r 2 14r 17

Algebra Tiles: The Visual Representation of Polynomials x 1 1 1 x x 2 tile x tile x 1 tile If the Tiles are Shaded In, they are the NEGATIVE representation That means that the Shaded and Non-Shaded CANCEL OUT and cancel out and cancel out and cancel out So here are a couple Examples: Add the following: + = x 2 + 2x 2 x 2 x + 3 x + 1 18

Subtract the following (this is tricky): x 2 x + 1 x 2 x + 1 2x 2 = Remember when we illustrated INTEGERS We need a NEGATIVE to take away and we don t have one so we bring in 0 Now we TAKE AWAY Multiply the following: x + 2 x x(x + 2) = x 2 2 19

Section 6.3 Practice Questions Multiply the following. Leave answer in DESCENDING order. 1. 3(x 7) 2. (t 2 7t + 4) 3. 4tp( 2t 2 + 3p) 4. 4k 2 (k 2 + 7k 2) 5. z(z + 4) 6. 2x(2y + x 3z) 7. xy(xyz + z xy) 8. 2st( 3s + 4t st) 9. 2x 2 (3x 2 2y 2 + 4z 2 ) 20

Divide the following. Leave answer in DESCENDING order. 10. 3x+12 3 11. t2 +4t t 12. 3x 2 9x+6 3 13. 5q3 +10q 2 5q 5q 14. 4t2 +2t 2t 15. a2 bc ab 2 c + abc 2 abc 16. 18z4 6z 3 +3z 2 3z 2 17. 4r12 +6r 3 8r 2 2r 2 18. a2 b 2 c+ab 2 c 2 a 2 b 2 c 2 ab 2 c 21

Perform the Combined Operations. Answer in DESCENDING order. 19. 3(x 2 + 4x) + 5x(x 6) 20. 2t(t2 4t) t 3t(4t 5) 21. 7q(3q2 +4q) 7 + 9q(6q2 3q) 3 22. 3z3 (z 3) 3 4z2 (3z+6z 2 ) 3 22

23. Add +1 1 +x 2 +x + x 2 +x 24. Multiply 23

Extra Work Space 24

Answer Key Section 6.1 1. Terms: 3 3x, 4x 2, 5 Degree: 1, 2, 0 5. Terms: 1 5 Degree: 0 2. Terms: 1 4xyz Degree: 3 6. Terms: 4 3x, 4y, 5z, x 2 Degree: 1, 1, 1, 2 3. Terms: 3 2xyz, 5xy, 4 Degree: 3, 2, 0 4. Terms: 3 5x 3 y, 4xy 3, 6xyz Degree: 4, 4, 3 7. 4x 2 5x + 3 8. 4t 3 3t 2 4t 9. 5x 2 x + 2 10. z 2 4z + 5 11. xy + xz + x y 12. x 2 5xy + 2x y 13. t 2 3t + 8 14. 7z 3 5z 2 + 5z 4 15. 5 16. 0 17. 1 6 i2 + 2i 5 18. 6.2x + y + 1 19. 8 x + 1 15 y + 10 20. 5 3 16 j2 3 j 2 Section 6.2 1. 2x 3 2. 5x 2 14 3. 5xy 4. 3t 2 + 6t + 2 5. 4j 3 6j + 19 6. 3x 2 + 4x + 4 7. 0 8. 5x 2 2x 3 9. 3x 2 3y 2 3x 10. x 2 + 4x 5 11. 2x 2 + 9x + 1 12. xy + 2x 4y 13. xy + x 3y 14. 5x 2 16 15. t 3 + 5t 2 + 2t 16. 2z + 3 17. w 11 18. 3r + 8 19. 6j + 7 20. 7k 2 21. t 2 2t 22. 7x 2y 23. 2x 12y + 5z 24. 5st 2s 25. x 12y + 5z 26. 2x 2 10xy + 9z Section 6.3 1. 3x + 21 2. t 2 + 7t 4 3. 8t 3 p + 12tp 2 4. 4k 4 + 28k 3 8k 2 5. z 2 4z 6. 2x 2 + 4xy 6xz 7. x 2 y 2 z x 2 y 2 + xyz 8. 2s 2 t 2 6s 2 t + 8st 2 9. 6x 4 + 4x 2 y 2 8x 2 z 2 10. x + 4 11. t + 4 12. x 2 3x + 2 13. q 2 + 2q 1 14. 2t + 1 15. a + b c 16. 6z 2 + 2z 1 17. 2r 14 3r 5 + 4r 4 18. ac a + c 19. 2x 2 42x 20. 10t 2 + 7t 21. 21q 3 5q 2 22. 9z 4 z 3 23. 4x + 1 24. 2x 25