CHAPTER 1 POLYNOMIALS

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1 CHAPTER 1 POLYNOMIALS 1.1 Removing Nested Symbols of Grouping Simplify. 1. 4x + 3( x ) + 4( x + 1). ( ) 3x + 4 5 x 3 + x 3. 3 5( y 4) + 6 y ( y + 3) 4. 3 n ( n + 5) 4 ( n + 8) 5. ( x + 5) x + 3( x 6) 6. 6( y 3) y ( y 1) 1.1 Removing Nested Symbols of Grouping

7. 7 3 5 ( 4 x) 6x 8. ( n ) 5 9 4 3 5n 9. 8n 3 n ( n + 4) 5 10. ( x + 5) x ( 3x 3) 11. ( x 5y ) + 3 4x 5( x + y 1) 1. ( 6) ( n m) m ( n 1) 1.1 Removing Nested Symbols of Grouping

3 13. 16 1( a b) + 3 ( a + b) 14. ( 9x 6y ) x ( y 3) 15. 8n ( n) ( m + n) + ( n m) 16. ( x) ( 3y x ) ( 3 y ) 17. 3 4a ( 5b) 3a 18. ( ) 5a a 1 3b 1 b + 4a 1.1 Removing Nested Symbols of Grouping

4 19. ( ) x x 4 x x + 5x 0. y y ( y y ) 8 3 3 + 1. 5ab 3 ( ab + a) + 1 ( ba a). 7 9 xy ( x y ) + 8 x ( x + xy ) { } { } 3. 15x + 3 + x + ( 3x + 5) 4. 18 4 + 3 ( y + 1 ) ( y 3 ) 1.1 Removing Nested Symbols of Grouping

5 { x } x ( x x ) 1 1 1 1 5. ( ) { } 6. a 3 ( a 3 a + a) + ( a a ) 7. 1 1 1 1 1 x x 3 x 1 3 + + 4 4 8. 3 1 1 1 1 y + y 1 y y 4 + 3 6 3 { } 9. 0.5a a 0.01( 90a 100) 30. 0.1 x ( 0.1 x 5 ) + 0.1 x + 3 ( x + 1 ) 1.1 Removing Nested Symbols of Grouping

6 1. Applying Integer Exponent Rules Simplify. 3 5 1.. 3 3 3 6 4 10 3. 3 0 4 x x x x 4. y y y y 0 5 3 4 5. ( n ) 3 5 6. ( m ) 5 7. ( 3x ) 3 8. ( y ) 4 3 9. ( 5a ) 10. ( 3b ) 3 5 11. ( 4x y ) 3 6 1. ( 3n m ) 4 13. ( 4x)( x ) 3 3 14. ( y ) ( 3y ) 4 3 4 3 15. ( 3a )( a ) ( a ) 16. ( n ) ( n 3 ) ( n 4 ) a a a a 4 0 17. ( ) 3 3x x x x 4 6 0 18. ( ) 4 3 3 5 1 0 0 19. ( x x y y ) ( x xyy ) 5 3 7 0 1 0. ( 3n n ) ( n n n) 1. Applying Integer Exponent Rules

7 1. x x 16 8. y y 4 0 3. a a 9 0 4. c c 5 40 5. 3 6. 4 7. a b 8. 3 a b 9. x 3y 30. 3 3a 4b 31. 3c a b 3 3. 4 4xy 3 5z 3 33. m nw 3 w x 4 0 4 34. a bc 0 5 3a b c 3 0 3 35. x y 1 x y 36. 3a b c 3 ab c 1 0 1 3 1. Applying Integer Exponent Rules

8 37. x y z 3x y z 3 5 4 1 38. 4ab c d 1 a b c 3 4 0 3 39. 0 3 ( 5x ) ( 5x ) 1 0 ( 5x ) ( 5 x ) 3 1 40. 1 0 1 ( a b ) ( 3 a b) 3 3a( ab ) 3 7 4 3 3 0 3 41. 3 a a a ( 3a) ( a ) a ( 4a) a 4. ( 3 ) ( 3 ) ( 1 ) 1 x y xy x y 1. Applying Integer Exponent Rules

9 0 3 4 6 1 1 43. 5 ( ) + ( 10 ) ( ) ( ) 0 1 0 x x y xy x y xy xy 44. 1 7 3 6a a + ( a ) 1 1 ( ) ( ) 3 a a a 45. ( ) 1 4 1 3 x y x y + 6x y 5 ( ) + ( ) 1 0 x y xy xy xy 46. 3 3 0 3( ab ) a b 9a b ( ) + ( ) ( ) 3 1 0 5 1 8 6a b 3 a b a b 1. Applying Integer Exponent Rules

10 1.3 Integer Exponents and Scientific Notation Use scientific notation to calculate the following. Write your answer in both scientific notation and standard notation. 1.,000 0.000001. 13,000 0.00000 3. 5,000,000 1,000,000 4. 1,000,000 00,000,000 5. 0.000008 0.0000001 6. 0.000005 0.00004 7. ( 3,000 )( 4,000,000 )( 0.000001 ) 8. ( 0.0000)( 10,000,000 )( 4,000 ) 9. ( 0,000,000 )( 4,000,000 )( 0.0000) 10. ( 0.000016)( 0.000)( 3,000 ) 11. 15,000,000 3,000 1. 40,000,000,000 6,000,000 1.3 Integer Exponents and Scientific Notation

11 13. 0.000009 0.0003 14. 0.000045 0.0005 15. 16,000 4,000,000 16. 81,000 90,000,000 17. ( 1,000) ( 0.0003) ( 1,000,000 ) 18. ( 0.0004)( 80,000,000 ) ( 40,000 ) 19. ( 50,000,000 )(,000,000 ) ( 5,000) 0. ( 0.006) ( 0.003) ( 0.0000) 1.3 Integer Exponents and Scientific Notation

1 1. ( 0.0001) ( 0.00) 3 ( 0.00) 3 4. (,000) (,000,000 ) ( 0.00) ( 00,000) 3 3 4 3. ( 0.00005) 4 ( 3,000) ( 0.0003)( 0.03) (,000,000 ) 4. 3 ( 0.000) ( 0.00000) 3 1.3 Integer Exponents and Scientific Notation

13 1.4 Adding or Subtracting Polynomials Perform the indicated addition and/or subtraction. 1. ( 9x + 6x + 13) + ( 1x 8x 19). ( 3x + 9x 1) + ( 5x 11x + 4) 3. ( x + 4x 6) ( x 5x + 7) 4. ( 6x 14x 5) ( x + x 30) 5. ( x 3 y 4x y 4 8xy + 1) + ( x 3 y + 3x y 4 xy 1) 6. ( 3x 4 y 3 x 3 y + x y 9) + ( 4x 4 y 3 + 3x 3 y x y + 4xy ) 1.4 Adding or Subtracting Polynomials

14 7. ( 7a 9ab + 5b + 16) ( 3a 6ab 8b + 3) 8. ( 5a 3 + 4a b 7ab + b 3 17) ( a 3 6a b + 9ab + b 3 3a b ) 9. ( m + 5n) + ( m + 4n 6) ( 3n m + 7mn 1) 10. ( 15m mn + n ) ( 7m + 5nm n 9) ( 3n 8mn m + 6) 1.4 Adding or Subtracting Polynomials

15 11. Subtract 1p 11q + 6w 15 from the sum of 9p + 4q 8w + 18 and 8p 9q 1w 7. 3 3 1. Subtract 6a ab + 9a b b + a from the sum of 3 3 10a + 3a b 4b 3ab 5a. 3 3 5a 4ab 3a b b a and 13. Add x + 5y 4z to the difference of 6x + 11y 16z and 10x 15y z. 14. Add 1m 4mn + 6n to the difference of m + mn 3n and 6m 8mn n. 1.4 Adding or Subtracting Polynomials

16 Write the polynomial expression for the unknown quantity. 15. Find the perimeter of a square whose side is given by x + x + 1. 16. Find the perimeter of a rectangle whose length and width are given by a 7 and respectively. a + a 3, 17. Find the perimeter of a triangle whose sides are given by m + n 3, m 8n + 7, and m 6n 11. 1.4 Adding or Subtracting Polynomials

17 18. Find the length of a rectangle whose perimeter is given by given by 3 y y y 9 + + +. 3 6y 4y 10 + + and whose width is 19. Find the width of a rectangle if its length is given by 4a b + 8ab + 8. 4a b ab 6 + and its perimeter is 0. Find the third side of a triangle if the other two sides are given by 3 3 x x + 4 and its perimeter is given by 5x + x x + 7. 3 x x x + + + 1and 1.4 Adding or Subtracting Polynomials

18 1.5 Multiplying Polynomials Use special products for binomials to multiply the following. ax 3by 5ax 4by 6nx 5y nx + 7y 1. ( )( ). ( )( ) 3. ( 3a + 4b) 4. ( 6x y ) 5. 1 1 a b 3 6. x y + 4 5 7. ( 4x y 3z) 8. ( 5ab + c ) 1.5 Multiplying Polynomials

19 9. ( 0.1x + 0.3y ) 10. ( 1.1a 0.9b) 11. ( 9a 4b)( 9a + 4b) 1. ( 8x + 7y )( 8x 7y ) 13. ( 5a 8b 3 )( 5a + 8b 3 ) 14. ( 6n + 11n 4 )( 6n 11n 4 ) 15. x y x y + 3 3 16. 3 3 a b a + b 5 4 5 4 1.5 Multiplying Polynomials

0 17. ( 0.6n 0.7m)( 0.6n + 0.7m) 18. ( 0.01x + 0.1y )( 0.01x 0.1y ) 19. 1 1 x y x + y 0. 4 3 4 3 a b + 3c a b 3c 5 5 1. x + ( y + 1) 3x + ( y ). ( ) ( ) 3x + 4y x 3 5y 3. ( a + 3) + ( b c ) ( a 1) ( b + 3c ) 4. ( x 1) + 3( y ) 3( x + ) + ( y + 1) 1.5 Multiplying Polynomials

1 5. 3a + ( b c) 6. ( ) 4x + y 5 7. ( x + y ) + ( z + 4) 8. ( m ) + ( n 3z ) 9. 3( a ) 4( b + 1) 30. 5 ( x + 3 ) ( y 4 ) 31. 3b ( 4a + 7) 3b + ( 4a + 7) 3. ( 5 ) ( 5 ) m n m + n 1.5 Multiplying Polynomials

33. ( x + 3y ) ( 4n + z) ( x + 3y ) + ( 4n + z) 34. ( a 3b) + ( x + y ) ( a 3b) ( x + y ) 35. 4( x 1) 5( y + ) 4( x 1) + 5( y + ) 36. 5( n m) ( y z) 5( n m) + ( y z) 37. ( 3n 4m + 6)( 3n 4m 8) 38. ( x 5y + )( 3x + 5y ) 39. ( 4a b + c + d ) 40. ( 3x y n 4)( 3x y + n + 4) 1.5 Multiplying Polynomials

3 1.6 Expanding Binomials Draw Pascal s Triangle. Expand the following using Pascal s Triangle. 1. ( x + 1) 3. ( y 1) 3 3. ( a + ) 4 4. ( b ) 4 5. ( x ) 5 6. ( y + ) 5 1.6 Expanding Binomials

4 7. ( n 3) 6 8. ( m + 3) 6 9. ( + x) 7 10. ( a) 7 x + y 11. ( ) 3 a b 1. ( ) 3 3a b 13. ( ) 4 x + 3y 14. ( ) 4 1.6 Expanding Binomials

5 a + 3b 15. ( ) 6 3a b 16. ( ) 6 17. 1 1 x y 3 4 18. 4 1 1 a + b 5 4 1.6 Expanding Binomials

6 0.1a + 0.b 19. ( ) 4 0.x 0.1y 0. ( ) 4 Write the Binomial Formula. Expand # 1 0 using the Binomial Formula. 1. ( x + 1) 3. ( y 1) 3 3. ( a + ) 4 1.6 Expanding Binomials

7 4. ( b ) 4 5. ( x ) 5 6. ( y + ) 5 7. ( n 3) 6 8. ( m + 3) 6 9. ( + x) 7 30. ( a) 7 x + y 31. ( ) 3 a b 3. ( ) 3 1.6 Expanding Binomials

8 3a b 33. ( ) 4 x + 3y 34. ( ) 4 a + 3b 35. ( ) 6 3a b 36. ( ) 6 1.6 Expanding Binomials

9 37. 1 1 x y 3 4 38. 1 1 a + b 5 4 4 0.1a + 0.b 39. ( ) 4 0.x 0.1y 40. ( ) 4 1.6 Expanding Binomials

30 1.7 Dividing Polynomials Divide. 1. ( x 3 x + ) ( x 1). ( y 3 + y + 4) ( y + 1) 3. ( y 4 + y + 4) ( y + ) 4. ( x 4 3x 1) ( x ) 1.7 Dividing Polynomials

31 5. ( x 4 x 1) ( x + x + 1) 6. ( y 4 + y 3 ) ( y y 1) 7. ( n 5 + n 3 n 1) ( n + n + ) 8. ( m 5 m + m 3) ( m 4) 1.7 Dividing Polynomials

3 3 4 9. ( x + 9) ( x + 1) 10. ( y 4) ( y 3) 11. ( x x + x 3 + ) ( 3 + x) 1. ( 6 n + n 4 + n) ( n 1+ n ) 1.7 Dividing Polynomials

33 13. ( y + y 3 3 + 3y 4 ) ( y + y 3 ) 14. ( m + 3m 4 + 6m 3 ) ( m + 1 m) 6 5 15. ( 1+ x ) ( x + 1) 16. ( y 3) ( y ) 1.7 Dividing Polynomials

34 3 4 17. ( a + 8) ( + a) 18. ( y 16) ( y ) 19. ( n 4 + n 4 ) ( + n) 0. ( x 3 + x 4 ) ( 3 + x) 1.7 Dividing Polynomials

35 1. ( x + x x 3 + x 4 1) ( x 1). ( n + n 3 + 4n + 3n 4 4) ( + n ) 3. ( x + 7xy + 4y ) ( x + 4y ) 4. ( 6y + xy 15x ) ( 3x + y ) 1.7 Dividing Polynomials

36 5. ( x 3 + x y 3xy + y 3 ) ( x y ) 6. ( 3a 3 a b + ab + b 3 ) ( 3a + b) 1.7 Dividing Polynomials

37 7. ( a 4 + b 4 ) ( a + b) 8. ( y 4 x 4 ) ( y x) 1.7 Dividing Polynomials

38 9. ( x 5 3y 5 ) ( x y ) 30. ( 64a 6 b 6 ) ( a b) 1.7 Dividing Polynomials

39 1.8 Chapter Review Simplify. { } 1. 8 4 x + 3( x 1) ( x + 5) { }. 6 ( x y 3) + 5 x 5y 3( x y ) ( x + y ) 1.8 Chapter Review

40 Simplify. Your answer should contain no zero or negative exponents. 3. x y z 3xy z 3 0 3 4 1 3 5a b c 4. 3a b c 1 0 3 5. 1 1 3 15a b ( a b c ) 1 1 0 0 ( 3 abc ) ( 4 a bc ) 1 ( 3 1 1 ) ( 4 0 x y z 3x y ) 6. 0 4 0 1 ( 3x yz) ( x yz ) 3 1 1.8 Chapter Review

41 Use special products for binomials to multiply. a + b a + + b 7. ( 3) ( 3) 8. x ( y 5) x + ( y 5) 9. ( n ) 3 + + 4m 1.8 Chapter Review

4 10. ( ) 5n 6 m 11. ( ) ( ) x + 1 + y w 1. ( ) ( ) 3x 5 y + p 1.8 Chapter Review

43 13. 3( c + d ) + ( e f ) 4( c d ) + 3( e 3f ) 14. 4( x x) + 3( x + 1) ( x + x) 5( x 1) 15. ( a + b + 3c d )( 4a b 5c + d ) 1.8 Chapter Review

44 16. ( x 4y z w )( x 4y + z + w ) x + 5y 17. ( ) 3 4a 3b 18. ( ) 3 1.8 Chapter Review

45 Use Pascal s Triangle to expand the following: a + b 19. ( ) 7 x y 0. ( ) 6 m + 3n 1. ( ) 5 3x 4y. ( ) 4 1.8 Chapter Review

46 Use the Binomial Theorem to expand the following: a + b 3. ( ) 4 x y 4. ( ) 5 3a + b 5. ( ) 6 x y 6. ( ) 7 1.8 Chapter Review

47 Divide using long division. 7. ( x 5 x 3 + x 4) ( x + x ) 8. ( 5y + y 5 + 5) ( y + 5 + y ) 1.8 Chapter Review

48 9. ( a 5 + b 5 ) ( a + b) 30. ( x 6 y 6 ) ( x y ) 1.8 Chapter Review

49 Use scientific notation to evaluate. Write your answer in scientific notation and standard notation. 0,000 5,000,000 0.00 000 000 4 16,000,000,000 0.00 000 3 31. ( )( )( ) ( )( ) 3. (,000,000 )( 0.00 000004) Simplify. 33. 1 1 1 ( ) 1 x + x + ( x + 3) 3 4 3 34. 0.01( y + 0.3) 0.1( 0.3y 0.04) 0.( y ) 1.8 Chapter Review

50 Simplify. Your final answer should contain no zero or negative exponents. 1 1 3 3 4 5y 3 3 35. a b ( a b ) 0 0 1 4 3 3y 3 1 36. x y ( x y ) 37. 0. 0. xy x y 1 1 38. 0.1a b 0.3 a b 3 3 1 1.8 Chapter Review

51 Use special products to multiply. 1 1 1 1 39. ( x y ) + z ( x y ) z 3 3 3 1 3 1 4 4 40. a ( b c) a + ( b c ) 41. 0.4( a + 1) 0.0b 0.4( a + 1) + 0.0b 4. 0.1 y 0.01 ( x + ) 0.1 y + 0.01 ( x + ) 1.8 Chapter Review

5 43. 3 1 1 x y + 4 5 44. 1 1 1 a d 3 + 4 45. 0.1( a 1) + 0.b 46. ( ) 0. x + 1 0.1y 1.8 Chapter Review

53 47. 1 1 x + y 3 3 48. 1 1 a b 4 3 49. ( 0.a 0.3b) 3 50. ( 0.1x + 0.y ) 3 1.8 Chapter Review

54 Expand using either Pascal s Triangle or the Binomial Theorem. 51. 1 1 x + y 3 5 5. 1 1 a b 3 4 0.1x 0.y 53. ( ) 6 0.3a + 0.1b 54. ( ) 5 1.8 Chapter Review

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