NOTE: In addition to the problems below, please study the handout Exercise Set 10.1 posted at http://www.austincc.edu/jbickham/handouts. 1. Simplify: 5 7 5. Simplify: ( ab 5 c )( a c 5 ). Simplify: 4x yz 7wxyz 4. Simplify: y 1 y 9 5. Simplify:. Simplify: 10 x 14w 1w 8 4 x 4 y z 0 y z 5 7. Simplify: (11 4 ) 8. Simplify: ( 4xy 4 ) 9. Simplify: ( a b 5 ) 4 10. Simplify: (5v 5 w )( v w 5 ) 11. Simplify: 4 0 1. Simplify: 5x 0 1. Simplify: (5x) 0 14. Identify each polynomial below as a monomial, a binomial, or a trinomial and indicate its degree. a. 10 + z 4 9z b. 19m 15 c. x + x 5 4 d. 100 5y e. 8 15. Evaluate u v 5uv + u v + 10 when u = and v = 5. 1. Add and simplify: (11x + 4x 5y y) + (x 4x + y) 17. Subtract and simplify: (7y x + y) (4y y + x)
18. Multiply and simplify: xy (5x 7xy + xy y) 19. Divide: 0. Divide: 1xy + 4 4xy x 5 y 9 0r s t 4rs t 1r st 1. Divide: (1x 7x 1) (x + ). Divide: (x 11x + ) (x ). Divide: (x + 4x + 8) (x + ) 4. Multiply and simplify: (w 5)(w 7) 5. Multiply and simplify: (8y + 1)(y 7). Multiply and simplify: (y + ) 7. Multiply and simplify: (w 4) 8. Multiply and simplify: (r 7s) 9. Multiply and simplify: (x 4)(x + 4) 0. Multiply and simplify: (w + v)(w v) 1. Multiply and simplify: (5x )(y + 9). Multiply and simplify: (4a )(a 5a + 1). Factor completely: a b 4. Factor completely: a + b 5. Factor completely: a ab + b. Factor completely: a + ab + b 7. Factor completely: 9ab a b 8. Factor completely: 10a + a b a 9. Factor completely: 1x y z 4 8x 5 y
40. Factor completely: 1x xy 8x + y 41. Factor completely: 5x + 5xy x y 4. Factor completely: x 9x + 0 4. Factor completely: x 15x 14 44. Factor completely: y + 4y 48 45. Factor completely: x + 7x + 4. Factor completely: y + 7y 0 47. Factor completely: x 7x 0x 48. Factor completely: x 1x + 5 49. Factor completely: 9a + 4a + 49 50. Factor completely: y 1y + 51. Factor completely: 5x 10x + 1 5. Factor completely: x 18x 5. Factor completely: 49a 81b 54. Factor completely: 5x + y NOTE: The equations in Problems 55 9 are quadratic equations, which means they can be written in the form ax + bx + c = 0 where a 0. Solve these equations using one of the following three methods: by using the square root property, by factoring, or by using the quadratic formula (when all else fails use this latter method - it should always work on any of these problems). The quadratic formula will be provided on your test. To solve using the square root property, refer to lesson 10.1. To solve by factoring or by using the quadratic formula, simplify each side of the equation if possible, then move all terms to one side of the equation so that one side of the equation equals zero, and then factor the other side of the equation or use the quadratic formula. Please refer to lesson 10.1 and the handout Exercise Set 10.1 posted at http://www.austincc.edu/jbickham/handouts for more information. If you have trouble, please ask for help. 55. Solve for b: b = 81 5. Solve for x: x = 98
57. Solve for a: 5a 180 = 0 58. Solve for w: (w ) = 5 59. Solve for y: y 8y 4 = 0 0. Solve for x: x + x = 15 1. Solve for a: a 4a + 10 = a. Solve for m: 10 m = m 9(m 1). Solve for n: n + 1 = 9 n 4. Twice a number is fifteen less than the square of that number. Find the number. 5. The product of two consecutive integers is 10. Find the two integers.. The sum of the squares of two consecutive odd integers is three less than eleven times the larger. Find the two integers. 7. A triangle has a base that is 10 cm more than its height. The area of the triangle is 1 square cm. Find the height and the base. 8. Find the dimensions of a rectangular picture whose length is inches shorter than twice its width and whose area is 5 square inches. 9. Find the lengths of the sides of a right triangle if the long leg is 7 cm longer than the short leg and the hypotenuse is 1 cm longer than the long leg. 70. The length of one leg of a right triangle is 14 feet. The length of the hypotenuse is 7 feet longer than the other leg. Find the lengths of the hypotenuse and the other leg. 71. A 114 meter rope is divided into two pieces so that one piece is three times as long as the other piece. Find the length of each piece.
ANSWERS: 1. 5 9. 18a b 5 c 8. 8wx y z 5 4. y 7 5. 4. y w z 5 7. 11 4 8. 4x y 1 9. 1a 8 b 0 10. 40v 11 w 18 11. 1 1. 5 1. 1 14. a. trinomial, 4 b. monomial, 15 c. trinomial, 5 d. binomial, 1 e. monomial, 0 15. 9 1. 1x 5y y 17. y 4x + 4y 18. 15x 4 y 1x y 5 + x y 4 xy 4 19. y + x 4 y
ANSWERS: 0. 5rs s t rt 1. - 4x 5 + x, or 4x 5 + x +. x + x. x x + 4 4. w 1w + 5 5. 1y 54y 7. 4y + 1y + 9 7. w 8w + 1 8. r 84rs + 49s 9. 9x 1 0. w v 1. 10xy + 45x y 7. 4a 0a 4 + a + 15a. a b = (a b)(a + b) [NOTE: Know this formula, and use it for factoring a difference of two squares.] 4. a + b is prime (not factorable) [NOTE: Know that a sum of two squares in which each term has a single variable raised to the exponent is not factorable unless it has a common factor. If it has a common factor, factor out the GCF. Example of common factor: In 18x + 1y, factor out the GCF, which is, to get (x + y ).] 5. a ab + b = (a b) [NOTE: Know this formula, and use it for factoring a perfect square trinomial with a negative middle term.]. a + ab + b = (a + b) [NOTE: Know this formula, and use it for factoring a perfect square trinomial with a positive middle term.] 7. ab (b a) 8. a(5a + ab 1)
ANSWERS: 9. 7x y (z 4 4x y) 40. (4x y)(x ) 41. (x + y)(5x 1) 4. (x 4)(x 5) 4. Prime (Not Factorable) [NOTE: (x 14)(x 1) = x 15x + 14, not x 15x 14] 44. (y + )(y 4) 45. (x + )(x + ) 4. (y + 5)(y 4) 47. x(x + 5)(x ) 48. (x 1)(x 5) 49. (a + 7) 50. (y ) 51. (5x 1) 5. x(x + )(x ) 5. (7a + 9b)(7a 9b) 54. Prime (Not Factorable) - Sum of Squares [NOTE: If you got a different answer, carefully multiply it back together to see that it is not equal to the original problem.] 55. b = 9, or b = 9 5. x = 7, or x = 7 57. a =, or a = 58. w = 8, or w = 59. y =, or y = 0. x =, or x = 5 1. a = 5, or a =
ANSWERS: + 9 9 9 9. m =. 107, or m = 0. 107 4. n =, or n = 4. The number is 5 or. (NOTE: You must give both parts of the answer.) To set up this problem, let x = the number. Then the equation is x = x 15. 5. The integers are 14 and 15, or the integers are 15 and 14. (NOTE: You must give both parts of the answer.) To set up this problem, let x = the first integer. Then x + 1 = the next consecutive integer. The equation is x(x + 1) = 10.. The integers are 5 and 7. (NOTE: 1.5 and 0.5 are not integers.) To set up this problem, let x = the first integer. Then x + = the next consecutive odd integer. The equation is x + (x + ) = 11(x + ). 7. The height of the triangle is cm and the base is 1 cm. To set up this problem, let x = the base. Then x 10 = the height. The formula for area of a triangle is A = 1 bh so the equation is 1 x(x 10) = 1. Hint: Multiply both sides of the equation by to clear the fraction, which makes the equation x(x 10) = 4. 8. The width of the rectangle is 5 inches and the length is 7 inches. To set up this problem, let x = the width. Then x = the length. The formula for area of a rectangle is A = lw so the equation is (x )x = 5, or x(x ) = 5. 9. The lengths of the sides of the triangle are 5 cm, 1 cm, and 1 cm. (NOTE: The hypotenuse of a right triangle is always the longest side.) To set up this problem, let x = the length of the short leg. Then x + 7 = the length of the long leg, and x + 7 + 1 = x + 8 = the length of the hypotenuse. Use the Pythagorean Theorem a + b = c to write the equation x + (x + 7) = (x + 8). Please also remember how to square a binomial such as (x + 7) = (x + 7)(x + 7). Multiply all terms together and combine like terms to get x + 14x + 49. 70. The length of the other leg is 10.5 ft and the length of the hypotenuse is 17.5 ft. To set up this problem, let x = the length of the other leg. Then x + 7 = the length of the hypotenuse. Use the Pythagorean Theorem to write the equation x + 14 = (x + 7). 71. The lengths of the pieces of rope are 8.5 m and 85.5 m. To set up this problem, let x = the length of the short piece. Then x = the length of the long piece. The sum of the two pieces is the total length of rope, which means that the equation is x + x = 114.