1. Show that the volume of the solid shown can be represented by the polynomial 6x x.

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7.3 Dividing Polynomials by Monomials Focus on Afte this lesson, you will be able to divide a polynomial by a monomial Mateials algeba tiles When you ae buying a fish tank, the size of the tank

The volume of the ectangula tank can be epesented by the polynomial epession 7.5w 2-3w. How could you detemine a polynomial epession that epesents the length of the tank in tems of w? 0.

Show that the volume of the solid shown can be epesented by the polynomial 6 2 + 1. 2. Use algeba tiles to epesent the aea of the ectangula base. w 3 3.

1 7.3 Dividing Polynomials by Monomials Focus on Afte this lesson, you will be able to divide a polynomial by a monomial Mateials algeba tiles When you ae buying a fish tank, the size of the tank depends on the size and habits of the fish. A tank fo a jagua cichlid, o Paachomis managuensis, should have the minimum dimensions shown, in metes. The volume of the ectangula tank can be epesented by the polynomial epession 7.5w 2-3w. How could you detemine a polynomial epession that epesents the length of the tank in tems of w? 0.6 m Eploe Dividing a Polynomial by a Monomial A ectangula solid has a width of, a height of 3, and an unknown length. The aea of the base of the solid is epesented by the polynomial Show that the volume of the solid shown can be epesented by the polynomial Use algeba tiles to epesent the aea of the ectangula base. w 3 3. Count the numbe of -tiles and 1-tiles equied fo the missing dimension of the ectangle. What epession epesents the missing dimension? What happens if you multiply you epession by the width of the ectangula solid? Reflect and Check 4. Show that you epession fo the missing dimension in #3 is coect. 5. Descibe the steps you would take to find an epession fo the atio of the volume to the side measuing. 272 MHR Chapte 7

2 Link the Ideas Eample 1: Divide a Polynomial by a Monomial Using a Model Detemine the quotient Solution You can use algeba tiles.? Use 6 positive 2 -tiles and 8 negative -tiles to epesent the polynomial The vetical side of the ectangle epesents the monomial diviso,. Stategies Model It Count the numbe of positive -tiles and negative 1-tiles equied to complete the hoizontal side of the ectangle. Thee ae 3 positive -tiles and 4 negative 1-tiles, o 3-4. This epession epesents the esult of dividing the polynomial, 6 2-8, by the monomial, = 3-4 Check: Multiply the quotient, 3-4, by the diviso,. ()(3-4) = ()(3) - ()(4) = How do you know that the answe is coect? You can also detemine the quotient algebaically = 62-8 = = 3-4 Show You Know Detemine each quotient. a) b) Dividing Polynomials by Monomials MHR 273

3 The fomula fo the suface aea of a cylinde is 2π 2 + 2πh. Eample 2: Dividing a Polynomial by a Monomial Algebaically a) What is the atio of the suface aea to the adius of the cylinde? Wite the atio in simplified fom. b) If the height, h, of the cylinde is the same as the adius,, what is the atio of the suface aea to the adius? Wite the atio in simplified fom. Solution a) suface aea adius = 2π2 + 2πh The epession can be boken down into two pats. suface aea adius = 2π2 + 2πh = 2π π 1 h 1 = 2π + 2πh b) Substitute h = into the atio fom pat a). suface aea adius = 2π + 2πh = 2π + 2π() = 2π + 2π = 4π h Show You Know Detemine each quotient. a) b) -2t 2 + 4t 2t Key Ideas You can divide a polynomial by a monomial using a model The unknown side length of the ectangle is made up of - 3 tiles = - 3 When you divide a polynomial by a monomial algebaically, you can divide the numeical coefficients and apply the eponent laws to the vaiables = = = - 4? 274 MHR Chapte 7

4 Check You Undestanding Communicate the Ideas 1. Eplain how you would pefom the following division: Anita used the following method to simplify an epession: a) Show that Anita s solution is incoect. b) How would you coect he solution? 3. Use a model to show a polynomial division statement with a quotient of Pactise 9k 2-3k 3 = 9k2 3-3k 3 = 3k - 1 Fo help with #4 to #7, efe to Eample 1 on page What division statement is epesented by the algeba tiles? Detemine the quotient. a)? 5. Detemine the division statement epesented by the algeba tiles and give the quotient. a)? b)? b)? c)? c)? 7.3 Dividing Polynomials by Monomials MHR 275

5 6. Divide each epession, using a model. a) b) Use a model to divide each epession. a) b) Fo help with #8 and #9, efe to Eample 2 on page Divide. a) 2y y 2y b) 12m2-6.2m c) -18y2-6y -6y 3cv - 2.7c d) 3c 9. Detemine each quotient. a) 2.7c c 3c b) 2 + 8y c) -s2-1.5st 5s d) -14w2-7w A ectangula fish tank has the dimensions shown, in metes. The volume of the tank can be epesented by 7.5w 2-3w. 0.6 m w a) What polynomial epession epesents the aea of the base of the tank? b) What polynomial epession epesents the length of the tank? c) What is the length of the tank if the width is 0.6 m? What is the volume of the tank? 12. Fo thei Valentine s Day dance, the gade 9 students want to decoate the end wall of the gym with ed poste pape. The aea of the wall is given by the polynomial One sheet of poste pape coves an aea given by the monomial 5. What polynomial epession epesents the numbe of sheets of pape the students will need to cove the wall? 13. A ectangle has an aea of squae units. The width of the ectangle is 3 units. What is the length? Apply 10. A dump tuck holds 10 m 3 of soil. You ae filling a ectangula space in a yad with the dimensions of ( + 3) by 5 by 2, in metes. What polynomial epession epesents the numbe of tuck loads of soil you will need? MHR Chapte 7

6 14. The fomula used to pedict the distance an object falls is d = 4.9t 2 + vt. In the fomula, d is the distance, in metes, t is the time, in seconds, and v is the stating velocity of the object, in metes pe second. a) The aveage speed of a falling object is calculated as s = d, whee s is the aveage t speed, in metes pe second. Use this infomation to develop a fomula fo the aveage speed of a falling object in tems of t and v. b) What is the aveage speed of an object that falls fo 5 s, if it stats fom a esting position? Etend 15. Divide. a) b) 3.6gf g 0.3g 2 3 b ab b 1 3 b c) Two ectangles have common sides with a ight tiangle, as shown. The aeas and widths of the ectangles ae shown. What is a simplified epession fo the aea of the tiangle? A = A = What is the atio of the aea of the shaded ectangle to the aea of the lage ectangle? If a ectangle has length y and aea 1 2 y + 6y 2, what is its peimete? You ae designing a pak that includes a lage paking lot that will be coveed with gavel. m a) Design two diffeent-shaped paking lots using any single shape o combination of egula shapes. Include the dimensions on a dawing of each paking lot design. Note that you will need to be able to calculate the aea of you paking lots. Each aea should be a diffeent shape. Make them no less than 200 m 2 and no geate than 650 m 2. 1 m ( + 4) m b) A tuck with dimensions simila to those shown in the pictue will delive the gavel. Wite an epession fo the appoimate aea that a single load of gavel will cove to a depth of 5 cm. c) Thee ae thee sizes of tucks that can delive the gavel. The widths ae 1.5 m, 2 m, and 3 m. Appoimately how many tuckloads would it take fo each tuck size to delive the equied amount of gavel fo each of you paking lots? You will cove each paking lot to a depth of 5 cm. Show you wok. d) Which tuck size do you think would be the most efficient to use fo each of you paking lots? Eplain you easoning. 7.3 Dividing Polynomials by Monomials MHR 277

3.6 Applied Optimization

3.6 Applied Optimization .6 Applied Optimization Section.6 Notes Page In this section we will be looking at wod poblems whee it asks us to maimize o minimize something. Fo all the poblems in this section you will be taking the