See Problem 6.

39. Metalwork A metalworker wants to make an open box from a sheet of metal, by cutting equal squares from each corner as shown.

    

    1. Write expressions for the length, width, and height of the open box.
    2. Use your expressions from part (a) to write a function for the volume of the box. (Hint: Write the function in factored form.)
    3. Graph the function. Then find the maximum volume of the box and the side length of the cut-out squares that generates this volume.

B Apply

Write each function in factored form. Check by multiplication.

40. y = 3 x 3 − 27 x 2 + 24 x y equals , 3 x cubed , minus , 27 x squared , plus 24 x
41. y = − 2 x 3 − 2 x 2 + 40 x y equals negative 2 , x cubed , minus , 2 x squared , plus 40 x
42. y = x 4 + 3 x 3 − 4 x 2 y equals , x to the fourth , plus 3 , x cubed , minus , 4 x squared

43. Think About a Plan A storage company needs to design a new storage box that has twice the volume of its largest box. Its largest box is 5 ft long, 4 ft wide, and 3 ft high. The new box must be formed by increasing each dimension by the same amount. Find the increase in each dimension.

    • How can you write the dimensions of the new storage box as polynomial expressions?
    • How can you use the volume of the current largest box to find the volume of the new box?

44. Carpentry A carpenter hollowed out the interior of a block of wood as shown below.

    

    1. Express the volume of the original block and the volume of the wood removed as polynomials in factored form.
    2. What polynomial represents the volume of the wood remaining?
45. Geometry A rectangular box is 2 x + 3 2 x plus 3  units long, 2 x − 3 2 x minus 3  units wide, and 3 x 3 x  units high. What is its volume, expressed as a polynomial?

46. Measurement The volume in cubic feet of a CD holder can be expressed as V ( x ) = − x 3 − x 2 + 6 x , v open x close equals negative , x cubed , minus , x squared , plus 6 x comma  or, when factored, as the product of its three dimensions. The depth is expressed as 2 − x . 2 minus x .  Assume that the height is greater than the width.

    1. Factor the polynomial to find linear expressions for the height and the width.
    2. Graph the function. Find the x-intercepts. What do they represent?
    3. What is a realistic domain for the function?
    4. What is the maximum volume of the CD holder?

Find the relative maximum, relative minimum, and zeros of each function.

47. y = 2 x 3 − 23 x 2 + 78 x − 72 y equals , 2 x cubed , minus , 23 x squared , plus 78 x minus 72
48. y = 8 x 3 − 10 x 2 − x − 3 y equals , 8 x cubed , minus , 10 x squared , minus x minus 3
49. y = ( x + 1 ) 4 − 1 y equals . open x plus 1 close to the fourth . minus 1
50. Open-Ended Write a polynomial function with the following features: it has three distinct zeros; one of the zeros is 1; another zero has a multiplicity of 2.
51. Writing Explain how the graph of a polynomial function can help you factor the polynomial.

For each function, determine the zeros. State the multiplicity of any multiple zeros.

52. f ( x ) = x 3 − 36 x f open x close equals , x cubed , minus 36 x
53. y = ( x + 1 ) ( x − 4 ) ( 3 − 2 x ) y equals open x plus 1 close open x minus 4 close open 3 minus 2 x close
54. y = ( x + 7 ) ( 5 x + 2 ) ( x − 6 ) 2 y equals open x plus 7 close open 5 x plus 2 close open x minus 6 , close squared

Table of Contents