6. If the product of three consecutive integers is 60, what are the integers?
Solution:
Let x be the first integer; x+1 be the second integer; and x+2 be the third integer.
𝑝
𝑞
(x) (x + 1) (x + 2) = 60
=
± 60,±30,±20,±15,±12,±10,±6,±5,±4,±3,±2,±1

±1
3 is the rational root
(x2 + x) (x + 2) = 60
x3 + 2x2 + x2 + 2x = 60
x3 + 2x2 + x2 + 2x – 60 = 0
1
1
3
2
-60
3
18
60
6
20
0
 3 is a rational root or x =3
3
Substituting 3, we get:
x = 3  1st integer
x + 1 = (3) + 1 = 4  2nd Integer
x + 2 = (3) + 2 = 5  3rd Integer
The three consecutive integers whose product is 60 are 3, 4 and 5.
7. The volume of a rectangular prism is 252 cm3. If its width is 5 cm less than its length and its height is 2 cm less than its
length, what are the dimensions of the prism?
Given: Vrectangular prism = 252 cm3
Let length (L) be x, width (W) = x – 5 cm, and height (H) = x – 2 cm.
Formula of the volume of a rectangular prism: L × W  H = Vrectangular prism
𝑝
𝑞
x (x – 5)(x – 2) = 252
x3 – 7x2 + 10x = 252
x3 – 7x2 + 10x -252 = 0
9
1
±1,± 2,± 3,± 4,± 6,± 7,± 9,± 12,± 14,± 18,± 21,± 28,± 36,± 42,± 63,± 84,± 126,±252
±1
 9 is the rational root or x = 9
x (x2 – 7x + 10x) = 252
1
=
-7
10
-252
9
18
252
2
28
0
Substituting 9, we get:
L = x = 9 cm
W = x – 5 cm = 9 cm - 5 cm = 4 cm
H = x – 2 cm = 9 cm – 2 cm = 7 cm
8. Find four whole numbers such that the product of the first, third and fourth is 160. Also, the second is 1 more than the
first, the third is 2 more than thrice the first and the fourth is 8 more than the first.
Let:
1st whole number = x
2nd whole number = x + 1
3rd whole number = 3x + 2
4th whole number = x + 8
= 1st × 3rd × 4th = 160

𝑝
𝑞
x (3x + 2)(x + 8) = 160
=
±1,± 2,± 4,± 5,± 8,± 10,± 16,± 20,± 32,± 40,± 80,±160
±1,±3
(3x2 + 2x) (x + 8) = 160
3x3 + 24x2 + 2x2 + 16x = 160
3x3 + 26x2 + 16x – 160 = 0
3
3
26
16
-160
6
64
160
32
80
0
2
Substituting 2, we get:
1st = x = 2
2nd = x + 1 = (2) + 1 = 3
3rd = 3x + 2 = 3(2) + 2 = 8
4th = x + 8 = (2) + 8 = 10
9. The length of a rectangular tank is 3 times its width. The height is 2 more than twice the width. If the volume is 216
cubic units, what are the dimensions of the tank?
Given: Vrectangular tank = 216 units3
Let width(W) be x, length(L) be 3x, and height 2x + 2.
Formula of the volume of a rectangular tank: L × W × H = Vrectangular tank
𝑝
𝑞
x (3x) (2x+2) = 216
=
±1,± 2,± 3,± 4,± 6,± 8,± 9,± 12,± 18,± 24,± 27,± 36,± 54,± 72,± 108,± 216
±1,±2,±3
3x2 (2x+2) = 216
6x3 + 6x2 = 216
6x3 + 6x2 – 216 = 0
Simplify: 6x3 + 6x2 – 216 = 0
6
x3 + x2 – 36 = 0
1
1
1
0
-36
3
4
36
4
12
0
3
Substitute 3, we get:
W=x=3
L = 3x = 3(3) = 9
H = 2x + 2 = 2(3) + 2 = 8
10. Find four consecutive even integers such that the product of the first, second, and fourth is 1920.
Let:
1st even integer: x
2nd even integer: x + 2
3rd even integer: x + 4
4th even integer: x + 6
(1st) (2nd) (4th) = 1920
x (x + 2)(x + 6) = 1920
𝑝
𝑞
=
(x2 + 2x)(x + 6) = 1920
x3 + 6x2 + 2x2 + 12x – 1920 = 0
x3 + 8x2 + 12x – 1920 = 0
1
1
8
12
-1920
10
180
1920
18
192
0
Substitute 10, we get:
1st = x = 10
2nd = x + 2 = (10) + 2 = 12
3rd = x + 4 = (10) + 4 = 14
4th = x + 6 = (10) + 6 = 16
10
±(1,2,3,4,5,6,8,10,12,15,16,…,320,384,480,640,960,1920)
±1