Chapter 8 Review

Direct, Inverse, and Joint Variation

100

If a is directly proportional to b, and a = 3.6 when b = 10.8, find b when a = 5.

15

200

If y varies inversely as the square root of x, and if y = 2 when x = 16, find y when x = 4.

4

300

The number of containers filled on an assembly line varies directly as the speed of the conveyor and inversely as the square of the radius of the container. If 30 5-cm containers can be filled when the speed is 2m/min, how may 3 cm container can be filled when the speed is 3m/min?

125

400

The volume of a cone is jointly proportional to its height and the square of the radius of its base. A cone with height 18cm and base radius 4 cm has the same volume as another cone with base radius 6 cm. What is the height of the second cone?

8 cm

500

Ten workers can pave a stretch of road in 35 days. How many additional workers would be needed to finish the project 10 days early?

4 workers

Dividing Polynomials

100

(12x³ - 7x² - 14x + 5) / (3x² + 2x - 1) 

4x - 5

200

(x⁴ + 3x² - 2x - 7)/(x² - x - 2)

x² + x +6 + (6x+5)/(x² - x - 2)

300

(6x² + 13x - 5)/(3x - 1)

2x + 5

400

(3x³ - 8x² + x - 6) / (x - 3)

3x² + x + 4 + (6)/(x - 3)

500

(8x³ - x + 2)/(3 - x)

-8x² - 24x - 71 + (215)/(3-x)

Remainder & Factor Theorems

100

If P(x) = 2x³ + x² - 3x - 2, find P (-2)

-8

200

If P(x) = 2x³ + 4x² - x - 3, find P(-3)

-18

300

Is x+3 a factor of 2x³ + x² - 3x - 2? If not, find the remainder.

NO, -38

400

is x + 2 a factor of 2x³ + 4x² - x - 3? If not, find the remainder

NO, -1

500

Find a polynomial equation with integral coefficients that has 1, 1/2, and 1/4 as roots.

8x³ - 14x² + 7x - 1 = 0

3 Other Theorems

100

x³+5x²-29x-105 = 0 

If -3 is one of the roots, what are all the roots?

-3, 5, -7

200

x³-5x²+36x-180 = 0

If 6i is a root, what are the roots of the equation

5, 6i, -6i

300

Find a cubic equation with integral coefficients that has -4 and 3 + i as roots.

x³ - 2x² - 14x + 40 = 0

400

Find a cubic equation with integral coefficients that has -3 and 1 - 2i as roots.

x³ + x² - x +15 = 0

500

List all the possibilities for the # of positive, negative, and imaginary roots of the equation x⁶ - 2x⁵ + x³ +3x² - 4 = 0

3, 3, 0; 3, 1, 2; 1, 3, 2; or 1, 1, 4

Finding Rational Roots

100

List every possible rational root of 4x⁴+3x²-1 = 0

1, - 1, 1/2, -1/2, 1/4, -1/4

200

List every possible rational root of 2x³+5x²-6x-4 = 0

1, -1, 2, -2, 4, -4, 1/2, -1/2

300

List every possible rational root of 2x³-3x²+4x-6 = 0

1, -1, 2, -2, 3, -3, 6, -6, 1/2, -1/2, 3/2, -3/2

400

Solve by first finding the rational roots: 

x⁴-x³-2x-4 = 0

-1, 2, i*root 2, -i*root2

500

Solve by first finding the rational roots:

2x⁴+3x³-7x²+3x-9=0

-3, 3/2, i, -i