Algebra
Chapter 8 Answers
Multiplying and Factoring Polynomials
Let's take a look at some ways algebra applies to space and space travel.
Part B, Multiplying Monomials
1. Open the StarDate Online Web site. Go to the Resources section and click on Solar System Guide. At this site you can find information about the various planets of our solar system.
a. Make a chart to keep track of the Mass, Diameter, and Average Distance in Miles from the Sun of the other eight planets. Express each entry in both standard and scientific notation.
Planet Avg.distance from sun
StandardAvg. distance from sun
ScientificMass (times Earth's mass) Diameter standard (miles) Diameter scientific (miles) Mercury 36,000,000
3.6 x 107
.055
3032
3.032 x 103
Venus 67,200,000 6.72 x 107
.815
7520
7.52 x 103
Earth 93,000,000 9.3 10 107
1.0
7926
7.926 x 103
Mars 142,000,000 1.42 x 108
.107
4222
4.222 x 103
Jupiter 484,000,000 4.84 x 108
318
88,846
8.8846 x 104
Saturn 888,000,000
8.88 x 108
95.2
74,898
7.4898 x 104
Uranus 1.800,000,000
1.8 x 109 14.5
31,763
3.1763 x 10 4
Neptune 2,800,000,000
2.8 x 109
17.2
30,775
1.43 x 103
Pluto 3,750,000,000
3.75 x 109
.0025
1430
1.43 x 103
Comment: Pluto's distance from Earth varies, depending on its orbit. The average is 3.75 x 10*9 or 3,750,000,000 miles.
c. Compute your weight on each of the planets using the surface gravity information.
PLANET Surface Gravity
(times weight on Earth)Mercury 0.38
Venus 0.91
Earth 1
Mars 0.377
Jupiter 2.53
Saturn 1.14
Uranus 0.9
Neptune 1.14
Pluto 0.08
Part C, Dividing Monomials
2. Use the Planets Web site to look up the Distances from
the Sun for Earth and Jupiter. How many times farther is Jupiter
than Earth from the sun? Use scientific notation.
[4.84 x 108 divided by 9.3
x 107 = 5.2]
3. Density is equal to the mass of an object divided by the
volume of the object. If the planets are each spherical in shape
and the formula for the volume of a sphere is (4/3)(3.14)(r3),
what is the density of Earth? Express your answer in scientific
notation.
[Density of Earth is 5.974 x 1024
kg divided by 1.08 x 1012 km, which equals 5.5 x 1012]
Part D, Making Connections
4. You are to plan the next space mission to Mars.
a. What are some things that you must consider?
[Possible answers include: the temperature on Mars, food sources, how long it will take a space shuttle to get you there, etc.]b. If you are to orbit Mars, how many Earth days will it take you?
[687 Earth days]c. What is the length in miles of the orbit of Mars if the circumference is equal to π times the diameter? Express your answer in scientific notation.
[Since the orbit of Mars is nearly circular, the circumference is equal to 2(3.14)(radius of the orbit). The radius of the orbit is the distance from the center of the sun to the edge of Mars or 432,000 miles plus 142 x 106 miles between the sun and Mars plus 4,222 miles for the diameter of Mars. The total radius of the orbit is 1.42436222 x 108 miles. Therefore, the length in miles of the orbit of Mars is approximately 8.945 x 108 miles.]
Blaise Pascal made many interesting discoveries after studying a pattern of numbers previously discovered by Omar Khayyam, a mathematician and astronomer who lived from 1048-1122. Let's see how Pascal's triangle can help us multiply and factor polynomials!
Part A, Multiplying and Factoring
1. Open the Pascal's Triangle Web site and read the page.
a. Draw a model of the binomials x + 3 and x2 + 1. You may find it helpful to remember your previous workwith algebra tiles.
b. What do you think models of 2x + 3 and 4x2 + 4x would look like?
c. What are the factors of each binomial in b above?
[(2x + 3) (1) and (4x) (x + 1) ]
Part B, Multiplying Binomials
2. Continue exploring the information at the Pascal's Triangle site.
a. Make a model of (x + y)2. You may use the chart at the Web site to give you a hint. Algebra tiles can be a big help too!
b. Push your Algebra tiles together and try to make a common geometric shape. What geometric figure is it? Why do you think that is reasonable?
[The figure is a square. This is reasonable since both factors are the same. In other words, the length and width of the rectangle are the same.]
Part E, Making Connections
3. Use the information from the Pascal's Triangle Web site to solve the following problems.
a. How does the binomial expansion of (x + y)2 relate to the triangle?
[The coefficients of the trinomial terms are in Row 2 of Pascal's Triangle.]b. How does the binomial expansion of (x + y)5 relate to the triangle?
[The coefficients of the trinomial terms are in Row 5 of Pascal's Triangle.]c. What would the binomial expansion of (x + y)6 look like? Explain the pattern.
[The coefficients of the trinomial terms are in Row 6 of Pascal's Triangle. The powers of x in each term go from 6 to 0 and the powers of y in each term go from 0 to 6. The binomial expansion would yield x6 + 6x5y + 15x4y2 + 20x3y3 + 15 x2y4+ 6xy5 + y6.]