Math 3
Name:
Extra Practice with Polynomials
Date:
1. Suppose you roll two six sided dice, but they don’t have the usual numbers. The
first die has the following numbers {3, 4, 4, 4, 5, 5} and the second die has the
numbers {2, 2, 2, 6, 6, 6}. Use the completed table below to answer the questions.
3
4
4
4
5
5
2
5
6
6
6
7
7
2
5
6
6
6
7
7
2
5
6
6
6
7
7
6
9
10
10
10
11
11
6
9
10
10
10
11
11
6
9
10
10
10
11
11
a. What is the probability that the sum is a 6?
b. What is the probability that the sum is a 5?
c. What is the probability that the sum is 4?
d. What is the probability that the sum is between 8 and 12?
e. What polynomial multiplication could you have use to answer the question?
Carry out the multiplication. Use the resulting polynomial to answer the same
questions. Make sure you get the same answers!
2. Consider the multiplication (4 x 5 + 2x 7 )(2x 2 + 2x 4 + 2x 6 ) .
a. How is this multiplication related to rolling two six sided dice? What are the
numbers on each of the dice?
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b. Find the product of (4 x 5 + 2x 7 )(2x 2 + 2x 4 + 2x 6 ) .
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c. In part a, you should have found that the two dice had the following numbers:
{5, 5, 5, 5, 7, 7} and {2, 2, 4, 4, 6, 6}. How many ways are there for the sum to
be 7?
d. How many ways are there for the sum to be 9?
e. What is the probability that the sum is 9?
3. Suppose you roll two eight sided dice with the following numbers:
{2, 2, 2, 4, 4, 4, 5, 5} and {3, 3, 3, 3, 3, 8, 8, 8}.
a. What polynomial multiplication problem is associated with these dice?
b. Do the multiplication problem you wrote down in the last problem. (In other
words, find the product.)
c. How many ways are there to get a sum of 7?
d. What is the probability that the sum is 7? Hint: When we were rolling two six
sided dice, the probabilities had a 36 in the denominator. What should the
denominator be when we are rolling two eight sided dice?
e. What is the probability that the sum is 13?
f. Find the sum of the coefficients for the product you found in part b. Why does
this make sense?
4. Suppose you have a number spinner with just two equal area wedges with the
numbers 1 and 0.
a. Suppose you spin it twice. What is the probability that the sum will be a 2?
Hint: Start by doing the associated polynomial multiplication. Do this
multiplication by hand.
b. You should have found that the associated polynomial multiplication was
(x1 + x 0 )2 . Notice that this is the same as (x +1)2 . Now suppose we spin the
spinner three times. What is the probability that the sum will be a 1? Note: Try
explaining the polynomial with the Binomial Theorem.
c. Now suppose we spin it five times. What is the probability that the sum will be
a 3? Note: Try explaining the polynomial with the Binomial Theorem.
d. Now suppose we spin it 15 times. What is the probability that the sum will be
an 8? Hint: Do you need to find every term of (x +1)15 ?
5. You are going to flip a coin 5 times. You want to know the probability of
getting 4 heads.
a. Let’s start by thinking about two flips of a coin. Complete the table below that
we’ve seen before.
H
T
H
T
b. What polynomial multiplication problem would look really similar to the table
in part a?
c. Now what polynomial multiplication problem would model 5 flips? Expand
your polynomial.
d. What’s the probability of flipping a coin 5 times and getting 4 heads?
e. What’s the probability of flipping a coin 5 times and getting at least 4 heads?
6. Assume you are taking a multiple-­‐choice test with 5 questions. Each question has one correct answer and 3 wrong answers. Assume you guess randomly on each question. You want to know the probability of getting at least 4 questions correct. a. Again, let’s start by thinking about just two questions. Fill in the table below. R
W
W
W
R
W
W
W
b. What’s the probability of getting exactly one question right? c. What polynomial multiplication problem looks really similar to the table above? d. What polynomial multiplication problem should we do to model guessing on 5 questions? e. What is the probability of getting exactly 4 questions right? (Hint: Don’t expand the whole polynomial) Answers
1. a.
9
36
= 14
b.
3
36
= 121
c. 0 d.
18
36
= 12 e. (3x 2 + 3x 6 )(x 3 + 3x 4 + 2x 5 )
2. b. 8x 7 +12x 9 +12x11 + 4x13 c. 8 d. 12 e.
12
36
= 13
3. a. (3x 2 + 3x 4 + 2x 5 )(5x 3 + 3x 8 ) € b. 15x 5 +15x 7 +10x 8 + 9x10 + 9x12 + 6x13 c. 15
d.
e. 646 = 323 f. 15 +15 +10 + 9 + 9 + 6 = 64 (There are 64 total possibilities,
counting repeats.)
15
64
4. a.
1
4
b.
3
8
c.
10
32
d.
6435
32768
5. b. (ℎ + 𝑡)! c. (ℎ + 𝑡)! = ℎ! + 5ℎ! 𝑡 + 10ℎ! 𝑡 ! + 10ℎ! 𝑡 ! + 5ℎ𝑡 ! + 𝑡 !
d.
!
!
e. !"
!"
6. b.
!
!
c. (𝑟 + 3𝑤)!
d. (𝑟 + 3𝑤)!
e.
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