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loris [4]
3 years ago
6

Let a, b, c, d be four integers (not necessarily distinct) in the set {1, 2, 3, 4, 5}. The number of polynomials x4 + ax3 + bx2

+ cx + d which is divisible by x + 1 is(
a. between 55 and 65. (
b. between 66 and 85.(
c. between 86 and 105. (
d. more than 105.
Mathematics
1 answer:
fgiga [73]3 years ago
4 0
By the polynomial remainder theorem, x+1 will be a factor of f(x)=x^4+ax^3+bx^2+cx+d if the remainder upon division is 0, and this remainder is given by f(-1):

f(-1)=(-1)^4+a(-1)^3+b(-1)^2+c(-1)+d
0=1-a+b-c+d
a+c=1+b+d

Since a,c\in\{1,\ldots,5\}, it follows that a+c\in\{2,\ldots,10\}. But notice that if a+c=2, then we have

2=1+b+d\implies 1=b+d

and since b,d\in\{1,\ldots,5\}, the equation above requires that either b=0 or d=0, which is impossible. So a+c\in\{3,\ldots,10\}.

So we have 8 cases to check:

(1) Notice that if a+c=10, we have b+d=9. This is only possible for (b,d)\in\{(4,5),(5,4)\}.

(2) If a+c=9, then b+d=8, and so we can have (b,d)\in\{(3,5),(4,4),(5,3)\}.

(3) If a+c=8, then b+d=7, and so (b,d)\in\{(2,5),(3,4),(4,3),(5,2)\}.

(4) If a+c=7, then (b,d)\in\{(1,5),(2,4),(3,3),(4,2),(5,1)\}.

(5) If a+c=6, then (b,d)\in\{(1,4),(2,3),(3,2),(4,1)\}.

(6) If a+c=5, then (b,d)\in\{(1,3),(2,2),(3,1)\}.

(7) If a+c=4, then (b,d)\in\{(1,2),(2,1)\}.

(8) If a+c=3, then (b,d)\in\{(1,1)\}.

At the same time, we have 8 cases to consider to find how many options there are for (a,c).

(1) a+c=10. We have only one choice of (a,c)=(5,5).

(2) a+c=9. This is the same as when b+d=9, which we found to be 2 choices.

(3) Same as b+d=8; 3 choices.

(4) Same as b+d=7; 4 choices.

(5) 5.

(6) 4.

(7) 3.

(8) 2.

In total, there are

2\times1+3\times2+4\times3+5\times4+4\times5+3\times4+2\times3+1\times2
=2(2\times1+3\times2+4\times3+5\times4)
=2\displaystyle\sum_{n=1}^4n(n+1)
=80

ways to choose a,b,c,d such that x+1 is a factor of x^4+ax^3+bx^2+cx+d, so the answer is B.

Note the symmetry of the sum above. You can easily give a slightly briefer combinatorial argument for this answer, but I figured a more brute-force approach would be easier to follow.
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Answer:

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Step-by-step explanation:

5 0
3 years ago
Why isn’t the maximum ACT score (36) at the 100th percentile?
irga5000 [103]

Answer and Explanations

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3 years ago
An audiometer produces a pure tone with a frequency f of 1200 hertz (cycles per second). The maximum pressure P produced from th
Scilla [17]

Answer: Pressure(t) = 5mP*sin(191.08*t)

Step-by-step explanation:

The information we have:

Maximum pressure P = 5 mP

frequency = 1200Hz

Period = 1/frequency = (1/1200) seconds

when t = 0, P = 0, so we should use a sin function (because sin(0) = 0)

Then we can write this as:

Pressure(t) = P*sin(c*t)

where t is time in secnds, P is the maximum pressure 5mP, and c is a constant that depends on the period.

Pressure(t) = 5*sin(c*t)

knowing that the period is 1/1200 seconds, we have that:

c*1/1200 s = 2*3.14

c = 1200/(2*3.14) 1/s= 600/3.14 1/s

Our function is

Pressure(t) = 5mP*sin(191.08*t)

3 0
3 years ago
Compare partial products and regrouping. describe how the methods are alike and different
svetlana [45]
<span><em>Partial product </em>multiplication is the process of multiplying the numbers partially (respectively to ones, tens and hundreds) and adding them together in the end. For example, in order to find the product of 3 8 × 6 we should write that,
1) 3 8
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  ___
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2)  3 8
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 _____
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   1 8 0

3) 3 8
    × 6
______
+  4 8
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 ______
 2 2 8

<em>Regrouping </em>is the multiplication process when we add the partial products to the next tens and hundreds and so on without writing them down. For example, in order to find the product of 3 8 × 6 with the help of regrouping, we write that
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228
, where the number 4 above 8 shows the tens of 4 (40), which is going to be added to the tens of the product of 30 times 6. The two processes are the same in a way that you are getting the same result. In the end, it is a multiplication process. The processes differ because of the methods we apply. In partial product multiplication, we break down the number in its ones, tens, hundreds steps and then calculate. However, in regrouping process we consider those steps without breaking them down. </span>
3 0
3 years ago
Read 2 more answers
Wendell is making punch for a party the recipe he is using says to mix for cups pineapple juice and cups orange juice and 12 cup
Sergeu [11.5K]

Answer:

if you meant 4 by "for cups" then 4+4+12=20 cups of punch

6 0
3 years ago
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