Answer:
1200
0.65
The answers are in the function
the expression 2 x ( x − 7 ) 2 is equivalent to x 2 b x 49 for all values of x . what is the value of b ?
To determine the value of b in the expression x^2b(x - 7)^2, we can compare it with the given equivalent expression x^2b49. By equating the two expressions, we can solve for b.
In the given expression x^2b(x - 7)^2, we can simplify it by multiplying the exponents:
x^2 * b * (x - 7)^2 = x^2b(x^2 - 14x + 49)
Comparing this with the equivalent expression x^2b49, we can equate the coefficients of the like terms:
x^2b(x^2 - 14x + 49) = x^2b49
From this equation, we can see that the coefficient of the x term is -14b. For it to be equivalent to 49, we have:
-14b = 49
Solving for b, we divide both sides by -14:
b = -49/14 = -7/2
Therefore, the value of b is -7/2.
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"
A Bernoulli differential equation is one of the form dy + P(x)y dx Q(x)y"" (*) Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = yl-n
For values of n other than 0 or 1 in a Bernoulli differential equation, the substitution [tex]u = y^{(1-n)[/tex] is used to transform it into a linear equation.
A Bernoulli differential equation is given by the form:
dy + P(x)y dx = Q(x)[tex]y^n[/tex] (*)
If we consider the case when n = 0 or n = 1, the Bernoulli equation becomes linear. Let's examine each case:
When n = 0:
Substituting[tex]u = y^{(-n) }= y^{(-0)} = 1[/tex], the differential equation becomes:
[tex]dy + P(x)y dx = Q(x)y^0[/tex]
dy + P(x)y dx = Q(x)
This is a linear differential equation of the first order.
When n = 1:
Substituting [tex]u = y^{(-n) }= y^{(-1)},[/tex] we have:
[tex]u = y^{(-1)[/tex]
Taking the derivative of both sides with respect to x:
[tex]du/dx = -y^{(-2)} \times dy/dx[/tex]
Rearranging the equation:
[tex]dy/dx = -y^2\times du/dx[/tex]
Now substituting the expression for dy/dx in the original Bernoulli equation:
[tex]dy + P(x)y dx = Q(x)y^1\\-y^2 \times du/dx + P(x)y dx = Q(x)y\\-y \times du + P(x)y^3 dx = Q(x)y[/tex]
This equation is also a linear differential equation of the first order, but with the variable u instead of y.
In summary, when n is equal to 0 or 1, the Bernoulli equation becomes linear. For other values of n, a substitution u = y^(-n) is typically used to transform the Bernoulli equation into a linear differential equation, allowing for easier analysis and solution.
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A coffee shop recently sold 8 drinks, including 2 Americanos. Considering this data, how many of the next 20 drinks sold would you expect to be Americanos?
Answer:
5 drinks will be americanos
Step-by-step explanation:
2:8 (2/8)
simplify
1:4 (1/4)
divide 20 by 4
5:20
The number of next 20 drinks which may be Americanos is 5.
What is Probability?Probability is simply the possibility of getting an event. Or in other words, we are predicting the chance of getting an event.
The value of probability will be always in the range from 0 to 1.
Given that,
Total drinks sold = 8
Number of drinks that is Americanos = 2
Probability of finding Americano = 2/8 = 1/4
If the total number of drinks next is 20,
Number of Americanos expected = Probability of Americanos × Number of drinks
= 1/4 × 20
= 5
Hence the number of Americanos expected is 5.
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(2/3)^2 without exponents
Answer:
[tex]\frac{4}{9}[/tex]
Step-by-step explanation:
[tex](\frac{2}{3} )^{2} =\frac{2^2}{3^2} =4/9[/tex]
Hope that helps :)
Convert the following base-ten numerals to a numeral in the indicated bases. a. 861 in base six b. 2157 in base nine C. 131 in base three a. 861 in base six is six
The values of base-ten numerals to the indicated bases are:
a. 861 in base six is 3553.
b. 2157 in base nine is 2856.
c. 131 in base three is 11221.
To convert the base-ten numerals to the indicated bases:
a. 861 in base six:
To convert 861 to base six, we divide the number by six repeatedly and note down the remainder until the quotient becomes zero.
861 ÷ 6 = 143 remainder 3
143 ÷ 6 = 23 remainder 5
23 ÷ 6 = 3 remainder 5
3 ÷ 6 = 0 remainder 3
Reading the remainders in reverse order, the base-six representation of 861 is 3553.
b. 2157 in base nine:
To convert 2157 to base nine, we follow a similar process.
2157 ÷ 9 = 239 remainder 6
239 ÷ 9 = 26 remainder 5
26 ÷ 9 = 2 remainder 8
2 ÷ 9 = 0 remainder 2
Reading the remainders in reverse order, the base-nine representation of 2157 is 2856.
c. 131 in base three:
To convert 131 to base three, we apply the same procedure.
131 ÷ 3 = 43 remainder 2
43 ÷ 3 = 14 remainder 1
14 ÷ 3 = 4 remainder 2
4 ÷ 3 = 1 remainder 1
1 ÷ 3 = 0 remainder 1
Reading the remainders in reverse order, the base-three representation of 131 is 11221.
Therefore:
a. 861 in base six is 3553.
b. 2157 in base nine is 2856.
c. 131 in base three is 11221.
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Plz help me out thanks
Answer:
the full answer is 215.859885inches cubed
Step-by-step explanation:
times the length, width and height together
The marked price of a radio is Sh. 12600. If the shopkeeper can allow a discount of 15% on the marked price and still make a profit of 25%.At what price did the shopkeeper buy the radio?
Answer:
13388
Step-by-step explanation:
12600 will be 100% so we want to get at what price its sold when there is a 15%dicount
So will minus 15% from the 100% of the Mp
100%-15%=85%
so if 100%=12600
what about 85%=?
we crossmultiply
85%×12600/100%=10710
so10710 is what the radio will be sold if a 15% dicount is given but we want to get wat price the shopkeeper got in that he made a profit of25%
so if 100%=10710
what about 125%
125%×10710/100%=13387.5 which is 13388/=
Consider Z is the subset of R with its usual topology. Find the subspace topology for Z.[r2]
The subspace topology for Z, which is a subset of R with its usual (standard) topology, is the set of open sets in Z.
In other words, the subspace topology on Z is obtained by considering the intersection of Z with open sets in R.
To find the subspace topology for Z, we need to determine which subsets of Z are open. In the usual topology on R, an open set is a set that can be represented as a union of open intervals. Since Z is a subset of R, its open sets will be the intersection of Z with open intervals in R.
For example, let's consider the open interval (a, b) in R. The intersection of (a, b) with Z will be the set of integers between a and b (inclusive) that belong to Z. This intersection is an open set in Z.
By considering all possible open intervals in R and their intersections with Z, we can generate the collection of open sets that form the subspace topology for Z. This collection of open sets will satisfy the axioms of a topology, including the properties of openness, closure under unions, and closure under finite intersections.
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The values of certain types of collectibles can often fluctuate greatly over time. Suppose that the value of a limited-edition flamingos riding alligators lawn ornament set is found to be able to be modeled by the function V(t) = 0.06t4 – 1.05t3 + 3.47t? – 8.896 +269.95 for Osts 15 where V(t) is in dollars, t is the number of years after the lawn ornament set was released, and t = 0 corresponds to the year 2006. a) What was the value of the lawn ornament set in the year 2009? b) What is the value of the lawn ornament set in the year 2021? c) What was the instantaneous rate of change of the value of the lawn ornament set in the year 2013? d) What is the instantaneous rate of change of the value of the lawn ornament set in the year 2021? e) Use your answers from parts a-d to ESTIMATE the value of the lawn ornament set in 2022.
The value of the lawn ornament set in the year 2009 was $51.375. The value of the lawn ornament set in the year 2021 was $558.181. The instantaneous rate of change of the value of the lawn ornament set in the year 2013 was $230.986. The instantaneous rate of change of the value of the lawn ornament set in the year 2021 was $351.076. The estimated value of the lawn ornament set in 2022 was $909.257.
a)
To find the value of the lawn ornament set in the year 2009, we have to plug in t = 3, as t = 0 corresponds to the year 2006.
V(3) = 0.06(3)4 – 1.05(3)3 + 3.47(3) – 8.896 + 269.95
V(3) = 51.375
So, the value of the lawn ornament set in the year 2009 was $51.375.
b)
To find the value of the lawn ornament set in the year 2021, we have to plug in t = 15, as t = 0 corresponds to the year 2006.
V(15) = 0.06(15)4 – 1.05(15)3 + 3.47(15) – 8.896 + 269.95
V(15) = $558.181
So, the value of the lawn ornament set in the year 2021 is $558.181.
c)
To find the instantaneous rate of change of the value of the lawn ornament set in the year 2013, we have to find V'(7), where V(t) is the given function.
V(t) = 0.06t4 – 1.05t3 + 3.47t – 8.896 +269.95 for Osts 15
V'(t) = 0.24t3 – 3.15t2 + 10.41t + 269.95
V'(7) = 0.24(7)3 – 3.15(7)2 + 10.41(7) + 269.95
V'(7) = $230.986
So, the instantaneous rate of change of the value of the lawn ornament set in the year 2013 was $230.986.
d) To find the instantaneous rate of change of the value of the lawn ornament set in the year 2021, we have to find V'(15), where V(t) is the given function.
V(t) = 0.06t4 – 1.05t3 + 3.47t – 8.896 +269.95 for Osts
15V'(t) = 0.24t3 – 3.15t2 + 10.41t + 269.95
V'(15) = 0.24(15)3 – 3.15(15)2 + 10.41(15) + 269.95
V'(15) = $351.076
So, the instantaneous rate of change of the value of the lawn ornament set in the year 2021 is $351.076.
e)
To ESTIMATE the value of the lawn ornament set in 2022, we can use the formula
V(t) ≈ V(a) + V'(a)(t – a),
where a is the year 2021.
V(a) = V(15) = $558.181
V'(a) = V'(15) = $351.076t = 16 (as we need to estimate the value of the lawn ornament set in 2022)
V(t) ≈ V(a) + V'(a)(t – a)
V(t) ≈ 558.181 + 351.076(16 – 15)
V(t) ≈ $909.257
So, the estimated value of the lawn ornament set in 2022 is $909.257.
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Use the binomial formula to find the coefficient of the y^120x² term in the expansion of (y+3x)^22. ?
This coefficient is not defined, since k must be a non-negative integer. Therefore, the coefficient of the y¹²⁰ x² term in the expansion of (y + 3x)²² is 0.
The binomial formula is used to expand binomials of the form (a + b)ⁿ, where a, b, and n are integer.
In general, the formula is given by:
[tex]$(a+b)^n=\sum_{k=0}^{n}{n \choose k}a^{n-k}b^k$[/tex]
The coefficient of the y¹²⁰ x² term in the expansion of (y + 3x)²² can be found by using the binomial formula.
To find this coefficient, we need to determine the value of k for which the term [tex]y^{22-k} (3x)^k[/tex] has y¹²⁰x² as a product.
Let's write out the first few terms of the expansion of (y + 3x)²²:
[tex]$(y + 3x)^{22} = {22 \choose 0}y^{22}(3x)^0 + {22 \choose 1}y^{21}(3x)^1 + {22 \choose 2}y^{20}(3x)^2 + \cdots$[/tex]
Notice that each term in the expansion has the form {22 choose k}[tex]y^{22-k} (3x)^k[/tex]
Thus, the coefficient of the y¹²⁰ x² term is given by the binomial coefficient {22 choose k}, where k is the value that makes 22 - k equal to the exponent of y in y¹²⁰ (i.e., 120). Therefore, we have:
22 - k = 120k = 22 - 120k = -98
Thus, the coefficient of the y¹²⁰ x² term is given by the binomial coefficient {22 choose -98}.
However, this coefficient is not defined, since k must be a non-negative integer. Therefore, the coefficient of the y¹²⁰ x² term in the expansion of (y + 3x)²² is 0.
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Help meeeeeeeeeeeeee
Answer:
x = 120°
Step-by-step explanation:
this is a 7-sided polygon and the sum of the interior angles is (7-2)×180° = 900°
add all 7 angles together and set equal to 900
x + 150 + x - 20 + 140 + 120 + x + 20 + 130 = 900
combine 'like terms'
3x + 540 = 900
3x = 360
x = 120
Plzzzzz I need help asap thank you
No links plzzzzz
Answer:
-1.4 ; -0.7 ; 0.003 ; 3% ; 0.3 ; 2/3 ; 7/8 ; 100/50
Step-by-step explanation:
0.003 ; -1.4 ; 100/50 = 2 ; 0.85 ; 2/3 = 0.67 ; 3% = 3/100 = 0.03 ; 7/8 = 0.875 ;
0.3 , -0.7
0.003 ; -1.4 ; 2 ; 0.85 ; 0.67 ; 0.03 ; 0.875 ; 0.3 ; -0.7
Least to greatest:
-1.4 ; - 0.7 ; 0.003 ; 0.03 ; 0.3 ; 0.67 ; 0.85 ;2
-1.4 ; -0.7 ; 0.003 ; 3% ; 0.3 ; 2/3 ; 7/8 ; 100/50
Negative numbers have least value.Then in decimal numbers, the number having the least value in tenth is the least
(q18) The average time to get your order at a restaurant is 15 minutes. What is probability that you will receive your order in the first 10 minutes?
Note:
where µ is the average value.
The correct answer is option (C): 0.487
Given that the average time to receive an order at a restaurant is 15 minutes, we can use the exponential distribution to calculate the probability of receiving the order in the first 10 minutes.
The exponential distribution is defined by the probability density function (PDF): f(x) = (1/µ) * e^(-x/µ), where µ is the average value or mean.
In this case, the mean (µ) is 15 minutes. We want to find P(a ≤ X ≤ b), where a is 0 (the lower bound) and b is 10 (the upper bound).
To calculate this probability, we need to integrate the PDF from a to b:
P(0 ≤ X ≤ 10) = ∫[0 to 10] (1/15) * e^(-x/15) dx
Integrating this expression gives us:
P(0 ≤ X ≤ 10) = [-e^(-x/15)] from 0 to 10
Plugging in the values, we get:
P(0 ≤ X ≤ 10) = [-e^(-10/15)] - [-e^(0/15)]
Simplifying further:
P(0 ≤ X ≤ 10) = -e^(-2/3) + 1
Using a calculator, we can evaluate this expression:
P(0 ≤ X ≤ 10) ≈ 0.487
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What is the y-intercept for the equation y= 11x + 1?
-11
-1
1
11
Answer:
1
Step-by-step explanation:
The y-intercept in the equation is 1 because the equation uses the format y=mx+b. The b in y=mx+b represents the y-intercept So, in this equation the y-intercept is 1 because b=1.
a museum gift shop sold 215 sets of dinosaurs. there were 9 dinosaurs in each set how many dinosaurs did they sell?
After driving 50 miles, you get caught in a storm and have to slow down by 10 mph. You then drive 75 miles at this slower speed all the way home. Find an equation for the time t of the trip as a function of the speed s of your car before slowing down.
The equation for the time of the trip, t, as a function of the speed, s, is t = (50/s) + (75/(s-10)).
To find the equation for the time of the trip as a function of the speed of the car before slowing down, we need to consider two parts of the journey. The first part is driving 50 miles at the original speed, which takes (50/s) hours, where s is the speed. The second part is driving 75 miles at a slower speed of (s-10) mph, which takes (75/(s-10)) hours.
To calculate the total time, we add the times for both parts: t = (50/s) + (75/(s-10)). This equation allows us to determine the time of the trip for any given speed before slowing down.
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jesse has never used the sliding board at his daycare because he is afraid. His teacher has encouraged him, but he refuses to slide down. one day his mother stands at the bottom And say's Jesse let's go and slides into his mother's arms Laughing The situation is a example of....
A Resiliency
B Adaptability
C Conditioning
D Social referencing
jesse has never used the sliding board at his daycare because he is afraid. His teacher has encouraged him, but he refuses to slide down. one day his mother stands at the bottom And say's Jesse let's go and slides into his mother's arms Laughing The situation is a example of Social referencing. Option (d) is correct.
What do you mean by Situation?Situation refers to a group of conditions or a current state of events.
Social reference is the method through which newborns control their behavior toward surrounding items, people, and circumstances by observing the emotive displays of an adult.
For adaptive social functioning to occur, one must recognize and make use of the emotional communication of others. The ability to negotiate complicated and frequently ambiguous settings is known as social referencing in the developmental literature and social appraisal in adult studies.
Therefore, Option (d) is correct. The situation is a example of Social referencing.
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PLSSS HELP IMMEDIATELY!!!!! i’ll give brainiest, i’m not giving brainiest if u leave a link tho. (pls check whole picture!!)
Answer:
(4,2)
Step-by-step explanation:
Answer:
(4, 2)
Step-by-step explanation:
The solution to a logistic differential equation corresponding to a specific hyena population on a reserve in A western Tunisia is given by P(t)= The initial hyena population 1+ke-0.57 was 40 and the carrying capacity for the hyena population is 200. What is the value of the constant k? (A) 4 (B) 8 (C) 10 (D) 20 6. Which of the following differential equations could model the logistic growth in the graph? AM 50 40 30/ 20 10 t (A) (B) dM =(M-20)(M-50) dt dM = (20-MM-50) dt dM = 35M dt dM = 35M(1000-M) dt (C) (D)
The logistic differential equation for the hyena population is given by:
dP/dt = r * P * (1 - P/K)
where P(t) is the hyena population at time t, r is the growth rate, and K is the carrying capacity.
We are given that:
P(t) = 40 + k * e^(-0.57t)
K = 200
To determine the value of k, we can plug in these values into the logistic differential equation and solve for k:
dP/dt = r * P * (1 - P/K)
dP/dt = r * P * (1 - P/200)
dP/dt = r/200 * (200P - P^2)
dP/(200P - P^2) = r dt
Integrating both sides, we get:
-1/200 ln|200P - P^2| = rt + C
where C is a constant of integration.
Using the initial condition P(0) = 40 + k, we can solve for C:
-1/200 ln|200(40+k)-(40+k)^2| = 0 + C
C = -1/200 ln|8000-480k|
Plugging in this value of C and simplifying, we get:
-1/200 ln|200P - P^2| = rt - 1/200 ln|8000-480k|
ln|200P - P^2| = -200rt + ln|8000-480k|
|200P - P^2| = e^(-200rt) * |8000-480k|
200P - P^2 = ± e^(-200rt) * (8000-480k)
Since the population is increasing, we choose the positive sign:
200P - P^2 = e^(-200rt) * (8000-480k)
Using the initial condition P(0) = 40 + k, we get:
200(40+k) - (40+k)^2 = (8000-480k)
8000 + 160k - 2400 - 80k - k^2 = 8000 - 480k
k^2 + 560k - 2400 = 0
(k + 60)(k - 40) = 0
Thus, k = -60 or k = 40. Since k represents a growth rate, it should be positive, so we choose k = 40. Therefore, the value of the constant k is option (A) 4.
For the second part of the question, the logistic equation that could model the growth in the graph is option (B) dM/dt = (20-M)*(M-50). This is because the carrying capacity is between 20 and 50, and the population growth rate is zero at both of these values (i.e. the population does not increase or decrease when it is at the carrying capacity).
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Can someone help me ill give you 25 points!!! no wrong answers or ill have brainly take all your points and band you forever Uhm yeah so......... plz help
Answer: Mean = 2.36 Median = 4 Range = 0
Step-by-step explanation:
Mean - the sum of the data values divided by the number of data values
Median - the middle number in an ordered set of data
Range - the difference between the greatest and least numbers in a data set
Use the data set and line plot below. Jerome studied the feather lengths of some adult fox sparrows.
How long are the longest feathers in the data set?
A.
2
2
inches
B.
2
1
4
214
inches
C.
2
1
2
212
inches
D.
2
3
4
234
inches
Answer: 2 1/2
Step-by-step explanation:
the answer is D i took the test here is proof
Trigonometry question help,,, NO LINKS
Answer:
87 ft
Step-by-step explanation:
SohCahToa is your best friend here.
You have two values you need to pay attention to:
The length that is adjacent to the 74°C, 25 ft. And the length opposite of the 74°C, the height of how high the rocket traveled.
So adjacent and opposite, O & A. "Toa", find the tangent of 74°C.
tan(74) = [tex]\frac{x}{25}[/tex]
x = (tan(74))(25)
x = 87 ft
What is the geometric mean of 4 and 3? Your answer should be a reduced radical, NOT A DECIMAL.
Answer:
[tex] 2 \sqrt{3} [/tex]
Step-by-step explanation:
Geometric mean of 4 and 3
[tex] = \sqrt{4 \times 3} \\ = \sqrt{ {2}^{2} \times 3 } \\ = 2 \sqrt{3} [/tex]
In real-life applications, statistics helps us analyze data to extract information about a population. In this module discussion, you will take on the role of Susan, a high school principal. She is planning on having a large movie night for the high school. She has received a lot of feedback on which movie to show and sees differences in movie preferences by gender and also by grade level. She knows if the wrong movie is shown, it could reduce event turnout by 50%. She would like to maximize the number of students who attend and would like to select a PG-rated movie based on the overall student population's movie preferences. Each student is assigned a classroom with other students in their grade. She has a spreadsheet that lists the names of each student, their classroom, and their grade. Susan knows a simple random sample would provide a good representation of the population of students at their high school, but wonders if a different method would be better. a. Describe to Susan how to take a sample of the student population that would not represent the population well. b. Describe to Susan how to take a sample of the student population that would represent the population well. c. Finally, describe the relationship of a sample to a population and classify your two samples as random, cluster, stratified, or convenience.
a. To take a sample of the student population that would not represent the population well, Susan could use a biased sampling method.
For example, she could choose students only from specific classrooms or grade levels that she believes have a certain movie preference, or she could select students based on her personal biases or preferences. This would introduce sampling bias and potentially skew the results, leading to a sample that does not accurately reflect the overall student population.b. To take a sample of the student population that would represent the population well, Susan should use a random sampling method. Random sampling ensures that every student in the population has an equal chance of being selected for the sample.c. A sample is a subset of the population that is selected for analysis to make inferences about the entire population. The relationship between a sample and a population is that the sample is used to draw conclusions or make predictions about the population as a whole.To know more about Random samples:- https://brainly.com/question/30759604
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Coronary bypass surgery: A healthcare research agency reported that
41% of people who had coronary bypass surgery in 2008
were over the age of 65. Twelve coronary bypass patients are sampled.
Part 1 of 2
(a) What is the mean number of people over the age of 65 in a sample of 12
coronary bypass patients? Round the answer to two decimal places.
The mean number of people over the age of 65 is ?
Part 2 of 2
(b) What is the standard deviation of the number of people over the age of 65
in a sample of 12 coronary bypass patients? Round the answer to four decimal places.
The standard deviation of the number of people over the age of 65 is ?
The standard deviation of the number of people over the age of 65 in a sample of 12 coronary bypass patients is 1.6487.
Given that a healthcare research agency reported that 41% of people who had coronary bypass surgery in 2008 were over the age of 65 and twelve coronary bypass patients are sampled.
To determine the mean number of people over the age of 65 in a sample of 12 coronary bypass patients, we use the formula below:
Mean = np
Where n = 12 and p = 0.41.
Mean = 12(0.41)
Mean = 4.92
Therefore, the mean number of people over the age of 65 in a sample of 12 coronary bypass patients is 4.92.
To determine the standard deviation of the number of people over the age of 65 in a sample of 12 coronary bypass patients, we use the formula below:
Standard deviation, σ = √(n p q)
Where n = 12, p = 0.41, and q = 1 - p.
Standard deviation, σ = √(12 × 0.41 × 0.59)
Standard deviation, σ = √2.71948
Standard deviation, σ = 1.6487 (rounded to four decimal places).
Therefore, the standard deviation of the number of people over the age of 65 in a sample of 12 coronary bypass patients is 1.6487.
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Enter values to complete the table below.
Please help me.
Answer:
6
0
2
Step-by-step explanation:
I assume we are taking the y-value and dividing it by x, as indicated by the y/x
-6/-1
6
0/1
0
6/3
2
Concession stand sales for each game in season are $320, $540, $230, $450, $280, and $580. What is the mean sales per game? Explain how you got your answer.
Answer:
$400
Step-by-step explanation:
all you do is add 320+540+230+450+280+580/6 and the asnwe comes out to 400
I NEED HELP WITH MATH PLS
screenshot is posted below
Answer: The correct answer is A or B
`
Step-by-step explanation:
Factor this expression using the GCF (greatest common factor) and then explain how you can verify your answer:
6ab+8a
Answer:
2ax(3b+4)
Step-by-step explanation:
there you go your answer
Determine the number of zeros of the function f(z) = Z^4 – 2z^3 + 9z^2 + z – 1 in the disk D[0,2].
Given the function f(z) = z^4 - 2z^3 + 9z^2 + z - 1. We have to determine the number of zeros of the function in the disk D[0,2].
According to the Fundamental Theorem of Algebra, a polynomial function of degree n has n complex zeros, counting multiplicity. Here, the degree of the given polynomial function is 4. Therefore, it has exactly 4 zeros.Let the zeros of the function f(z) be a, b, c, and d. The function can be written as the product of its factors:$$f(z) = (z-a)(z-b)(z-c)(z-d)$$$$\Rightarrow f(z) = z^4 - (a+b+c+d)z^3 + (ab+ac+ad+bc+bd+cd)z^2 - (abc+abd+acd+bcd)z + abcd$$
According to the Cauchy's Bound, if a polynomial f(z) of degree n is such that the coefficients satisfy a_0, a_1, ..., a_n are real numbers, and M is a real number such that |a_n|≥M>|a_n-1|+...+|a_0|, then any complex zero z of the polynomial satisfies |z|≤1+M/|a_n|.
We can write the polynomial function as $$f(z) = z^4 - 2z^3 + 9z^2 + z - 1 = (z-1)^2(z+1)(z-1+i)(z-1-i)$$The zeros of the function are 1 (multiplicity 2), -1, 1 + i, and 1 - i. We have to count the zeros that are in the disk D[0,2].Zeros in the disk D[0,2] are 1 and -1.Therefore, the number of zeros of the function f(z) = z^4 - 2z^3 + 9z^2 + z - 1 in the disk D[0,2] is 2.
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