Write a polynomial f(x) that satisfies the given conditions. Express the polynomial with the lowest possible leading positive
integer coefficient.
Polynomial of lowest degree with lowest possible integer coefficients, and with zeros 3-2i and 0 (multiplicity 4).
Step-by-step explanation:
Let use the following rules to construct a polynomial,
Rule 1:
If r is a zero of some polynomial, f(x), then
[tex](x - r)[/tex]
is a factor of the polynomial
We first know that one of our zeroes are real, which is 0.
So since r=0
[tex](x - 0) = x[/tex]
is a factor
However, since the root ,0, has a multiplicty of 4, that basically means we have 0 as a root 4 times or basically we multiply the factor 4 times.
[tex]x \times x \times x \times x = x {}^{4} [/tex]
This leads to an another rule:
If a polynomial has a zero, r that has a multiplicty of k,
we represent the factor as
[tex](x - r) {}^{k} [/tex]
where k is all integers greater than 0.
So to reinforce myself, since 0 has a multiplicty of 4, we have
[tex](x - 0) {}^{4} = {x}^{4} [/tex]
So our first factor is
[tex] {x}^{4} [/tex]
Part 2: Complex Zeroes.
When dealing with complex zeroes, we must note this rule.
if (a+bi) is a zero of some polynomial, p then its conjugate (a-bi) is a factor as well.
So if 3-2i is a factor, then
3+2i is a factor as well.
Using Rule 1, our factors now become
[tex](x - (3 + 2i))[/tex]
and
[tex](x - (3 - 2i))[/tex]
So our factors are
[tex] {x}^{4} (x - (3 - 2i)(x - (3 + 2i))[/tex]
To simplify use FOIL Method,
[tex] {x}^{4} ( {x}^{2} - x(3 + 2i) - x(3 - 2i) + 9 - 4 {i}^{2} )[/tex]
[tex] {x}^{4} ( {x}^{2} - 6x + 13)[/tex]
[tex] {x}^{6} - 6 {x}^{5} + 13 {x}^{4} [/tex]
So our polynomial is
[tex] {x}^{6} - 6 {x}^{5} + 13 {x}^{4} [/tex]
If using the method of completing the square to solve the quadratic equation
x^2+ 15x - 23 = 0, which number would have to be added to "complete the
square"?
Step-by-step explanation:
you would take half of 15 and then square it to get to number, which is 56.25
Answer:
[tex](\frac{15}{2} )^2=\frac{225}{4}[/tex]
Step-by-step explanation:
[tex]x^2+15x-23=0\\x^2+15x+(\frac{15}{2} )^2-(\frac{15}{2})^2-23=0\\(x^2+15x+(\frac{15}{2})^2)=23+(\frac{15}{2})^2\\(x+\frac{15}{2})^2=(23*4+15^2)/4=317/4\\x+\frac{15}{2}=+\frac{\sqrt{317}}{4}\\, -\frac{\sqrt{317}}{4}\\\\x=-\frac{15}{2}-\frac{\sqrt{317}}{4}\\\\, -\frac{15}{2}+\frac{\sqrt{317}}{4}\\\\[/tex]
Please help me I was sick today and missed out on class and I don’t understand this question
Answer:
see explanation
Step-by-step explanation:
(a)
the 2 angles are vertically opposite angles and are congruent
(c)
since they are congruent , then
7x + 6 = 6x + 18 ( subtract 6x from both sides )
x + 6 = 18 ( subtract 6 from both sides )
x = 12
(d)
then
7x + 6 = 7(12) + 6 = 84 + 6 = 90
6x + 18 = 6(12) + 18 = 72 + 18 = 90
NO LINKS!!! Part 1 Please help me with this similarity practice
Answer:
C' = (13, 4)
Dilation by a scale factor of 2 with the origin (0, 0) as the center of dilation, followed by a translation of 3 units right.
Step-by-step explanation:
Given vertices of triangle ABC:
A = (0, 0)B = (1, 4)C = (5, 2)If two triangles are said to be similar:
Their corresponding angles are congruent.Their corresponding sides are in the same ratio.To keep the corresponding angles of both triangles the same and maintain similarity, dilate triangle ABC.
If we compare the coordinates of B (1, 4) and B' (5, 8), the y-value has been multiplied by 2.
Therefore, the first step of the transformation is a dilation by a scale factor 2 with the origin (0, 0) as the center of dilation.
⇒ (1 × 2, 4 × 2) = (2, 8)
After multiplying the coordinates of B by two, the difference between the x-values of the dilated point and B' is 3, so the second step of the transformation is a translation of 3 units right.
Therefore, the series of transformations is:
Dilation by a scale factor of 2 with the origin (0, 0) as the center of dilation.Translation of 3 units right.Mapping Rule: (x, y) → (2x+3, 2y)
Therefore, the coordinates of point C' are:
⇒ C' = (2(5) + 3, 2(2)) = (13, 4)
Answer: (13, 4)
Infinitely many answers are possible.
====================================================
Explanation:
See this similar question
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because I'll be using the same steps and ideas. There are also diagrams that might help visualize what's going on.
---------
The translation vector from B(1,4) to C(5,2) is <4,-2> since we move to the right 4 units, and down 2 units. It's the shorthand way of writing [tex](x,y) \to (x+4,y-2)[/tex]
Double the coordinates of the translation vector <4,-2> to get <8,-4>
If we started at point B'(5,8) and followed the pathway "right 8, down 4" then we arrive at C'(13,4) as one possible answer.
We could express the steps like this
[tex](\text{x},\text{y})\to(\text{x}+8,\text{y}-4)\\\\(5,8)\to(5+8,8-4)\\\\(5,8)\to(13,4)\\\\[/tex]
As I mentioned in the link above, there are infinitely many answers because we can change the scale factor to any nonnegative real number. For instance we could triple the coordinates of <4,-2> to get <12,-6> and follow the same outline of steps to get C'(17,2) as another possible answer.
The equation of a line is written below.
y=2x-3
The slope of the line is____
The y-intercept of the line is_____
Answer:
Slope: 2
Y-intercept: -3
Step-by-step explanation:
y=2x-3
The slope is always the number(s) before the x and the y-intercept is what is added or subtracted
Need some help!!!can you please explain?
iven:
he fucnction is,
[tex]f(x)=\frac{3}{4}\sqrt[3]{x+1}[/tex]xplanation:
he graph fo the function cnan be obtained by graph of cube root of x.
If we shift the graph of cube root by x by one unit left then then function changed to,
[tex]\sqrt[3]{x+1}[/tex]If the obtained graph shrink vertically by a factor of 3/4 then graph changed to function,
[tex]\frac{3}{4}\sqrt[3]{x+1}[/tex]This is the requied function for which we need the graph.
So graph can be obtained by horizontal shift of 1 unit to left, shrink by a factor of 3/4 and there is no vertical shift and graph will not flip over any axis.
Thus answer is, All statements are true.
What is the distance, d, between the points(2,7/2) and (5/3,1)?
Enter your answer in the box. Enter your answer in simplest radical form.
[tex]~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{2}~,~\stackrel{y_1}{\frac{7}{2}})\qquad (\stackrel{x_2}{\frac{5}{3}}~,~\stackrel{y_2}{1})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ d=\sqrt{(\frac{5}{3}-2)^2 + (1-\frac{7}{2})^2}\implies d=\sqrt{(\frac{-1}{3})^2 + (-\frac{5}{2})^2} \\\\\\ d=\sqrt{\cfrac{1}{9}~~ + ~~\cfrac{25}{4}}\implies d=\sqrt{\cfrac{229}{36}}\implies d=\cfrac{\sqrt{229}}{\sqrt{36}}\implies d=\cfrac{\sqrt{229}}{6}[/tex]
Score: 0 of 1 pt1 of 16 (0 complete)Instructor-created questionA rectangle is 3times as long as it is wide. If the width of the rectangle is2inches, what is the rectangle's area in square inches?6O A. 10OB. 12O C. 16OD. 6O E. 7Click to select your answer and then click Check AnswerAll parts showingClear All0
The length of the rectangle is 3 times it width. The width size is given as 2 inches.
Therefore,
width = 2 inches
length = 3(2) = 6 inches
The area of the rectangle can be calculated below
[tex]\begin{gathered} \text{area}=\text{length}\times\text{width} \\ \text{area}=6\times2 \\ \text{area}=12inches^2 \end{gathered}[/tex]The answer is B.
how much does $3000 earn in 6 months at in interest rate of 4%, compounded quarterly?
The amount earned by $3000 deposited in the account after 6 months is $60.30
What does quarterly compounding mean?
Quarterly compounding means that the interest is computed and credited to the account every quarter, that is every 3 months, in other words, our future value formula would indicate quarterly interest rate rather than an annual interest rate
FV=PV*(1+r)^N
FV=the amount in the account after 6 months=unknown\
PV=initial amount deposited in the account=$3,000
r=quarterly interest rate=4%/4=0.01
N=number of quarters in 6 months=2
FV=$3000*(1+0.01)^2
FV=$3000*(1.01)^2
FV=$3000*1.0201
FV=$ 3,060.30
amount earned=FV-PV
amount earned=$3060.30-$3000
amount earned=$60.30
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I8. Last night, you spent a total of 51 minutes working on your homework problems for math andbiology. Each math problem takes 5 minutes to complete and each biology problem takes 3minutes to complete. You completed a total of 15 problems. Write a system of equations thatdetermines how many math problems and how many biology problems you completed lastnight.
Let x represent the number of math problems that you completed.
Let y represent the number of biology problems that you completed
From the information given,
You completed a total of 15 problems. This means that
x + y = 15
If each math problem takes 5 minutes to complete, it means that the time to complete x math problems is 5 * x = 5x
If each biology problem takes 3 minutes to complete, it means that the time to complete y biology problems is 3 * y = 3y
If you spent 51 minutes working on your homework problems for math and
biology, it means that
5x + 3y = 51
Thus, the system of equations is
x + y = 15
5x + 3y = 51
4. The radius of a hydrogen atom is
0.000000000025 meter. How would you
express this radius in scientific notation?
Answer:
2.5×10^-11
Step-by-step explanation:
Put the point in the middle of 2 and 5 and count how many times you need to get there ...
like 0.05 to put point after 5 you have to jump 2 steps so it is 5×10^-2.
Jason is canoeing on a river that flows
downstream at a rate of 1 mile per hour. After
paddling 5 miles downstream, he turns around
and paddles 6 miles upstream to report the
condition of the river to the rest of his group. The
whole trip takes 3 hours.
a) Fill out the last column in table below by
writing an expression for time downstream and
for time upstream
An expression exists as a number, a variable, or a combination of numbers and variables, and function symbols. The speed in still water exists 4 mph.
What is meant by expression?The acronym BODMAS, which stands for brackets of, division, multiplication, addition, and subtraction, might be used to simplify the expression. It specifies how to answer a mathematical expression in a particular order.
Let x be the speed in still water
[tex]$&\frac{5}{x+1}+\frac{6}{x-1}=3[/tex]
LCD = (x + 1)(x - 1)
[tex]$&(x+1)(x-1) \cdot \frac{5}{x+1}+(x+1)(x-1) \cdot \frac{6}{x-1}=(x+1)(x-1) \cdot 3 \\[/tex]
simplifying the above equation, we get
5(x - 1) + 6(x + 1) = 3(x + 1)(x - 1)
5x - 5 + 6x + 6 = 3 x² - 3
3x² - 11x - 4 = 0
(3x + 1)(x - 4) = 0
x = -1/3 (speed is not negative);
then we get x = 4 mph
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A survey of 600 school employees yielded the following information: 449 were instructors, 325 were full-time faculty, and 260 of the instructors were full-time faculty. How many of the 600 surveyed school employees were instructors or were full-time faculty?
Answer:
260+325 = 585
260 where instructors that were full time and 325 were already full time. this is correct if I am reading your question correctly if not the answer is likely 260
Find the sum 40 +41 +42 +43 +...+ 124 + 125.
The sum is
Please help I’ll mark you as brainliest if correct!
Answer: 415
Step-by-step explanation:
??
Jose wants to earn at least S61 trimming trees. He charges $7 per hour and pays $9 in equipment fees. What are the possible numbers of hours Jose could trim
trees?
Use for the number of hours.
Write your answer as an inequality solved for t.
Answer:
That depends on what exactly is meant by "$3 in equipment fees". Is that per hour, per day or what?
Assuming per hour, he makes $7 per hour but pays $3 for a profit of $4 per hour. So, to make at least $39, he'd have to work 39/4 or 9 3/4 hours.
Now, if his equipment fee is daily, then he would need to earn at least $42 in a day to profit at least $39. Thus, he'd have to work at least 42/7 or 6 hours to make that much profit.
Step-by-step explanation:
An actress has a probability of getting offered a job after a try-out of 0.11. She plans to keep trying out for
new jobs until she gets offered one. Assume outcomes of try-outs are independent.
Find the probability she will need to attend more than 3 try-outs. Round your answer to 4 decimal places.
Submit Question
The probability she will need to attend more than 3 try-outs is 0.7049.
The probability of an actress getting a job after a tryout is 0.11.
The actress wants to keep trying out for new jobs until she gets offered one.
The number of trials necessary for success in a series of independent trials follows a geometric distribution with parameter p, where p is the probability of success in a single trial.
The probability mass function is given as:
[tex]P(X=k)=q^{k-1}p[/tex] where k = 1, 2, 3
We have p = 0.11
q = 1 - 0.11 = 0.89
If X is the number of tryouts,
[tex]P(X=k)=q^{k-1}p\\P(X=k)=(0.89)^{k-1}(0.11)[/tex]
[tex]P(X\leq r)=1-q^r\\P(X\leq r)=1-(0.89)^r[/tex]
Therefore, the probability that she will need to attend more than 3 try outs will be:
P = P( X > 3 )
P = 1 - P( X ≤ 3 )
P = 1 - [ 1 - 0.89³ ]
P = 1 - 1 + 0.89³
P = 0.704969
P = 0.7049
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The equation |2x + 1 = 9 has two solutions, a and b, where a < b. Find the two solutions.
Answer:
Write down one equation or inequality that has more than two solutions, but not infinitely many solutions
Step-by-step explanation:
12. Evaluate: 0.0512 x 104.
O 51.2
O 5120
O 512
O 0.00000512
Use subtraction
14x25x4
There are two possibilities for answers none of which require subtraction based on the expression given:
With the question none of which involves subtraction. First is that this is a simple multiplication in which case the answer is 1,400;
Second that the above expression actually means 14x * 25x * 4 in which case the answer is 1400x²
Explanation 1
14 x 25 x 4
= 350 x 4
= 1,400
Explanation 2
14x * 25x * 4
= 1400 * x * x
= 1400x²
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1.Write a linear equation of the form y1 = mx + b for your first set of data.2.Write a linear equation of the form y2 = mx + b for the other equation in your system.3.Explain the Point of Intersection. (X,Y)
First, find the actual number of miles per round trip in each case.
-Hyperloop
[tex]distanceH=speed*time=760*0.583=443.08miles[/tex]-Airplane
[tex]distanceA=535*1.25=668.75miles[/tex]Then, the cost (round trip) would be related to the constant b in both cases since it is a fixed rate. Multiply it by the maximum number of passengers,
[tex]\begin{gathered} b_H=40*28=1120 \\ b_A=88*360=31680 \end{gathered}[/tex]Then, multiply the rate of consumption by the fuel costs for each mean of transportation,
[tex]\begin{gathered} \frac{1}{m_H}=2*16.8*1=33.6 \\ \frac{1}{m_A}=2*2736*2.98=16306.56 \end{gathered}[/tex]Notice that it is multiplied by 2 since the rate of consumption row is given in one-way unitsThus,
[tex][/tex]Fargus deposits $366 in a savings account at City BankThe account pays an annual interest rate of 5 percent She makes no other deposits or withdrawalsAfter three months the interest is calculated. How much simple interest does her money
The Simple interest after three month $423.70
What is Simple Interest ?
Simple interest is a quick and simple formula for figuring out how much interest will be charged on a loan. The daily interest rate, the principle, and the number of days between payments are multiplied to calculate simple interest.
If she deposits $366 in the account and you receive interest, you must first calculate the interest for the current year before adding the sum to calculate the interest for the following month.
for instance, deposit plus interest equals the new amount for month one, month one amount plus interest equals amount for month two, and month two amount plus interest equals amount for month three.
The total would thus be $423.70 for the following month:
month 1
366+ (366*.05)=x
366+18.3=x
month 2
384.30+(384.30*.05)=x
384.30+19.22=x
month 3
403.52+(403.52*.05)=x
403.52+20.18=x
423.7 = x
Thus, total amount then would be $423.70
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I need help with all these questions as I don't know what to do with any of them.
answer: 4) -10
The entire graph of the function f is shown in the figure below.Write the domain and range of fusing interval notation.
Domain = the set of possible values along the x axis.
Domain = (-5 , 3 ]
Range = the set of possible values along the y axis.
Range = (-3 , 5 ]
please help you just solve the equation and list properties
EXPLANATION
To solve the given equation:
[tex]-4=\frac{v+6}{2}[/tex]Step 1:
Multiply both-side of the equation by 2 in other to eliminate the denominator.
[tex]-4\times2=\frac{(v+6)}{2}\times2[/tex]At the right-hand side of the equation, 2 add the numerator will cancel-out 2 at the denominator.
[tex]-8=\frac{(v+6)}{\cancel{2}}\times\cancel{2}[/tex]- 8 = v+ 6
step 2:
Subtract 6 from both-side of the equation.
That is;
-8 - 6 = v + 6 - 6
-14 = v
Hence, the value of v = -14
The full-time year-round median salary for U.S. workers in
2000 was $30,846, compared to $52,800 in 2020.
The linear function that gives the full-time year-round median salary for U.S. workers in t years after 2000 is given by:
S(t) = 30846 + 1097.7.
What is a linear function?A linear function, in slope-intercept format, is modeled according to the following rule:
y = mx + b
In which:
The coefficient m is the slope of the function, which is the constant rate of change.The coefficient b is the y-intercept of the function, which is the initial value of the function.Linear functions are used in situations when the rate of change is constant, or can be interpreted as a constant, and this rate is called the slope, as we stated above.
In this problem, we have that:
The reference year is of 2000, hence the intercept is of b = 30846, as it was the median salary during the year of 2000.In 20 years, the median salary increased to $52,800, hence the slope is calculated as follows: m = (52800 - 30846)/20 = 1097.7.Hence the linear function that models the situation is given as follows:
S(t) = 30846 + 1097.7.
What is the missing information?The complete problem could not be found on any search engine, hence we are going to suppose that it asks for a linear function for the median salary in t years after 2000.
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there are 60 minutes in one hour and 60 seconds in one minute using this information explain how you could convert 20 miles per hour to feet per second
20 miles per hour converted to feet per second is 88/3 feet per second.
It is given in the question that there are 60 minutes in one hour and 60 seconds in one minute.
We have to convert 20 miles per hour in feet per second.
We know that,
1 mile = 5280 feet
1 hour = 60 minutes
1 minute = 60 seconds
Hence, we can write,
1 hour = 60*60 = 3600 seconds
1 mile per hour = 5280/3600 feet per second = 22/15 feet per second
Using unitary method, we can write,
20 miles per hour = 20*(22/15) = 88/3 feet per second or 29.3334 feet per second.
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A game is played using one die. If the die is rolled and shows 2, the player wins $25. If the die shows any number other than 2, the player wins nothing Complete parts (a) through (b) below
a. If there is a charge of $5 to play the game, what is the game's expected value?
$(Round to the nearest cent)
The game's expected value is -$0.83.
The game is played using one die.If the die is rolled and it shows 2, then the player wins $25.If the die is rolled and it shows any number other than 2, then the player wins nothing.The probability of getting 2 on the die is 1/6.The probability of not getting 2 on the die is 5/6.The cost of playing the game is $5.If the die shows 2, the player has a profit of $20.If the die doesn't show 2, the player has a loss of $5.The expected value is the sum of the product of the value of individual events and their respective probabilities.The expected value is 20(1/6) - 5(5/6).The expected value is - 5/6.The expected value is -$0.83.To learn more about probability, visit :
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Jackie has 9 stamps. Kevin has 36 stamps. Kevin says he has 5 times as many stamps as Jackie. Is Kevin correct? Explain your reasoning.
The most appropriate choice for linear equation will be given by -
Kevin is not correct
What is linear equation?
At first it is important to know about equation
Equation shows the equality between two algebraic expressions by connecting the two algebraic expressions by an equal to sign.
A one degree equation is known as linear equation.
Here,
Let Kevin has x times as many stamps as stamp
Jackie has 9 stamps
Number of stamps Kevin has = 9x stamps
By the problem,
9x = 36
[tex]x = \frac{36}{9}[/tex]
x = 4
So Kevin has 4 times as many stamps as Jackie
But Kevin says he has 5 times as many stamps as Jackie.
So Kevin is not correct.
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What is the equation of the quadratic function represented by this graph?
-10-
Type the correct answer in each box. Use numerals instead of words,
Step-by-step explanation:
9x + 4y is where (9,4) is the coordinates
Scampini Technologies is expected to generate $75 million in free cash flow next year, and FCF is expected to grow at a constant rate of 7% per year indefinitely. Scampini has no debt or preferred stock, and its WACC is 12%, and it has zero nonoperating assets. If Scampini has 40 million shares of stock outstanding, what is the stock's value per share? Do not round intermediate calculations. Round your answer to the nearest cent
If Scampini has no debt or preferred stock, and its WACC is 12%, and it has zero nonoperating assets in which Scampini has 40 million shares of stock outstanding, the stock's value per share is $37.50.
Stock priceNext year CF = $75 millions
Growth rate of CF = 7%
WACC = 12%
Number of shares outstanding = 40 million shares
Estimated business valuation:
Valuation = Next year CF/ (WACC -growth rate)
Valuation = $75 million / (12% - 7%)
Valuation = $75 million /5%
Valuation =$1500 million
Estimated stock price:
Stock price = Valuation /Shares outstanding
Stock price = $1500 million / 40 million shares
Stock price = $37.50
Therefore the stock price is $37.50.
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