Find a polynomial f(x) of degree 4 with real coefficients and the following zeros.3 (multiplicity 2) , -i
Answers
We are told that we want a polynomial f(x) with the given zeros.
Recall that if we know the zeros oa polynomial, we can write the polynomial by writing the factors (x - zero of the polynomial) and multiply them all together.
For example, if we want a polynomial of degree 2 with zeros at 2 and 3, then the polynomial would be
[tex](x\text{ -2)}\cdot(x\text{ -3)}[/tex]In this case, we have a polynomial f(x) of degree 4. So far, we know that 3 is a zero and that -i is a zero. So we write the following
[tex]f(x)=(x\text{ -a)}\cdot(x\text{ -b)}\cdot(x\text{ -c) }\cdot\text{ (x -d)}[/tex]where a,b,c and d are the zeros of f(x). We know that 3 is a zero and that -i is a zero. So we have
[tex]f(x)=(x\text{ -3)}\cdot(x\text{ -b)}\cdot(x\text{ -( -i)) }\cdot(x\text{ -d)}[/tex]So to fully describe f(x) we need to find the values of b and d. We are told that 3 is a zero of multiplicity 2. This means that the factor (x -3) appears two times in the factorization of f(x). So we can say that b=3. So we have
[tex]f(x)=(x\text{ -3)}\cdot(x\text{ -3) }\cdot(x\text{ +i ) }\cdot(x-d)=(x-3)^2\cdot(x\text{ +i)}\cdot(x\text{ -d)}[/tex]Now, we need to find the value of d. Note that we are told that -i is a zero of the function. -i is a complex number, so one important property of polynomials is that if a complex number a+bi is a zero of the polynomial, then the number a-bi (which is called the complex conjugate) is also a zero. Note that the complex conjugate of a complex number is calculated by leaving the real part intact and multiplying the imaginary part by -1.
In our case we have the complex number -i. So we can write -i= 0 - 1i . Then, its complex conjugate is i.
So, we have that d=i.
Then our polynomial would look like this
[tex]f(x)=(x-3)^2\cdot(x+i)\cdot(x\text{ -i)}[/tex]Note that
[tex](x+i)\cdot(x-i)=x^2\text{ -i}\cdot x\text{ + i}\cdot x+1=x^2+1[/tex]So our polynomial ends up being
[tex]f(x)=(x-3)^2\cdot(x^2+1)[/tex]Related Questions
114. If plane X averages 800 mph and plane Y averages 400 mph, how manyhours will plane X travel before it overtakes plane Y if plane Y has a 2 hourand 30 minute head start?a.1b. 2c. 5d. 72
Answers
To determine the time taken for the plane X to travel:
If plane X averages 800 mph and plane Y averages 400 mph
Distance column is found by multiplying the rate
by time.
The time taken for plane Y to travel = 2hr 30 minutes = 2.5hrs head start
Be sure to distribute the 400(t +2.5)for Plane Y,
and Plane X to distribute 800t
As they cover the same distance
[tex]\begin{gathered} Dis\tan ce\text{ is equal ,} \\ 800t=400(t+2.5) \\ 800t=400t+1000 \\ 800t-400t=1000 \\ 400t=1000 \\ t=\frac{1000}{400} \\ t=2\frac{1}{2}hr \end{gathered}[/tex]Therefore the plane X will travel for 2 1/2 hours
Hence the correct answer is Option B
Jenny wants to earn $1,300by the end of the summer. How much more will she need to earn to meet her goal?
Answers
The most appropriate choice for subtraction of natural numbers will be given by-
Jenny needs $1172.95 to earn her goal.
What is subtraction?
At first, it is important to know about natural numbers.
Natural numbers are integers which are greater than or equal to 1
One of the operations on natural number is subtraction
The process of reducing one number from another number is called subtraction. Subtraction is used to find the difference between two numbers. The larger number is called minuend and the smaller number is called subtrehend.
Amount of money Jennyy had before = $127.05
Amount of money Jenny wants to earn = $1300
Amount of money Jenny needs to earn her goal = $(1300 - 127.05)
= $1172.95
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Complete Question
Jenny wants to earn $1,300 by the end of the summer. How much more will she need to meet her goal?
(Jenny had $127.05 before.)
Reflects the given the coordinates points across the y - axis
Answers
Answer:
Explanation:
The reflection over the line y = x gives the following transformation of coordinates
[tex](x,y)\to(y,x)[/tex]therefore, for our case the transformation gives
[tex]\begin{gathered} S(-2,5)\to S^{\prime}(5,-2) \\ T(-3,0)\to T^{\prime}(0,-3) \\ U(1,-1)\to U^{\prime}(-1,1)_{} \end{gathered}[/tex]which are our answers!
The graphical representation of a point and its reflection about the line y =x is the following:
Triangle LMN is drawn with vertices at L(−2, 1), M(2, 1), N(−2, 3). Determine the image vertices of L′M′N′ if the preimage is rotated 90° clockwise. L′(1, 2), M′(1, −2), N′(3, 2) L′(−1, 2), M′(−1, −2), N′(−3, 2) L′(−1, −2), M′(−1, 2), N′(−3, −2) L′(2, −1), M′(−2, −1), N′(2, −3)
Answers
ANSWER
L'(1, 2), M'(1, -2), N'(3, 2)
EXPLANATION
The rule for rotating a point (x, y) 90° clockwise is,
[tex](x,y)\rightarrow(y,-x)[/tex]So, the vertices of triangle LMN will be mapped to,
[tex]\begin{gathered} L(-2,1)\rightarrow L^{\prime}(1,2) \\ M(2,1)\rightarrow M^{\prime}(1,-2) \\ N(-2,3)\rightarrow N^{\prime}(3,2) \end{gathered}[/tex]Hence, the image has vertices L'(1, 2), M'(1, -2), N'(3, 2).
I need help with this practice problem solving It is trigonometry I will send another picture with the graph that is included in the problem, it asks to use the graph to solve
Answers
Given the function
[tex]f(x)=\sin (\pi x+\frac{\pi}{2})[/tex]The graph of the function is as shown below:
Susan has a job selling cars, and earns 1.25% commission on the first $100,000 in sales,The commission increases to 4.95% on the next $200,000. Last month her total sales were$387,000. How much was her commission if she received 7.25% for any sales over $300,000
Answers
Solution:
Susan earns a commission based on car sales made.
Given:
Total sales made for the month = $387,000
Her commission is calculated based on levels and commision rate for each level.
On the first $100,000, she earns 1.25% commission.
Total sales at this level is $100,000
[tex]\begin{gathered} \text{Commision made on the first \$100,000 is;} \\ \frac{1.25}{100}\times100000=\text{ \$1,250} \\ =\text{ \$1,250} \end{gathered}[/tex]On the next $200,000, she earns 4.95% commission.
Total sales at this level is $300,000
[tex]\begin{gathered} \text{Commision made on the next \$200,000 is;} \\ \frac{4.95}{100}\times200000=\text{ \$9,900} \\ =\text{ \$9,900} \end{gathered}[/tex]On the next $87,000, total sales at this level is $387,000. She earns 7.25% commission for sales above $300,000.
[tex]\begin{gathered} \text{Commision made on the next \$87,000 is;} \\ \frac{7.25}{100}\times87000=\text{ \$6,}307.50 \\ =\text{ \$6,}307.50 \end{gathered}[/tex]Therefore, Susan's total commission received for the month is;
[tex]\begin{gathered} \text{ \$1250 + \$9900 + \$6307.50} \\ =\text{ \$17,457.50} \end{gathered}[/tex]Hence, her commission received in total for the sales made is $17,457.50
Exercise 2 Find a formula for Y in terms of X
Answers
Given:
y is inversely proportional to square of x.
The equation is written as,
[tex]\begin{gathered} y\propto\frac{1}{x^2} \\ y=\frac{c}{x^2}\ldots\ldots\ldots c\text{ is constant} \end{gathered}[/tex]Also y = 0.25 when x = 5.
[tex]\begin{gathered} y=\frac{c}{x^2} \\ 0.25=\frac{c}{5^2} \\ 25\times0.25=c \\ c=\frac{25}{4} \end{gathered}[/tex]So, the equation of y interms of x is,
[tex]y=\frac{25}{4x^2}[/tex]When x increases,
[tex]\begin{gathered} \lim _{x\to\infty}y=\lim _{x\to\infty}(\frac{25}{4x^2}) \\ =\frac{25}{4}\lim _{x\to\infty}(\frac{1}{x^2}) \\ =0 \end{gathered}[/tex]Hence, the value of x increases then y decreases.
Solve the inequality and graph the solution set.3 ≤ 4x + 1 < 9
Answers
Okay, here we have this:
Considering the provided inequality, we are going to solve it and graph the solution set, so we obtain the following:
3 ≤ 4x + 1 < 9
3 -1≤ 4x + 1 -1< 9-1
2 ≤ 4x < 8
2/4 ≤ 4x/4 < 8/4
1/2 ≤ x < 2
In interval notation the solution set will be: [1/2, 2)
And if we plot this solution interval we get:
Where the solution set will be the purple part.
tell whether you can prove that each quadrilateral is a parallelogram. Explain.
Answers
WE know that in any quadrilateral the interior sum of its angles is 360. The missing angle in this case is:
[tex]360-121-59-59=121[/tex]Now, we also know that if the oposite angles in a quadrilateral are equal then the quadrilateral is a parallelogram.
Therefore the figure shown is a parallelogram.
Cisco Enterprises in Ontario purchased the following in a single month all-inclusive of taxes:
16,000 units of network routers at $79.25 each
Answers
Answer:
1268000
Step-by-step explanation:
16000x79.25=1268000
Please tell me if these are correct if theyre not please help and tell me which ones are the right answers
Answers
Answer:
They're correct
Step-by-step explanation:
hello I don't know if you can help me with this but I no am doing something wrong. because at the bottom its not spelling right
Answers
8. The difference of three and a number means x+3 or x-3 because in both equations you have three units plus or minus the number X.
10. '"4 times the sum of a number and three" means
[tex]4\cdot(x+3)=4x+12[/tex]Letter D
Real number between 0 and 6 will be picked according to the probability distribution shown in the figure. Regions under the curve are liable with A, B, C, and D. The area of each is shown in the table. Use the figure and table to answer the parts
Answers
Part A
The probability that a real number between 1 and 4 is picked
P=PB+PC
P=0.15+0.50
P=0.65Part B
The probability that a real number between 2 and 6 is picked
P=PC+PD
P=0.50+0.30
P=0.80Simplify the numerical expression (3^2 * 5^-1)^2
Answers
Simplify the numerical expression
[tex](3^2\cdot5^{-1})^{2}[/tex][tex]\begin{gathered} (9\cdot\frac{1}{5})^{2}= \\ (\frac{9}{5})^{2}= \\ \frac{81}{25} \end{gathered}[/tex]Find the total amount in the compound interest account $2650 is compounded annually at a rate of 11% for 1 year
Answers
The compound interest formula is:
[tex]A\text{ = P}(1+i)^t[/tex]where:
A is the final amount including the principal
P is the principal amount
i is the interest rate (as a decimal)
t is time in years
Replacing with P = $2650, i = 0.11, and t = 1, we get:
A = 2650*(1 + 0.11)
A = 2650*1.11
A = $2941.5
The cost of 5 gallons of ice cream has a varianceof 36 with a mean of 36 dollars during the summer.What is the probability that the samplean would differ from the true mean by more than 0.6 dollars if a sample of 107 5-gallon pails is randomly selected? Roundyour answer to four decimal places.
Answers
Given:
[tex]\begin{gathered} Variance=36 \\ mean=36 \end{gathered}[/tex]To Determine: The samplean would differ from the true mean by more than 0.6 dollars
Solution
Please note that standard deviation is the square root of variance
[tex]\begin{gathered} SD=\sqrt{Variance} \\ SD=Standard-deviation \\ SD=\sqrt{36}=6 \end{gathered}[/tex][tex]\begin{gathered} S.E=\frac{SD}{\sqrt{n}} \\ S.E=Standard-Error \\ n=107 \\ S.E=\frac{6}{\sqrt{107}}=0.5800 \end{gathered}[/tex]Please note that Z is the number of SE(standard error away from the mean. Therefore
[tex]\begin{gathered} Z=\frac{0.6}{0.5800} \\ Z=1.0345 \end{gathered}[/tex][tex]P(|Z|<1.0345)[/tex][tex]P(|Z|<1.0345)=1-P(Z<-1.0345)=1-0.1515=0.8485[/tex]Hence the probability is 0.8485
Find the area round to two decimal places as needed
Answers
To find the area of an obtuse triangle you have to multiply the base of the triangle by the vertical height and divide the result by 2 following the formula:
[tex]A=\frac{b\cdot h}{2}[/tex]The base of the triangle is b= 7 miles and the height is h= 8 miles, using these lengths calculate the area as follows:
[tex]\begin{gathered} A=\frac{7\cdot8}{2} \\ A=\frac{56}{2} \\ A=28mi^2 \end{gathered}[/tex]The area of the triangle is 28 square miles.
The cargo of the truck welghs no more than 2,800 pounds.Use w to represent the weight (in pounds) of the cargo.
Answers
We know that
• The truck weighs no more than 2,800 pounds.
This problem is about inequalities.
"no more" indicates an inequality sign, specifically, it shows that we should use "less than or equal to", because this sign indicates the same as the problem do.
Therefore, the expression of the truck weight is
[tex]w\leq2,800[/tex]The odds in favor of a horse winning a race are 7:4. Find the probability that the horse will win the race.A. 7/12B. 4/7C. 7/11D. 4/11
Answers
We have a reason for 7:4,
i.e. the total probability of winning is 7+4=11
If the horse has a probability of winning of 7 between 11
We can say that the Pw of the horse is as follows
[tex]\frac{7}{11}[/tex]The answer is the option C
A survey stopped men and women at random to ask them where they purchased groceries, at a local grocery store or online.Grocery OptionsStoreOnlineTotalWomen231235Men221537Total452772What percent of the people surveyed shop at a local grocery store? Round your answer to the nearest whole number percent
Answers
Given:
A survey stopped men and women at random to ask them where they purchased groceries, at a local grocery store or online. Grocery Options
Store Online Total
Women 23 12 35
Men 22 15 37
Total 45 27 72
Required:
To find the percentage of the people surveyed shop at a local grocery store.
Explanation:
The total number of people is 75.
And the total number of people surveyed shop at a local grocery store is 45.
Now the percentage of the people surveyed shop at a local grocery store is,
[tex]=\frac{45}{72}\times100[/tex][tex]\begin{gathered} =62.5\% \\ \\ \approx63\% \end{gathered}[/tex]Final Answer:
63% of the people surveyed shop at a local grocery store.
calculate the slope of a line passing through the given points (5,-2) and (5,-3)
Answers
Given the points (5,-2) and (5,-3), we can find the slope of the line that passes through them with the following formula:
[tex]\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ \Rightarrow m=\frac{-3-(-2)}{5-5}=\frac{-1}{0} \end{gathered}[/tex]since we have that the slope is not defined, the line that passes through the points (5,-2) and (5,-3) is the vertical line x=5
Find the equation of the linear function x 1 2 3 4 y 1 6 11 16
Answers
We solve as follows:
*We determine first the slope (m), that is:
[tex]m=\frac{6-1}{2-1}\Rightarrow m=5[/tex]Now, using the slope and one point of the line, we replace in the following expression:
[tex]y-y_1=m(x-x_1)[/tex]We can use any point of the line, but I will be using the point (1, 1), that is:
[tex]y-1=5(x-1)[/tex]Now, we solve for y:
[tex]\Rightarrow y-1=5x-5\Rightarrow y=5x-4[/tex]A construction crew is lengthening a road. Let y represent the total length of the road (in miles). Let x represent the number of days the crew has worked. Suppose that x and y are related by the equation y=53 + 2x. Answer the questions below. Note that a change can be increased or a decrease. For an increase, use a positive number. For a decrease, use a negative number. What was the road’s length when the crew started working?What is the change per day in the road’s length?
Answers
Given the equation:
[tex]y=53+2x[/tex]Where y represents the total length of the road (in miles), and x represents the number of days the crew has worked.
(a) What was the road’s length when the crew started working?
When the crew started working we have x = 0. Then:
[tex]\begin{gathered} y=53+2\cdot0 \\ \therefore y=53\text{ miles} \end{gathered}[/tex]The road's length is 53 miles.
(b) What is the change per day in the road’s length?
The change per day in the road's length is 2 miles/day.
A random variable X follows a normal distribution with a mean of 150 and a standard deviation of sigma. If we know that P(120 < X < 180) = 0.95, then, according to the 68-95-99.7 rule, the value of sigma is approximately:
a.
20
b.
15
c.
40
d.
30
e.
60
Answers
The value of sigma according to the 68-95-99.7 rule is 15.
What is the 68-95-99.7 rule?This is the informal term that is used in statistics to remember the percentage of values that are in the interval of a distribution in statistics.
We have the mean = 150
the interval is given as P(120 < X < 180)
based on this rule, 95 percent of the data lies in the u - 20 and u + 20 region
Such that we would have
u - 2α < x < u + 2α = 0.95
we have
u - 2α = 120
150 - 2α = 120
2α = 150 - 120
2α = 30
divide through by 2
α = 15
Sigma is given as 15
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Rosa needs to build a wall. She has to start the wall with one postand then every 5.75 feet put another post. The wall will be166.75 feet long. How many posts will she need?
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For every 5.75 feet, here is one post.
Determine the number of posts in a wall of 166.75 feet.
[tex]\begin{gathered} p=\frac{166.75}{5.75} \\ =29 \end{gathered}[/tex]So 29 posts needed for the wall.
A car used 15 gallons of gasoline when driven 315 miles. Based on this information, which expression should be used to determine the unit rate of miles per gallon of gasoline?
Answers
Given trhat a car used 15 gallons of gasoline to cover 315 miles.
The expression that will be used to determine the unit rate of miles per gallon of gasoline is:
[tex]\frac{315\text{ miles}}{15\text{ gallons}}[/tex]ANSWER:
[tex]\frac{315\text{ miles}}{15\text{ gallons}}[/tex]it says i need to find the shortest distance between the point and the line for geometry honors, how would i figure it out
Answers
The given line equation is,
[tex]3x-y=-6[/tex]The given point is ,
[tex](5,1)[/tex]The graph will look like this,
let us rewrite the line equtaion as ,
[tex]3x-y+6=0[/tex]now, let us compare with the general equation of line,
[tex]Ax+By+C=0[/tex]then, A= 3,B=-1 and c= 6.
let us use the formula,
[tex]\begin{gathered} d=\frac{|Ax+By+c|}{\sqrt[]{A^2+B^2}} \\ d=\frac{|3\times5+(-1)\times1+6|}{\sqrt[\square]{3^2+(-1)^2}} \\ d=\frac{|15-1+6|}{\sqrt[\square]{9+1}} \\ d=\frac{20}{\sqrt[\square]{10}} \\ d=6.32 \end{gathered}[/tex]The shortest distance is 6.32 .
This is a non graded practice that I am doing. I don’t under these questions 5-11
Answers
7. The intersection of two intersecting lines is a point.
In the given image, we see that lines NQ and ML intersect at point P.
Therefore, the intersection of NQ and ML is P.
A teacher determines the linear equation y=12x + 40 best models the number of points a student should earn on a test, y, if the student studies for x hours. Which statement is true
Answers
Given the equation:
y = 12x + 40
Where x represents the number of hours and y represents the number of points the student should earn.
To find the correct statement substitute the number of hours and points given for x and y respectively. If the left hand side of the equation equals the right hand side then the statement is the true.
We have:
1. A student who studies for 3 hours should earn about 76 points.
x = 3
y = 76
Substitute 3 for x and 76 for y.
y = 12x + 40
76 = 12(3) + 40
76 = 36 + 40
76 = 76
This statement is true.
2.
Sketch the graph of each line.
24)
y=3/5x-4
Answers
The graph of the given line is attached below.
We are given the line:-
y = (3/5)x - 4
We will find the x and y intercepts of the line to plot in the graph.
As the equation is already in the slope intercept form, we can write,
The y-intercept of the line is -4.
Hence, the coordinates of the point will be (0,-4).
To find the x - intercept of the line we will put y = 0 in the given line.
0 = (3/5)x - 4
4 = 3x/5
x = 20/3
The coordinates of the point will be (20/3,0).
We can plot these points to get the desired graph.
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