100 Points! Write a polynomial function of least degree with integral coefficients that have the given zeros of -1,1,i√6. Photo attached. Please show as much work as possible. Thank you!
Answers
The required polynomial \(f(x) = x^4+5x^2-6\) has the zeros -1, 1, \(i\sqrt6\), and \(-i\sqrt6\).
What is a polynomial ?
A polynomial is a mathematical expression that consists of variables and coefficients, combined by addition, subtraction, and multiplication, but not division by a variable.
To write a polynomial function of least degree with integral coefficients that has zeros of -1, 1, and \(i\sqrt{6\), we need to include their conjugates as well. This is because complex roots always come in conjugate pairs.
The conjugate of \(i\sqrt{6\) is \(-i\sqrt{6\), so our polynomial function will have the following zeros: -1, 1, \(i\sqrt{6\), and \(-i\sqrt{6\).
To find the polynomial function, we can use the fact that the product of the factors of a polynomial is equal to the polynomial itself. So, we can start by multiplying out the factors:
\((x+1)(x-1)(x-i\sqrt6)(x+i\sqrt6)\)
Expanding this out gives:
\((x^2-1)(x^2+6)\)
Multiplying these two expressions gives:
\(x^4+5x^2-6\)
So the polynomial function of least degree with integral coefficients that has zeros of -1, 1, and \(i\sqrt{6\) is \(f(x) = x^4+5x^2-6\)
To verify that this polynomial has the desired zeros, we can factor it using the difference of squares:
\(x^4+5x^2-6 = (x^2-1)(x^2+6) = (x-1)(x+1)(x^2+6)\)
And the quadratic factor has no real roots, so it must be the factorization \((x-i\sqrt6)(x+i\sqrt6)\).
Therefore, \(f(x) = x^4+5x^2-6\) has the zeros -1, 1, \(i\sqrt6\), and \(-i\sqrt6\), as required.
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Related Questions
When I add 45 to a certain number and divide the sum by 2, the result is the same as 5 times the number. What is the number?
Answers
Answer:
5
Step-by-step explanation:
(45 + x) / 2
add 45 to a certain number and divide the sum by 2
= 5 × x
is 5 times the number
(45 + x)/2 = 5x
45 + x = 10x
45 = 9x
x = 5
y + 12 = -30
Please help
Answers
What do you want help with? You didn't ask a question...
18. Multiply, then check your work by switching factors.
a. 693 x 83
b. 910 x 45
c. 38 x 84
d. 409 x 89
Answers
The requried, Multiplies(with switching factors.) area given below,
a.
693 x 83 = 57489
83 x 693 = 57489
The answer is 57489.
b.
910 x 45 = 40950
45 x 910 = 40950
The answer is 40950.
c.
38 x 84 = 3192
84 x 38 = 3192
The answer is 3192.
d.
409 x 89 = 36401
89 x 409 = 36401
The answer is 36401.
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I am divisible by 3.
I am an even number.
I am the missing number
in 48/x=8.
Who am I?
Answers
Answer:
You are 6
Step-by-step explanation:
8×6=48 and 6/3=2
6 is even and fits in all of these areas.
Hope this helps.
Mark brainliest if correct.
–3(x – 14) + 9x = 6x + 42
Answers
Answer:
All real numbers
0 = 0
Step-by-step explanation:
Hope i'm right
Solving this equation results in the statement 0 = 0. Because this is a true statement, the equation has infinitely many solutions.
Answer:
Step-by-step explanation:
-3x + 42 + 9x = 6x + 42
6x + 42 = 6x + 42
0 = 0
infinitely many solutions
Example. Kennedy plans to completely cover the wall behind her kitchen sink with square tiles. The wall is a rectangle that is 15 in. high and 24in. long. Each square tile will have whole-number side lengths. What is the side length of the largest square tile that Kennedy can use?
The side length of the tile must divide both 15 in. and 24 in.
List the factors of 15: 1, 3, 5, 15
List the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
The only whole numbers that divide into both 15 and 24 are 1 and 3. The greatest common factor of 15 and 24 is 3. The side length of the largest tile Kennedy can use is 3 in.
1. Suppose the wall in the Example is 16 in. high and 28 in. long. a. What is the side length of the largest square tile that Kennedy can use? Show your work
Answers
9514 1404 393
Answer:
4 in
Step-by-step explanation:
Factors of the numbers are ...
16 = 2^4
28 = 2^2·7
The greatest common factor is 2^2 = 4.
The largest square tile Kennedy could use on a 16 in × 28 in space is 4 in square.
Answer:
Step-by-step explanation:
o
In an insect colony, there are 270 insects after 7 days. If there were initially 80 insects, how long will it take the population to grow to 700 insects?
Answers
Step-by-step explanation:
Total insects = 270
Days = 7
If 80 insects to 700 insects
Days = 700 ÷ 80 = 8.75
It will take 8.75 days
4. As part of your retirement planning, you purchase an annuity that pays 4 % annual
interest compounded quarterly
a. If you make quarterly payments of $900 how much will you have saved in 5
years?
b. Instead, if you make quarterly payments of $450, how much will you have saved
in 10 years?
Answers
9514 1404 393
Answer:
a. $19817.10
b. $21998.87
Step-by-step explanation:
The formula for the future value of an annuity with payments "A" and interest at rate r compounded quarterly for t years is ...
FV = A((1 +r/4)^(4t) -1)/(r/4)
The attachment shows this evaluated for ...
a. A = 900, r = 0.04, t = 5. FV = $19817.10
b. A = 450, r - 0.04, t = 10. FV = 21,998.87
Write an equation for a rational function with:
Vertical asymptotes at x=5 and x=2
X-intercepts at x=1 and x=4
y-intercept at 3
y=
Answers
The required equation of the rational function is f(x) = -8(x-1)(x-4) (ax + 3/2)/(3(a+3/2)(x-5)(x-2))
Equation of rational functionTo create a rational function with the given characteristics, we can start by setting up the general form for a rational function:
f(x) = (ax + b) / (cx + d)
To ensure that the function has vertical asymptotes at x=5 and x=2, we need the denominators of our function to be (x-5) and (x-2), respectively. So we can rewrite our function as:
f(x) = (ax + b) / ((x-5)(x-2))
To make sure that the function passes through the x-intercepts at x=1 and x=4, we need the numerator of the function to be equal to zero at those values of x. So we add two factors of (x-1) and (x-4) to the numerator:
f(x) = k(x-1)(x-4) (ax + b) / ((x-5)(x-2))
where k is a constant that we will determine shortly.
To find the value of k, we use the fact that the function has a y-intercept at (0,3). This means that when x=0, f(x) = 3. So we substitute these values into our equation and solve for k:
3 = k(0-1)(0-4) (a(0) + b) / ((0-5)(0-2))
3 = 20k b / 10
b = 3/2
Now our equation becomes:
f(x) = k(x-1)(x-4) (ax + 3/2) / ((x-5)(x-2))
To find the value of k, we can use one of the x-intercepts. Let's use x=1:
0 = k(1-1)(1-4) (a(1) + 3/2) / ((1-5)(1-2))
0 = -3k/4 (a + 3/2)
Solving for k:
k = -8/(3(a+3/2))
Now our equation becomes:
f(x) = -8(x-1)(x-4) (ax + 3/2) / (3(a+3/2)(x-5)(x-2))
Therefore, an equation for the rational function with vertical asymptotes at x=5 and x=2, x-intercepts at x=1 and x=4, and y-intercept at 3 is:
f(x) = -8(x-1)(x-4) (ax + 3/2)/(3(a+3/2)(x-5)(x-2))
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A regular hexagon is inscribed into a circle. Find the length of the side of the hexagon, if the radius of the circle is 12 cm.
A. 20 cm
B. 18 cm
C. 16 cm
D. 12 cm
E. None of these
Answers
Answer:
D, 12 cm.
Step-by-step explanation:
First, draw out the circle with a hexagon in it. I did this by dividing the circle into thirds and then halving each of those because it was easier to visualize. I ended up with something like the image below. Now, we know that the sum of all of the angles that come together at the center of the circle is going to be 360. We can divide 360 by 6 to get 60. The top angle of the triangle, the angle closest to the center, is 60 degrees. We also know that the length of all of these legs that are going out will be the same, because the radius of the circle is constant. This makes each triangle an isosceles triangle. Given the isosceles base angle theorem, both base angles of the triangle must be the same. The sum of all angles in a triangle is 180 degrees. So the equation for x, one of the angles, would be 60 (angle closest to center) + 2x = 180. Let's subtract 60 from both sides to get 2x = 120. Divide both sides by two to get x = 60. Okay, so now we know that all angles in the triangle are 60 degrees. By the equilateral triangle theorem, we know these triangles must be equilateral because all of their angles are 60 degrees. Equilateral triangles also have the same side lengths. Because 12 is one side length, then all side lengths = 12. Therefore, the length of one side of the hexagon is 12.
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The graph compares the weights in pounds of 100 dogs and cats that are brought in
to a veterinarian's office. Using the medians, how much more does a typical dog weigh than a typical cat?
Answers
Answer:20
Step-by-step explanation:this is wrong
Answer:
40
Step-by-step explanation:
subtract median of dogs and median of cats
In a bag there are 3 red marbles, 2 yellow marbles, and 1 blue marble. What is the likelihood of a yellow marble being selected on the first draw?
likely
unlikely
even chance
certain
Answers
Therefore, it is not even chance, but it is also not very unlikely. It is moderately likely that a yellow marble will be selected on the first draw.
What is Probability?Probability is a branch of mathematics that deals with the study of random events and their likelihood of occurring. It is a measure of the likelihood or chance of an event happening. Probability is expressed as a number between 0 and 1, where 0 means the event is impossible, and 1 means the event is certain.
In other words, probability is a way of quantifying uncertainty. It is used in various fields, such as statistics, physics, finance, engineering, and more, to help make predictions and decisions based on uncertain information. Probability theory provides a set of rules and tools for analyzing and manipulating random variables and events, and for calculating the probability of complex events.
The likelihood of a yellow marble being selected on the first draw can be calculated by dividing the number of yellow marbles by the total number of marbles in the bag:
likelihood of selecting a yellow marble = number of yellow marbles / total number of marbles
So, in this case, the likelihood of selecting a yellow marble on the first draw is:
likelihood of selecting a yellow marble = 2 / (3 + 2 + 1) = 2/6 = 1/3
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Identify the outlier in the data set, and determine how the outlier affects the mean, median, and mode of the data.
95, 88, 72, 26, 69, 78, 97
GROUP OF ANSWER CHOICES
A. 97; adding the outlier decreased mean by 5 and median by 8.2. The mode did not change.
B. 26; adding the outlier decreased mean by 8.2 and median by 5. The mode did not change.
C. 97; adding the outlier increased mean by 8.2 and median by 5. The mode did not change.
D. 26; adding the outlier increased mean by 5 and median by 8.2. The mode did not change.
Answers
The outlier in the data set is 97.
Correct. Adding an outlier of 26 increases the mean by \((95+88+72+69+78)/5 = 80.4 to (95+88+72+69+78+26)/6\) = 72.67. This is a decrease of 7.73. The median is 78, which is not affected by the addition of the outlier. The mode is not affected either. Then option C is correct.
A. Incorrect. Adding an outlier of 97 increases the mean by (95+88+72+26+69+78+97)/7 = 70.1 to (95+88+72+26+69+78+97)/8 = 73.38. This is an increase of 3.28. The median is the middle value when the data is arranged in order and adding an outlier of 97 does not affect the median of the data set, which is 78. The mode, which is the value that appears most frequently in the data, is There is no mode as no value is repeated.
B. Incorrect. Adding an outlier of 97 decreases the mean by (95+88+72+26+69+78)/6 = 70 to (95+88+72+26+69+78+97)/7 = 70.1. This is an increase of 0.1. The median is 78, which is not affected by the addition of the outlier. The mode is not affected either.
C. Incorrect. Adding an outlier of 97 increases the mean by (95+88+72+26+69+78)/6 = 70 to (95+88+72+26+69+78+97)/7 = 70.1. This is an increase of 0.1. The median is 78, which is not affected by the addition of the outlier. The mode is not affected either.
D. Correct. Adding an outlier of 26 increases the mean by (95+88+72+69+78)/5 = 80.4 to (95+88+72+69+78+26)/6 = 72.67. This is a decrease of 7.73. The median is 78, which is not affected by the addition of the outlier. The mode is not affected either.
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Which is greater, (1/2)⁴² or (1/2)⁴⁵? Why?
Answers
Answer:
1/2^42 because the 45 will be dividing it b half 3 more times
Step-by-step explanation:
gordon drives 220 miles from new york to boston in 4 hours. what is his car's average speed?
Answers
Answer:
55 mph
Because he drives to new york to boston in 4 hours and its 220 miles so 220 is the miles and you divide by the hours so 4 hours so he drove 55 mph
Find the equation of a straight line cutting off the y-intercept 4 from the axis of y and inclined to 60° with the positive direction of X-axis.
Answers
The linear function is given as follows:
\(y = \sqrt{x} + 4\)
How to define a linear function?The slope-intercept equation for a linear function is presented as follows:
y = mx + b
In which:
m is the slope.b is the intercept.The y-intercept is of 4, hence the parameter b is given as follows:
b = 4.
The line is inclined to 60° with the positive direction of X-axis, hence the slope m is given as follows:
m = tan(60º)
\(m = \sqrt{3}\)
Thus the function is given as follows:
\(y = \sqrt{x} + 4\)
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All pictures are below please help Sandi had the following problem on her math test: . She showed her work on the test as follows: Given: Step 1: Step 2: Step 4: Step 5: In at least two paragraphs, explain the steps Sandi took to arrive at her answer. If she made any errors in her process clearly explain what she did wrong and how she could have made the error. If there is an error, provide the correct steps in your explanation of Sandi’s work. Use correct math terminology wherever appropriate.
Answers
Answer:
18
Step-by-step explanation:
6×4 +2 :2 - 7
24 +1 - 7
25 - 7
18
1 step 6×4
2step 2:2
3step 24 +1
4step 25 - 7
Answer:
18
Step-by-step explanation:
6 × 4 + 2 ÷ 2 - 7
24 + 1 - 7
25 - 7
= 18
thanks
log2(1/128)=x, then x =
Answers
The value of x will be -7. The expression is solved using the identity.
What is the definition of a logarithm?Exponents can also be written as logarithms. The other number is equal to a logarithm with a number base. It's the exact inverse of the exponent function.
Given expression;
\(\rm log_2(\frac{1}{128}) = x\)
The expression is solved using the identity as;
\(\rm log_e1 = x\\\ e^x=1\)
Apply the identity;
\(\rm log_2(\frac{1}{128}) = x \\\\ 2^x = \frac{1}{128}\\\\ 2^x = \frac{1}{2^7} \\\\ 2^x = 2^{-7} \\\\ x = -7\)
Hence the value of x will bed -7.
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What is the mean for the set of data? Pls help i will give brainliest
Answers
Answer:
3.3
Step-by-step explanation:
all the numbers add up to 33
and there is 10 numbers so divide 33 by 10 and you get 3.3
I need help with this question.
Answers
Answer: \(y=-\frac{7}{4}x-\frac{3}{4}\)
Step-by-step explanation:
Rearranging the line into slope-intercept form,
\(4y+7x=-6\\\\4y=-7x-6\\\\y=-\frac{7}{4}x-\frac{3}{2}\)
Since the center of dilation does not lie on the line, we can multiply the y-intercept by the scale factor of the dilation.
So, the equation of the line is \(y=-\frac{7}{4}x-\frac{3}{4}\)
The circle graphs shows the percentages of student participation in sports a certain school.
E
F
Other
11%
Track &
Field
14%
Baseball
15%
A
Football
22%
Basketball
17%
m EF =
Find m EF. Do not round.
Soccer
21%
C
B
Answers
From the given below solution the value of m EF is 25%.
What is circle graphs?A circle graph, also known as a pie chart, is a type of data visualization that displays data as a circular graph, divided into sectors that represent numerical proportions.
Each sector of the circle represents a percentage or proportion of the total data set, and the size of each sector is proportional to its corresponding value.
To find m EF, we need to add the percentages of participation in sports E and F:
m EF = percentage of participation in sports E + percentage of participation in sports F
m EF = 11% + 14%
m EF = 25%
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Find each value given the following function:
Answers
Answer:
Step-by-step explanation:
1) f(-4) --> if x < or equal to 3
2) 1/(-4)-4
3) The answer is - 1/8
PLEASE HELP!!!! Mia has 50 feet of fencing to make a rectangular kennel for her dog. She decides to use part of her hbuse as one side (width)
of the kennel. She needs a minimum enclosed area of 300 square feet. Let/represent the length of the kennel.
Which inequality represent this situation, and what is the solution set?
Select all the correct answers.
Answers
Answer: C and D. Trust me bruh.
Step-by-step explanation:
Answer:
A C D
Step-by-step explanation: Got it correct
Please Help
There are several marbles of different colours in a bag in front of you. Your job is to select three marbles and record the colour of each marble you chose. In the bag, there are 22 blue marbles, 17 red marbles, 8 yellow marbles and 3 purple marbles. What are the chances of selecting one red marble, one yellow marble and a blue marble?
Assume that once each marble is chosen, it is not returned to the bag.
Answers
The probability is 2.54%.
What is probability?The formula of the favorable outcome over the total outcomes is used to describe probability.
given, There are several marbles of different colors in a bag in front of you. Your job is to select three marbles and record the color of each marble you chose. In the bag, there are 22 blue marbles, 17 red marbles, 8 yellow marbles, and 3 purple marbles.
The final probability if conditions are not dependent on each other is multiple possibilities of happening these events.
Probability of getting desired outcomes = (17 / 50) (8/49) ( 22/48)
Therefore the probability of selecting one red marble, one yellow marble, and a blue marble is 2.54%.
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Question 12 of 22
Select the action you would use to solve = 16. Then select the property that
justifies that action.
A. Property: Multiplication property of equality.
B. Action: Add 4 to both sides.
C. Property: Addition property of equality.
D. Action: Divide both sides by 4.
E. Property: Division property of equality.
OF Action: Multiply both sides by 4.
Answers
Answer:
The correct answer is:
D. Action: Divide both sides by 4.
E. Property: Division property of equality.
17.
Use Mathematical induction to prove:
4 + 9 +14 +…..+(5n-1) = ( n/2) (5n+3)
Answers
Answer: plse use the given formula in rs aggarwal book
Step-by-step explanation:
To prove by mathematical induction, let Pₙ be the equation:
Let Pₙ be 4 +9 +14 +...+(5n -1)= (n/2)(5n +3).
1) Base step: prove P₁ is trueWhen n=1,
LHS= 4
RHS
\( = \frac{1}{2} (5 + 3)\)
\( = \frac{1}{2} (8)\)
= 4
= LHS
∴ P₁ is true.
2) Assume Pₙ is true,i.e. 4 +9 +14 +...+(5n -1)= (n/2)(5n +3)
3) Induction step: prove Pₙ₊₁ is trueFind Pₙ₊₁ by substituting (n+1) into n on both sides.
Pₙ₊₁: 4 +9 +14 +...+(5n -1)+ [5(n+1) -1]=
\(( \frac{n + 1}{2} )[5(n + 1) + 3]\)
Prove that the LHS is equal to the RHS.
LHS
= 4 +9 +14 +...+(5n -1)+ [5n +5-1]
\(= \bf{4 +9 +14 +...+(5n -1)}+ (5n +4)\)
Substitute the value of Pₙ in (2) into bolded expression:
\( = \frac{n}{2} (5n + 3) + 5n + 4\)
\( = \frac{5}{2} n^{2} + \frac{3}{2}n + 5n + 4 \)
\( = \frac{1}{2} (5 {n}^{2} + 3n + 10n + 8)\)
\( = \frac{1}{2} (5 {n}^{2} + 13n+ 8)\)
\( = \frac{1}{2} (n + 1)(5n + 8)\)
\( = \frac{n + 1}{2} (5n + 8)\)
\( = \frac{n + 1}{2} [5(n + 1) + 3]\)
= RHS
(shown)
4) Write concluding statements∴ If Pₙ is true, Pₙ₊₁ is true
Since P₁ is true, Pₙ is true for all positive integers of n.
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https://brainly.com/question/24393371I need help again this making so tried so help me pls I beg I want to sleep
Answers
Answer:
32229 feetStep-by-step explanation:
The high and low points are
31098 feet- 1131 feetThe difference in elevation is:
31098 - (-1131) = 32229 feetStep-by-step explanation:
Answer:
32229 feet
Step-by-step explanation:
The high and low points are
31098 feet - 1131 feet
The difference in elevation is:
31098 - (-1131) = 32229 feet
Hope it will help you in your need
Select the correct answer.
Which equation is true for the value b= 10?
O A. 2(b + 4) = 16
OB.
2(b + 2) = 40
OC. 316 - 2) = 24
OD. 2(8 + b) = 42
O E. 3(b - 4) = 20
Answers
Answer:
C. 3(b - 2) = 24Step-by-step explanation:
b = 10
Substitute b in every equation below and compare the results
A. 2(b + 4) = 16
2(10 + 4) = 2(14) = 28 ≠ 16, incorrectB. 2(b + 2) = 40
2(10 + 2) = 2(12) = 24 ≠ 40, incorrectC. 3(b - 2) = 24
3(10 - 2) = 3(8) = 24, correctD. 2(8 + b) = 42
2(8 + 10) = 2(18) = 36 ≠ 42, incorrectE. 3(b - 4) = 20
3(10 - 4) = 3(6) = 18 ≠ 20, incorrectAnswer:
C. 3(b - 2) = 24..
A pools diving board is 12 feet above water in the pool. The water is 20 feet deep. How far is the diving board from the bottom of the pool?
Answers
32 feet
From the surface of the water to the bottom is 20 feet, and from the surface to the diving board is 12 feet. add 20 and 12 and you get 32
Hello help help please help help
Answers
Answer:
\(8x + 6 > 38 \\ 8x > 38 - 6 \\ 8x > 32 \\ x > \frac{32}{8} \\ \color{yellow} \boxed{x > 4}\)