The polynomial function of the least degree with integral coefficients that has the zeros [tex]1 + 3i[/tex], [tex]-2i[/tex] is f(x) = x⁴ - 2x³ + 14x² - 8x + 40.
Writing a polynomial function of the least degreeFrom the question, we are to write a polynomial function of the least degree with integral coefficients that have the given zeros.
The given zeros are [tex]1+3i[/tex],[tex]-2i[/tex].
If [tex]1 + 3i[/tex] and [tex]-2i[/tex]are zeros of a polynomial function with integral coefficients, then their conjugates [tex]1 - 3i[/tex] and [tex]2i[/tex] are also zeros of the function.
To find the polynomial function, we can use the fact that if r is a zero of a polynomial function, then (x - r) is a factor of the function. Thus, we can start by writing out the factors corresponding to each of the zeros:
[tex](x - (1 + 3i))(x - (1 - 3i))(x - (-2i))(x - 2i)[/tex]
Next, we can simplify these factors by multiplying them out:
[tex][(x - 1) - 3i][(x - 1) + 3i](x + 2i)(x - 2i)\\= [(x - 1)^2 - (3i)^2](x^2 - (2i)^2)[/tex]
= [(x - 1)² + 9](x² + 4)
Expanding the terms, we get:
(x² - 2x + 10)(x² + 4)
Multiplying out the factors, we obtain:
x⁴ - 2x³ + 10x² + 4x² - 8x + 40
Simplifying this expression, we get:
x⁴ - 2x³ + 14x² - 8x + 40
Hence, the polynomial function is f(x) = x⁴ - 2x³ + 14x² - 8x + 40
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How many 2-digit numbers contain the digit 7 only once
Answer:
18
Step-by-step explanation:
Counting:
07 17 27 37 47 57 67 70 71 72 73 74 75 76 78 79 87 97
notice we can ALSO use variation with no repetition equation with n=9 (numbers from 0 to 10) and p=1 (two digits) but the value can be in the first position or the second position
[tex]\frac{9!}{(9-8)!}[/tex] + [tex]\frac{9!}{(9-8)!}[/tex][tex]=9+9=18[/tex]
Jillian worked 62 hours this week and last week. She worked x hours last week and y hours this week. She worked 20% more hours this week than last week. What pair of equations can be used to find x and y?
The pair of equations that can be used to find x and y are:
x + y = 62
(1.02)x = 7
What are the pair of equations?The form of the pair of equations would be:
hours worked last week + hours worked this week = total hours worked both weeks equation 1
(1 + percent increase/100) x hours worked last week = hours worked this week equation 2
x + y = 62 equation 1
(1 + 20/100) ×x = y
(1.02)x = 7 equation 2
The two equations can be solved using either of these three methods:
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Mary, Nick, Jane have 60$ altogether.
Mary has a fourth part and Nick has a fifth.
HOW MUCH MONEY DOES JANE HAVE??
Mary has 60/4 = 15$
Nick has 60/5 = 12$
The total amount of money that Mary and Nick have is 15$+12$ = 27$
To find out how much money Jane has, we need to subtract the total amount that Mary and Nick have from the total amount of money they have altogether:
60$ - 27$ = 33$
Therefore, Jane has 33$.
Answer: $33
Step-by-step explanation:
1/5 of $60 is 12. 1/4 of $60 is 15. 15 + 12 = 27. To find out how much money Jane has, simply subtract 60 - 27 to get $33.
What is the ratio of the amount spent on staff salaries to the total budget for the youth programs?
1:10
1:8
1:5
1:4
1:2
Answer:1:2
Step-by-step explanation:
the length of a rectangle is 8in. more than 11 times the width. The perimeter of the rectangle is 184 inches. Find the measure of the length and width of the rectangle.
Considering the definition of perimeter, the length and width of the rectangle is 85 in and 7 in respectively.
Definition of perimeterThe perimeter of a two-dimensional figure is the distance around the figure. That is, the perimeter of a flat geometric figure is called the length of its contour.
The perimeter is the measurement obtained as a result of the sum of the sides of a flat geometric figure.
Perimeter of a rectangleA rectangle is a geometric figure that has two pairs of sides of equal length. The expression to calculate the perimeter of a rectangle is:
Perimeter= 2× length + 2× width
Length and width of the rectangleIn this case, you know
The length of a rectangle is 8 in. more than 11 times the width. → lenght= 11×width + 8The perimeter of the rectangle is 184 inchesReplacing in the definition of the perimeter of the rectangle:
184= 2× (11×width + 8) + 2× width
Solving:
184= 2×11×width + 2×8 + 2× width
184= 22×width + 16 + 2× width
184 - 16= 22×width + 2× width
168= 24×width
168÷ 24= width
7 in= width
So, the length can be calculated as:
lenght= 11×7 in + 8 in
lenght= 85 in
Finally, the length of the rectangle is 85 in and the width of the rectangle is 7 in.
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help me graph this pls
The frisbee would spend 2.62 seconds in air before touching ground.
What is an equation?An equation is an expression that shows the relationship between numbers and variables using mathematical operators such as addition, subtraction, exponent, multiplication and division.
Let h represent the height of the frisbee after t seconds.
h = -16t² + 40t + 5
The time the Frisbee is in air is at h = 0, hence:
-16t² + 40t + 5 = 0
t = 2.62 seconds
The frisbee would spend 2.62 seconds in air
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a) Note that given the above function the height of the Frisbee each second it is in the air are:
t=1; h=29 feett=2; h = 21 feett =3; h = -19b) The frisbee lasts t [tex]\approx[/tex] 2.6 (s) in the air.
What is the justification for the above response?Recall that the function given is -16t² + 40t + 5
t=0; and h = 5
a)
Where t is 1
h = -16t² + 40t + 5
h= -16(1)² + 40(1) + 5
h= -16 + 40 + 5
h= 24 + 5
h= 29, thus, where = 1, h = 29
Using a similar sequence above, where t is
t=2; h = 21 feet; and where
t =3; h = -19
b) in this case, we must set h(t) = 0
⇒ -16t² + 40t + 5 = 0
To solve for t in the equation -16t² + 40t + 5 = 0, we can use the quadratic formula:
t = (-b ± √(b² - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation.
In this case, a = -16, b = 40, and c = 5, so substituting these values into the quadratic formula, we get:
t = (-40 ± √(40² - 4(-16)(5))) / 2(-16)
Simplifying:
t = (-40 ± √(1600 + 320)) / (-32)
t = (-40 ± √(1920)) / (-32)
t = (-40 ± 8 √(30)) / (-32)
t = (5/4) ± (√(30)/4)
Thus we have:
t = (5/4) + (√(30)/4) or t = (5/4) - (√(30)/4)
t₁ = 1.25 + (5.477225575051661134569697828008/4) or
t₂ = 1.25 - (5.477225575051661134569697828008/4)
t₁ = 1.25 + 1.369306393762915283642424457002
t₁ [tex]\approx[/tex] 2.6
t₂ = 1.25 - 1.369306393762915283642424457002
t₂ [tex]\approx[/tex] -0.119
Thus the Frisbee lasts approximatley 2.6 seconds in the air.
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What is the outlier in the set of data below:
12, 8, 15, 45, 7, 13
a
7
b
8
c
15
d
45
Answer:
D
Step-by-step explanation:
3984 is what percent of 24.9?
Answer:
percentage = (part/whole) x 100%
where "part" is the value we are trying to express as a percentage of "whole". In this case, "part" is 3984 and "whole" is 24.9.
So we can substitute these values into the formula:
percentage = (3984/24.9) x 100%
percentage = 159.838% (rounded to three decimal places)
Therefore, 3984 is approximately 159.838% of 24.9.
Answer:
3984 is 16,000% of 24.9
Step-by-step explanation:
percentage = (3984÷24.9) x 100%
percentage = 159.838%
or percentage ≈ 16,000%
Susie is three times as older as her sister Jenny. Paula is 7 years younger
than Jenny.
Write an expression in simplest form that represents the sum of their ages.
P - 7 = J is an expression in simplest form that represents the sum of their ages.
What is a mathematical expression?
A mathematical expression is a phrase that includes at least two numbers or variables, at least one arithmetic operation, and the expression itself. This mathematical operation may be addition, subtraction, multiplication, or division.
Any mathematical statement made up of numbers, variables, and an operation between them is called an expression or an algebraic expression.
Susie is three times as older as her sister Jenny.
3S = J
Paula is 7 years younger than Jenny.
P - 7 = J
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Classifying a Parallelagram
The values are, ∠BCA = 65°, ∠B = 72°, x = 3
Define the term parallelogram?A quadrilateral with two sets of parallel sides is referred to as a parallelogram. As a result, a parallelogram's opposite sides are parallel and congruent in length, and its opposite angles are similarly congruent.
Given that the diagram ABCD is a parallelogram,
So, BC = AD (parallel sides are equal)
2x = 6
x = 3
BC ║ AD and AC is a transversal line,
then, ∠BCA = ∠DAC (alternate interior angle)
∠BCA = 65°
Opposite angles are equal for parallelogram then,
∠B = ∠D
∠B = 72°
Therefore, the values are, ∠BCA = 65°, ∠B = 72°, x = 3
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question is in the picture . please help me
Answer:
C m∠A + m∠B = 180°
Step-by-step explanation:
Because angle A and angle B are inscribed angles on the diameter they are half of 180° which is 90°
When both of these angles are added they equal 180°
Select the correct answer. Consider functions f and g. The picture shows a one-to-one function diagram. x has values of 1, 2, 3, and 4, and g of x has values of minus 1, minus 2, minus 4, and minus 8. Every x value has a relation in g of x. What is the value of x when (f o g)(x)= -8? A. -4 B. 0 C. 3 D. 4
Holly opened a savings account at a local bank. She deposited $1,300.00 into the account 6 years ago. If the account earns an annual simple interest rate of 6.8%, how much interest has she earned?
Answer:
Interest = $530.4
Step-by-step explanation:
We know:
Interest = Principal x Rate x Time
First, put in the numbers:
Interest = $1,300 x 6.8% x 6 years
Interest = $1,300 x [tex]\frac{6.8}{100}[/tex] x 6 years
Interest = $13 x 6.8 x 6 years
Interest = $88.4 x 6
Interest = $530.4
Consider the functions f and g.
Which statement is true about these functions ?
The correct statement regarding the average rate of change of the functions f(x) and g(x) on the interval [-2,2] is given as follows:
Over the interval [-2,2], the function f is increasing at a faster rate than function g is decreasing.
How to obtain the average rate of change?The average rate of change of a function is given by the change in the output of the function divided by the change in the input of the function. Hence we must identify the change in the output, the change in the input, and then divide then to obtain the average rate of change.
For the function f(x), the numeric values at x = -2 and x = 2 are given as follows:
f(-2) = (-2)³ + 5(-2)² - (-2) = 14.f(2) = (2)³ + 5(2)² - 2 = 26.Hence the rate is of:
(26 - 14)/(2 - (-2)) = 12/4 = 3. -> positive rate -> increasing.
For function g(x), the rate is given as follows:
(-16 - 4)/4 = -5.
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A farmer notices that there is a linear relationship between the number of bean stalks, n, she plants and the yield, Y. When she plants 3 stalks, each plant yields 115 ounces of beans. When she plants 8 stalks, each plant yields 190 ounces of beans.
Answer:
Step-by-step explanation:
We can use the information given to find the equation of the line that represents the relationship between the number of bean stalks and the yield. The equation of a line is typically given by the slope-intercept form, y = mx + b, where y is the dependent variable (in this case, the yield), x is the independent variable (the number of bean stalks), m is the slope, and b is the y-intercept.
To find the slope of the line, we can use the formula:
m = (Y2 - Y1) / (n2 - n1)
where (n1, Y1) and (n2, Y2) are two points on the line. We can use the two data points given in the problem to find the slope:
m = (190 - 115) / (8 - 3) = 15
To find the y-intercept, we can use the point-slope form of a line, which is:
y - Y1 = m(x - n1)
where (n1, Y1) is one of the points on the line. We can use the point (3, 115):
y - 115 = 15(x - 3)
Simplifying:
y = 15x - 20
Therefore, the equation that represents the relationship between the number of bean stalks and the yield is:
Y = 15n - 20
This equation tells us that for each additional bean stalk planted, the yield increases by 15 ounces, and the y-intercept of -20 indicates that even if no bean stalks were planted, there would still be a yield of -20 ounces (which doesn't make physical sense in this case, but is a mathematical artifact of the linear regression).
An open rectangular box is to be constructed by cutting square corners out of a 16- by 16- inch piece of cardboard and folding up the flaps. A box formed this way is shown on the right. Find the value of x for which the volume of the box will be as large as possible.
Answer:
the value of x for which the volume of the box will be as large as possible is approximately 2.67 inches.
Step-by-step explanation:
Let x be the length of the side of the squares that are cut out of the corners of the cardboard. Then the dimensions of the base of the rectangular box will be:
Length = 16 - 2x (since two squares of side x are cut out of each end of the 16-inch length)
Width = 16 - 2x (since two squares of side x are cut out of each end of the 16-inch width)
The height of the box will be x, since that is the height of the squares that were cut out.
The volume of the box can be expressed as:
V = Length × Width × Height
V = (16 - 2x) × (16 - 2x) × x
V = 4x^3 - 64x^2 + 256x
To find the value of x that maximizes the volume of the box, we can take the derivative of V with respect to x and set it equal to zero:
dV/dx = 12x^2 - 128x + 256 = 0
We can solve this quadratic equation for x using the quadratic formula:
x = [128 ± sqrt(128^2 - 4 × 12 × 256)] / (2 × 12)
x = [128 ± sqrt(16384)] / 24
x = [128 ± 128] / 24
x = 10.67 or x = 2.67
Since x represents the length of a side of a square, it must be non-negative. Therefore, the only valid solution is x = 2.67 inches.
So, the value of x for which the volume of the box will be as large as possible is approximately 2.67 inches.
Answer: Let x be the side length of the square corners that are cut out of the cardboard. Then the dimensions of the base of the box (after the corners have been cut out and the flaps folded up) are 16-2x by 16-2x, and the height of the box is x.
The volume V of the box is given by:
V = (16-2x)(16-2x)(x)
Expanding this expression gives:
V = 4x^3 - 64x^2 + 256x
To find the value of x that maximizes V, we can take the derivative of V with respect to x and set it equal to zero:
dV/dx = 12x^2 - 128x + 256 = 0
Dividing both sides by 4 gives:
3x^2 - 32x + 64 = 0
This quadratic equation can be factored as:
(3x - 16)(x - 4) = 0
So the solutions are x = 16/3 and x = 4.
Since x must be less than half of 16 (the side length of the cardboard), we can reject x = 16/3. Therefore, the value of x that maximizes the volume of the box is x = 4 inches.
Step-by-step explanation:
A ladder 17 feet long is leaning against a wall. The bottom of the ladder is 8 feet from base of the wall. How far up the wall is the top of the ladder?
Round to the nearest tenth if necessary.
ANSWER IN FEET!!!
The top of the ladder is 15 feet up the wall.
What is right-angle triangle?
If one of the triangle's angles is 90 degrees, the triangle is said to be right-angled. The combined angles of the other two are 90 degrees. The triangle's base and perpendicular sides both include the right angle. The longest side of the three sides, known as the hypotenuse, is the third side.
Given that the height of the ladder is 17 feet. The distance between the base of the ladder and the wall is 8 feet.
It forms a right-angle triangle.
The hypotenuse of the triangle is 17 feet and the base is 8 feet.
Apply Pythagorean theorem:
Altitude² = Hypotenouse² - Base²
Altitude² = 17² - 8²
Altitude² = 289 - 64
Altitude² = 225
Take square root on both sides:
Altitude = ± 15
Since height can't be a negative number, Altitude = 15 feet.
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A right triangle has side lengths….
Answer:
sin x = d/e, cos x = f/e, tan x = d/f
Step-by-step explanation:
We will use trigonometry and SOH CAH TOA for this problem.
sin x = (opposite/ hypotenuse)
sin x = d/e
cos x = (adjacent / hypotenuse)
cos x = f/e
tan x = (opposite/adjacent)
tan x = d/f
You take a random token from a bag that contains 8 red, 7 green, and 4 blue tokens. Let R be the set of red tokens, G green tokens, and B blue tokens.
What is the probability that your token is not in G? Enter your answer as an unsimplified fraction.
Answer:If my calculations are right it should be 18
Step-by-step explanation:
Is y= |x| + 4 a relation or function
Answer: probably a function
Step-by-step explanation:
The port hole in the side of a ship is in the
shape of a circle with a 2 foot diameter.
The top of the port hole is 6 feet below
the surface of the water and the density of
the water is 62.4 pounds per cubic foot.
Find the total force on this port hole due
to liquid pressure, accurate to the nearest
whole number.
[?] pounds
Answer:
the total force on the port hole due to liquid pressure is approximately 7400 pounds.
Step-by-step explanation:
The area of the circle is A = πr^2, where r = d/2 = 1 foot is the radius of the circle. So, the area is A = π(1 ft)^2 = π ft^2.
The port hole is submerged in water, with a height of 6 feet. The pressure of water at a depth of h feet is given by the formula P = ρgh, where ρ = 62.4 lb/ft^3 is the density of water, and g = 32.2 ft/s^2 is the acceleration due to gravity.
The total force on the port hole due to liquid pressure is the product of the pressure and the area of the circle, so we have:
F = P × A = ρgh × A = 62.4 lb/ft^3 × 32.2 ft/s^2 × 6 ft × π ft^2 ≈ 7400 lb
Therefore, the total force on the port hole due to liquid pressure is approximately 7400 pounds.
Answer: 578490-=356478e
Step-by-step explanation:
part of a circle is 75849=n so we use all the equations so 56748=4 we know the answer by the book page 356 then on chemestry rules.
60 people plan to attend the dance and each person will eat 2 slices of cake. If each cake contains 8 slices, how many cakes should you order?
Answer: 15 Cakes
Step-by-step explanation:
2 slices per person, and 8 slices in 1 cake means that 4 people can share 1 cake.
So, you would do the amount of people (60) divided by the number of people that can eat 1 cake (4) to end up with ordering 15 cakes
Match the vocabulary terms to the correct definitions.piecewise function
A piecewise function is a function that has different definitions, based on the interval of the input values of the function.
What is a piecewise function?A piecewise function is a mathematical function that is defined by different rules over different intervals or subdomains of its domain. These different rules can take on different forms or expressions, allowing the function to behave differently on different parts of its domain.
Hence one example of a piecewise function is the absolute value function, which is defined as follows, depending on the signal of the input:
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What percent of the customers purchased fewer than 6 flowers
Finding the percentage of consumers who bought fewer than six flowers requires figuring out what proportion of the data is below six. [tex]40[/tex] % of customers purchased fewer than 6 flowers.
What is the parameter for calculating percent?Based on the box plot, we can see that the minimum value is 2, and the maximum value is 12. The box plot also shows that the median (50th percentile) is between 6 and 8.
To find the percent of customers who purchased fewer than 6 flowers, we need to determine what percent of the data is below 6.
From the box plot, we can see that the lower quartile (25th percentile) is at 2, and the upper quartile (75th percentile) is at 10.
Therefore, the interquartile range (IQR) is [tex]10 - 2 = 8.[/tex]
To find the percentage of customers who purchased fewer than 6 flowers, we need to find the percentile rank of 6.
Percentile rank = ((value below target) / (total values)) x 100
Percentile rank of [tex]6 = ((2 / 5) \times 100) = 40[/tex]%
Therefore, [tex]40[/tex] % of customers purchased fewer than 6 flowers.
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The given question is incomplete. the complete question is given below.
Rodrigo works at a flower shop. He recorded how many flowers each customer purchased
yesterday. This box plot shows the results.
Flowers purchased
++
2
6
8
10
12
What percent of the customers purchased fewer than 6 flowers?
%
I will mark you brainiest!
To prove a polygon is a rectangle, which of the properties listed must be included in the proof?
A) All side lengths are equal.
B) Diagonals are perpendicular bisectors.
C) Opposite pairs of angles are supplementary.
D) All interior angles are right angles.
Answer:
Step-by-step explanation:
D) All interior angles are right angles.
To prove that a polygon is a rectangle, you must show that all of its interior angles are right angles. The other properties listed may be true for certain types of rectangles, but they are not sufficient to prove that a polygon is a rectangle.
Brian is at the grand opening celebration of a supermarket. He spins a wheel with 10 equal-sized slices, as shown below. The wheel has 3 black slices, 3 grey slices, and 4 white slices. When the wheel is spun, the arrow stops on a slice at random. If the arrow stops on the border of two slices, the wheel is spun again. (a) If the arrow stops on a black slice, then Brian wins a gift card. Find the odds against Brian winning a gift card.
The probability against Brian winning a gift card are approximately 3.118 to 1.
What is probability?Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event. The probability of an event happening is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Probability is used in various fields, such as statistics, finance, science, engineering, and gaming, to make predictions and decisions based on uncertain events.
In the given question,
The probability of the arrow stopping on a black slice is 3/10. Therefore, the probability of the arrow stopping on a non-black slice is 7/10.
If the arrow stops on the border of two slices, the wheel is spun again, so Brian has another chance to win. The probability of the arrow stopping on a non-black slice is 7/10, so the probability of the arrow stopping on a black slice on the second spin is 3/7.
To find the odds against Brian winning a gift card, we first find the probability of him not winning a gift card on either spin:
P(not winning) = P(non-black on first spin) + P(black on first spin and non-black on second spin)
P(not winning) = (7/10) + (3/10) × (4/7)
P(not winning) = 0.7571
The probability of not winning is 0.7571, so the odds against winning are:
Odds against winning = P(not winning) / P(winning)
Odds against winning = 0.7571 / 0.2429
Odds against winning = 3.118
Therefore, the odds against Brian winning a gift card are approximately 3.118 to 1.
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What fraction does each person get if 8 friends share 5 apples equally?
Answer:
.625
Step-by-step explanation: calculator
Calculate the volume of a prism with:
L = 9 in
W = 1 ¹⁄ ³ in.
H = 4 ²⁄ ³ in.
Select one:
A.
56 in³
B.
52 in³
C.
54 in³
Answer:
The volume of the prism is 54 in³.
This can be calculated by multiplying the length by the width by the height.
Which comes out to 9 in x 1 ¹⁄ ³ in x 4 ²⁄ ³ in = 54 in³.
In triangle ABC, let the angle bisectors be BY and Cz. Given AB = 8, $AY = 6, and CY = 3, find BZ and BC.
Therefore , the solution of the given problem of triangle comes out to be BZ = (8/9) * BC = 40/9 as a result.
Exactly what is a triangle?Due to its two or more extra sections, a trapezoid is a polygon. Its shape is a simple rectangle. Only the three edges A, B, but instead C distinguish a triangle from a regular triangle. Euclidean geometry produces a singular area rather than a cube when the borders are not perfectly collinear. Triangular shapes are defined as having three sides and three angles. Angles are formed by the meeting of a quadrilateral's 3 sides.
Here,
Let D represent the intersection of the angle bisector BY and the side AC, and let E represent the intersection of the angle bisector CZ and the side AB.
=>BC² = AB² + AC² - 2AB * AC * cos(BAC)
However, since BY and CZ are angle bisectors, we can calculate BAC as follows:
=> BC² = 8² + 9² - 2(8)(9)cos[(180° - (ABC + ACB))/2]
If we simplify, we get:
=> BC² = 145 - 144cos[(ABC + ACB)/2]
Using the angle bisector theory once more, we have the following:
=> AD/DC = 8/BC
=> DC = BC(AD/8)
Simplifying and substituting into the Law of Cosines equation yields:
=> BC² = 145 - 144cos[(ABC + ACB)/2]
=> 145 - 144cos[(180° - BAC)/2]
=> 145 - 144cos(B/2)
in which B Equals ABC.
We can now replace the value of BZ that we discovered earlier with:
=> BZ = (8/9) * BC
and solve for BC:
=> BC² = 145 - 144cos(B/2)
=> (9/8)² * (145 - 144cos(B/2)) + (8/9)² * BZ²
Simplifying and substituting BZ results in:
BC² = 25
Since BC is a length, we obtain: by taking the positive square root.
=> BC = 5
=> BZ = (8/9) * BC = 40/9 as a result.
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Ayudaaaaa no sé cuando vaya a venir a revisar estoooolo
huh um intentaría ayudar pero no tiene mucho sentido lo siento