Using the Factor Theorem, the non-real solutions are given as follows:
[tex]\frac{1 + \sqrt{5}i}{3}, \frac{1 - \sqrt{5}i}{3}[/tex]
What is the Factor Theorem?The Factor Theorem states that a polynomial function with roots [tex]x_1, x_2, \codts, x_n[/tex] is given by:
[tex]f(x) = a(x - x_1)(x - x_2) \cdots (x - x_n)[/tex]
In which a is the leading coefficient.
For this problem, the function is:
f(x) = 3x^5 + 25x^4 + 26x³ - 82x² + 76x - 48.
The function can be written as:
f(x) = g(x)h(x).
In which:
g(x) is the polynomial with the real solutions.h(x) is the polynomial with the non-real solutions.Using a calculator, as the graph is not given, the real solutions are given as follows:
[tex]x_1 = -6, x_2 = -4, x_3 = 1[/tex]
Hence:
g(x) = (x + 6)(x + 4)(x - 1)
g(x) = (x² + 10x + 24)(x - 1)
g(x) = x³ + 9x² + 14x - 24.
f(x) is of degree 5, g(x) of degree 3, hence h(x) is of degree 2 and:
f(x) = g(x)h(x)
3x^5 + 25x^4 + 26x³ - 82x² + 76x - 48 = (x³ + 9x² + 14x - 24)(ax² + bx + c)
Hence:
ax^5 = 3^5 -> a = 3.-24c = -48 -> c = 2.-24bx + 14cx = 76x
-24b + 28 = 76
-24b = 48
b = -2.
Then the equation with the non-real solutions is:
h(x) = 3x² - 2x + 2.
Which is a quadratic equation with coefficients a = 3, b = -2 and c = 2, thus:
[tex]\Delta = b^2 - 4ac = (-2)^2 - 4(3)(2) = -20[/tex][tex]x_1 = \frac{2 + \sqrt{-20}}{6} = \frac{2 + 2\sqrt{5}i}{6} = \frac{1 + \sqrt{5}i}{3}[/tex][tex]x_2 = \frac{2 - \sqrt{-20}}{6} = \frac{2 - 2\sqrt{5}i}{6} = \frac{1 - \sqrt{5}i}{3}[/tex]Hence the non-real solutions are:
[tex]\frac{1 + \sqrt{5}i}{3}, \frac{1 - \sqrt{5}i}{3}[/tex]
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2.5 of what is 3?
Need help ASAP
A certain plant grows 1 1/6 inches every week. How long will it take the plant to grow 6 1/6 inches?
It will take 6.14 weeks for the plant to grow the required length
The growth rate of the plant each week = 1 1/6 inches
Converting the mixed fraction into an improper fraction we multiply the whole number by the denominator and add the numerator to get the improper fraction, therefore
The growth rate of a plant in improper form = 7/6 inches
Required growth rate = 6 1/6 inches
The required growth rate in improper fraction = 43/6 inches
Number of weeks it will take the plant to grow the required length = Required length/Growth rate each week = (43/6)/(7/6)
= 43/7 weeks or 6.14 weeks
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Three line segments form the letter Z. Rotate the letter Z counterclockwise around the midpoint of segment BC by 180 degrees. Describe the result.
Answer:
i don't known the question you are asking
branlyueer099799893
3 minutes ago
Mathematics
College
ANSWER RIGHT AWAY PLEASE!
question one: simply the exponents. (show your work)
- 21^0 = ?
- (2/3)^-2 = ?
question two: Solve the problem. (leave your answer as a base number with an exponent and show your work)
- 4^6 / 4^2 = ?
question three: Convert the number into numerical form. (show your work)
- 8.723 x 10^-10 = ?
question four: Solve the problem. (leave your answer in scientific notion and show your work)
- 5.1 x 10^8 - 2.3 x 10^3 = ?
- 6 x 10^5 x 1.1 x 10^7 = ?
PLEASE ANSWER I WILL GIVE 100 POINTS TO ANYONE WHO ANSWERS AND MARK BRAINLIEST AT LEAST ANSWER THE ONES YOU KNOW
Determine the equation for the given line in slope-intercept form.
y=-5/3x-1
y=5/3x+1
y=3/5x+1
y=-3/5x-1
Answer:
The last answer is correct
y = [tex]\frac{-3}{5 }[/tex] x - 1
Step-by-step explanation:
You can see that the graph crosses the y axis at (0,-1). So, the graph crosses the y axis at -1. Count how much you go down and the to right from the point on the left to the point on the right. You go down 3 and left 5. This is a negative slope of [tex]\frac{-3}{5}[/tex].
y = mx + b Put in the slope for m and the y-intercept by b
y = [tex]\frac{-3}{5}[/tex]x = 1
PLSSS HELP ANSWER, THE QUESTION IS IN THE SCREENSHOT
[tex]y(\pi/3)=1 \implies 1=C-2\cos(\pi/3) \\ \\ 1=C-1 \\ \\ C=0 \\ \\ \therefore y=\boxed{-2\cos x}[/tex]
can someone help me with this
The table shows ordered pairs of the function y=8-2x. What is the value of y when x=8
Answer:
y = - 8
Step-by-step explanation:
substitute x = 8 into the equation
y = 8 - 2(8) = 8 - 16 = - 8
At a local pizzeria, a pizza with one topping is 9.99. You can add additional toppings for 1.25 each. The pizzeria's menu lists 10 different toppings. Write a function C(x) that represents the cost of a pizza with at least one topping, where x is the number of toppings. Find the domain and range.
The function C(x) that represents the cost of a pizza with at least one topping, where x is the number of toppings is C(x) = 9.99 + 1.25x
The domain is x >= 0 and the range is c(x) >= 9.99
Write a function C(x) that represents the cost of a pizza with at least one topping, where x is the number of toppings.The given parameters are:
One topping = 9.99
Additional topping = 1.25 each
We have x to represent the number of toppings
This means that
C(x) = One topping + Additional topping * x
So, we have
C(x) = 9.99 + 1.25x
How to find the domain and range?We have
C(x) = 9.99 + 1.25x
The smallest number of topping is 0
So, the domain is x >= 0
Substitute 0 for x in C(x) = 9.99 + 1.25x
C(0) = 9.99 + 1.25(0)
This gives
C(0) = 9.99
So, the range is c(x) >= 9.99
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Answer:
C(x) = 8.74 + 1.25x
Domain: 1 ≤ x ≤ 10
Range: [9.99, 21.24]
Step-by-step explanation:
a) We need to find the function C(x) that represents the cost of a pizza with at least one topping, where x is the number of toppings.
We know that a pizza with one topping is $9.99 and that you can add additional toppings for 1.25 each.
So the equation is C(x) = 9.99 + 1.25x right?
Not really... Since $9.99 is the price of a pizza and 1 topping, we need to subtract 1 from x otherwise we would be paying for a pizza with 2 toppings when we had ordered a pizza with 1!
So the new equation is:
C(x) = 9.99 +1.25(x - 1)
or, C(x) = 9.99 + 1.25x - 1.25
or, C(x) = 1.25x + 8.74
b) We need to find the Domain of C(x).
We know that the pizzeria's menu lists 10 different toppings and that a pizza has at least one topping.
So we can choose up to 10 different toppings. And we need at least 1 topping.
∴ Domain is 1 ≤ x ≤ 10.
b) We need to find the Range of C(x).
We know that the domain of C(x) is 1 ≤ x ≤ 10, and that a pizza with 1 topping is $9.99.
Since we already know the minimum (9.99) all we have to do is find the maximum. We can do that by plugging in 10 into C(x) since it is the maximum of our domain.
C(10) = 1.25(10) + 8.74
or, C(10) = 12.50 + 8.74
or, C(10) = 21.24
∴ Range of C(x) = [9.99, 21.24]
Answer:
C(x) = 8.74 + 1.25x
Domain: 1 ≤ x ≤ 10
Range: [9.99, 21.24]
Suppose you are heading to the
beach with a friend. The beach is 80
miles from your house and the drive
takes about 1 hour and 30 minutes.
Your friend's house is 64 miles from
the beach. Use direct variation to
determine how long it will take to
drive to the beach if you leave from
your friend's house.
The time it would take to drive to the beach if you leave from the friend's house is 1 hour 12 minutes
Calculating timeFrom the question, we are to calculate how long it would take to drive to the beach if you leave from your friend's house
From the given information,
The beach is 80 miles from your house and the drive takes about 1 hour 30 minutes
Distance, d, varies directly with time, t.
d ∝ t
d = vt
Where v is the proportionality constant.
Now, we will find v
d = 80 miles
t = 1.5 hours = 3/2 hours
Thus,
80 = (3/2)v
v = (2 × 80)/3
v = 160/3 miles/hour
Now, we will determine t when d is 64 miles
d = vt
64 = 160/3 × t
t = (64 × 3)/160
t = 192/160 hours
t = 6/5 hours
t = 1 hour 12 minutes
Hence, the time it would take to drive to the beach if you leave from the friend's house is 1 hour 12 minutes
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Use properties of operations to add (−3) and (−17)
14
20
-20
-14
Answer:
Step-by-step explanation:
-20
High divers use the height of the diving platform to calculate the speed at which they enter the
water. The formula 58-34-12 models the diver's speed s (in miles per hour) and the height
h (in feet) of the platform.
a. Solve the formula for 3 and for h
b. At what speed does a diver enter the water when the platform is 82 feet tall?
c. A diver enters the water at 48 miles per hour. How tall is the platform?
a. 3h = -12 + 5s
b. Speed = 51. 6 miles per hour
c. Height = 76 feet
How to determine the valueGiven the function;
5s - 3h = 12
Where;
s is the speed in miles per hourh is the height in feeta. To solve for 3h
We have;
- 3h = 12 - 5s
3h = -12 + 5s
b. If the height is 82 feet , then
5s - 3(82) = 12
5s - 246 = 12
collect like terms
5s = 12 + 246
5s = 258
Make 's' the subject
s = 258/ 5
s = 51. 6 miles per hour
c. If the speed is 48 miles per hour, we have;
5(48) - 3h = 12
240 - 3h = 12
collect like terms
-3h = 12 - 240
-3h = -228
Make 'h' the subject
h = -228/ -3
h = 76 feet
Thus, the speed and height are 51. 6 miles per hour and 76 feet respectively.
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a number z is divided by four added to one in mathematical translate?
[tex]\displaystyle\\Answer:\ \frac{z}{4} +1[/tex]
Use synthetic division to find the quotient and reminder when
X^3 - 7x^2 + 5x + 10 is divided by - 2
Hence , using synthetic division the quotient and remainder of given equation are x²- 9x + 23 and -36 respectively.
Synthetic division could be a methodology accustomed perform the division operation on polynomials once the divisor could be a linear issue. one in every of the benefits of mistreatment this methodology over the normal long methodology is that the artificial division permits one to calculate while not writing variables whereas performing arts the polynomial division, that additionally makes it a neater methodology compared to the division.We have given an equation , x³ - 7x² + 5x + 10 which is divided by -2
Steps for using synthetic division are
Step 1 - Write the coefficients of x within the dividend within associate inverted division image.
| 1 -7 5 10
|
Step 2 - As provides put = -2 place at the left.
- 2 | 1 -7 5 10
|
Step 3 - Drop the primary constant of the dividend below the division image.
- 2 | 1 -7 5 10
|
1
Step 4 - Multiply the result by the constant, and place the merchandise within the next column.
- 2 | 1 -7 5 10
| -2
1
Step 5 - Add down the column.
- 2 | 1 -7 5 10
| -2
1 -9
Step 6 - Repeat Steps Four and 5 till you'll go no further.
- 2 | 1 -7 5 10
| -2 18 - 46
1 -9 23 -36
Hence the quotient is x²- 9x + 23 and remainder is -36
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Brock, age 50, and Erin, age 49, are married with three dependent children. They file a joint return for 2021. Their income from salaries totals $50,000, and they received $8,000 in taxable interest, $5,000 in royalties, and $3,000 in other ordinary income. Brock and Erin's deductions for adjusted gross income amount to $2,500, and they have itemized deductions totaling $19,250.
Based on the filing status of Brock and Erin, their itemized deductions, and their income from their salaries, their income tax liability is $4,250.
What is the income tax liability?First, find the taxable income:
= Adjusted gross income - larger of Itemized deductions or standard deductions
= (50,000 + 8,000 + 5,000 + 3,000 - 2,500) - 24,800
= $38,700
The tax rate is 10 % for income between $0 and $19,750 and 12% for salaries between $19,751 and $80,250.
Brock and Erin therefore have a tax liability of:
= (10% x 19,750) + (12% x (38,700 - 19,750))
= $4,252
Rest of the question is:
Find the Income tax liability before any credits
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help me with this precal
Answer: Ok
Step-by-step explanation:
9-0-0
Which transformation would take Figure A to Figure B?
Answer:.......................
4²×3³×(-1)¹²² please solve this question tomorrow my exam of maths
The value of 4²×3³×(-1)¹²² is 432.
How to find the value of 4²×3³×(-1)¹²²?When a number is repeatedly multiplied by itself, the result is a power of that number.
A power of the original number would be produced if you multiplied a number by itself any number of times.
When a single number is added up several times, it is said to be multiplied.
given that 4²×3³×(-1)¹²²
now find the value of 4²×3³×(-1)¹²².
4² = 4 × 4
3³ = 3 × 3 × 3
(-1)¹²² = 1
then, multiply these numbers.16 X 27 X 1 = 432
Hence, the value of 4²×3³×(-1)¹²² is 432.
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PLEASE HELP FAST I ONLY HAVE ONE HOUR!!!!!!!
The proportion of users that park bike on the trail is 0.1728
How to determine the proportion of users that park bike on the trail?The table represents the given parameters
From the table, we have the following parameters
Suburban total = 243
Users that park bike on the trail = 42
The proportion of users that park bike on the trail is then calculated as
P = Users that park bike on the trail/Suburban total
This gives
P = 42/243
Evaluate
P = 0.1728
Hence, the proportion of users that park bike on the trail is 0.1728
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< Question 7, 1.5.25 >
The line graph to the right indicates that in 1960, 21% of taxes came from corporate income tax. For the period from 1960 through 2010,
this percentage decreased by approximately 0.32 each year. If this trend continues, by which year will corporations pay zero taxes?
The 21% of taxes and annual percentage decrease of 0.32, using arithmetic progression gives;
The year corporations will pay zero taxes as 2025How can arithmetic progression be used to find the required year?The percentage of tax that comes from corporate income tax = 21%
Amount by which the percentage decreased each year = 0.32
Period during which the tax reduced = Between 1960 and 2010
Required;
The year corporations pay zero taxes.
Solution;
The amount of tax paid each year can be given by the arithmetic progression formula,
[tex] a_{n} = a + (n - 1) \cdot d[/tex]
Where;
a = The first term = 21
[tex] a_{n} = value \: on \: the \: nth \: year[/tex]
d = The common difference = -0.32
When [tex] a_{n} = 0 [/tex], we have;
[tex] 0 = 21 + (n - 1) \times (-0.32) [/tex]
0.32•(n - 1) = 21
n - 1 = 21/0.32 = 65.625
n = 65.625 - 1 = 64.625 ≈ 65
The year the corporations will pay no taxes is 1960 + 65 = 2025
Corporations will pay no taxes in the year 2025
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Triangle XYZ has translated 4 units up and 3 units left to yield ΔX'Y'Z'. What is the distance between any two corresponding points on ΔXYZ and ΔX'Y'Z′?
The distance between any two point is 5 unit.
What is distance?
The measurement of distance between two objects or points can be quantitative or occasionally qualitative. Distance in physics or common language can refer to a physical length or an assumption based on other factors.
As per the question
Triangle xyz is translated = 4units
And triangle xyz left to yield=3 unit
The distance between the corresponding is hypotenuse of the triangle is-
Buy applying Pythagorus theorem we can get C^2=a^2+b^2
As we know a,b=legs
c=hypotenus
C^2=a^2+b^2C^2
=3^2+4^2
=9+16
=25C =√25 =5 unit
Therefore, the distance between any two point is 5 unit.
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21 The average person at rest takes approximately 16 breaths per minute. Approxir many breaths do they take in 70 years of 365 days each? Use a calculator. Write in exponential notation using powers of 10.
A person will take 5.88672 × [tex]10^{8}[/tex] breaths in 70 years.
Here, we are given that on average a person at rest takes approximately 16 breaths per minute.
1 hour = 60 minutes
Thus, in 1 hour a person will take-
60 × 16
= 960 breaths
Now, 1 day = 24 hours
Thus, in 1 day, a person will take-
24 × 960
= 23,040 breaths
Further, 1 year = 365 days
Thus, in 1 year, a person will take-
23,040 × 365
= 8,409,600 breaths
In 70 years, a person will take-
8,409,600 × 70
= 588,672,000 breaths
We can express this in exponential notation as-
5.88672 × [tex]10^{8}[/tex]
Thus, a person will take 5.88672 × [tex]10^{8}[/tex] breaths in 70 years.
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You are lubricating several motors in the shop. One motor must be lubricated with of 3/16 an ounce of grease, and one larger motor must be lubricated with 3/8 of an ounce of grease. How many ounces of grease will you require? Your answer should be a fraction of an ounce, reduced to its lowest terms.
4x+3y=7 x+2y=-2
put both in slope-intercept form
Answer:
Second equation Answer: [tex]y=-\frac{1}{2}x-1[/tex]
First equation Answer: [tex]y=-\frac{4}{3}x~+\frac{7}{3}[/tex]
Step-by-step explanation:
Concept:
The general equation of a line in slope-intercept form is given below as
[tex]y=mx+c[/tex]
where
[tex]m=slope[/tex]
[tex]c=intercept[/tex]
First equationThe first equation is given below as
[tex]4x+3y=7[/tex]
We will make y the subject of the formula to change the equation in slope-intercept form
[tex]4x+3y=7[/tex]
subtract 4x from both sides
[tex]4x-4x+3y=7=-4x[/tex]
[tex]3y=-4+7[/tex]
divide both sides by 3
[tex]\frac{3}{y} =\frac{-4x}{3} +\frac{7}{3}[/tex]
[tex]y=-\frac{4}{3}x~+\frac{7}{3}[/tex]
Hence,
The equation of 4x+3y=7 in slope intercept form is given below as
[tex]y=-\frac{4}{3}x~+\frac{7}{3}[/tex]Second EquationThe second equation is given below as
[tex]x+2y=-2[/tex]Subtract x from both sides[tex]x-x+2y=-2-x[/tex][tex]2y=-x-2[/tex]divide through by 2[tex]\frac{2y}{2} =-\frac{x}{2} =-\frac{2}{2}[/tex][tex]y=-\frac{1}{2}x-1[/tex]Hence,
The equation of x+2y=-2 in slope-intercept form is
[tex]y=-\frac{1}{2}x-1[/tex]Hope this Helps! <3Answered by:AshTheNeko
Which of the following systems of equations has the same
solution as
-2x+2y=-2;3x-y=9
a.
A. 4x = 16;3x-y=9
B. [-2x+2y=-2;16x-2y = 9
C. -3x-7y=3;3x-y=9
D. [-2x+2y=-2;x+y=11
Answer:
Step-by-step explanation:
16/4
←
Assume log x=0.27 and log by = 0.81. evaluate the following expression logb x/y
By evaluating following logarithm equation and we we get the value of required expression is -0.54.
What is logarithm?
The logarithm is the opposite of exponentiation as a function. This indicates that the base b of another fixed number, the logarithm of a given number x, is the exponent that must be raised in order to obtain the given number x.
As per the question
we have to assume x
log x=0.27
log by=0.81
We have to calculate log bx/y by following expression
We know loge (a/b)= log_e a-log_e b
We know (x)= logb y
=0.27-0.81
= -0.54
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To make barstools, Federico rents a shop for $1510 a month and calculates the materials for each stool are $45. Find the total cost for him to make 98 stools in a month.
Answer:
4410
Step-by-step explanation:
45 x 98 = 4410
Wilson hires a financial analyst to analyse costs and profits for his cereal production business. The analyst determines that Wilson’s eventual profit function is given as: , where x is the number of bags of cereal produced. At what point or number of bags of cereal will Wilson’s profit start decreasing
The company's profit will start to decline if it only produces one bag.
Given f(x) = 2x⁴ ₋ 4x³ ₊ 7
Differentiate it with respect to x.
therefore, f'x) = 8x³ ₋ 12x²
= 4x²(2x ₋ 3)
For decreasing, f'(x) ≤ 0
⇒ 4x²(2x ₋ 3) ≤ 0
⇒ 2x ₋ 3 ≤ 0
⇒ x ≤ 3/2
Since the number of bags are represented by x.
Hence, x ∈ N
Therefore, x = 1 is only possibility.
Therefore, if the company only produces one bag, its profit begins to decline.
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Your question was incomplete. Please find the missing the content here.
Wilson hires a financial analyst to analyse costs and profits for his cereal production business. The analyst determines that Wilson's eventual profit function is given as: pi = 2x⁴ ₋ 4x³ ₊ 7 where x is the number of bags of cereal produced. At what point or number of bags of cereal will Wilson's profit start decreasing?
a car rental company charges a daily rate of $26 plus $0.10 per mile for a certain car. Suppose that you rent that car for a day and your bill (before taxes) is $71. How many miles did you drive?
Answer: 450 miles
Step-by-step explanation: we can put the info we got from the question into an equation. since the car was only rented for a day, the daily rate that was paid was $26. and $0.10 for every mile the car was driven. So we can put into the equation 71 = 26 +0.10m, where m represents miles. Now subtract 26 from both sides and we get 45 = 0.10m. Divide both sides by 0.10 and we get m= 450
Im struggling on finding the arewa on this and how to give my answer as one decimal can somebody help me please
Answer:
5.8
Step-by-step explanation:
Area = (h x b)/2
A = (2.5 x 4.6)/2 = 5.75 = 5.8