Answers
As there is the highest degree 4 there are four zeros .
Using calculator
the roots are(attached)
The graph is also attached
Answer:
a) 4 roots
b) x = 1 and x = -3
[tex]\textsf{c)} \quad x=1, \quad x=-3, \quad x=\dfrac{-1 + \sqrt{3}i}{2}, \quad x=\dfrac{-1 - \sqrt{3}i}{2}[/tex]
d) see attached
Step-by-step explanation:
Given function:
[tex]f(x)=x^4+3x^3-x-3[/tex]
Part (a)
From inspection of the function, the highest exponent of the polynomial is 4. Therefore, it is expected that the polynomial will have 4 roots.
Part (b)
To find the rational roots, find values of x where [tex]f(x)=0[/tex]
[tex]f(1)=(1)^4+3(1)^3-1-3=0[/tex]
[tex]f(-3)=(-3)^4+3(-3)^3-(-3)-3=0[/tex]
Therefore, the rational roots are: x = 1 and x = -3
Part (c)
As there are only 2 rational roots, the other 2 roots must be complex roots. To find these, first factor the polynomial using the 2 rational roots found in part (b):
[tex]\implies f(x)=(x-1)(x+3)(x^2+bx+1)[/tex]
[tex]\implies f(x)=(x^2+2x-3)(x^2+bx+1)[/tex]
[tex]\implies f(x)=x^4+bx^3+x^2+2x^3+2bx^2+2x-3x^2-3bx-3[/tex]
[tex]\implies f(x)=x^4+(2+b)x^3+(2b-2)x^2+(2-3b)-3[/tex]
Compare coefficients to find b:
[tex]\implies (2+b)x^3=3x^3[/tex]
[tex]\implies b=1[/tex]
Therefore the fully factored function is:
[tex]\implies f\:\!(x)=(x-1)(x+3)(x^2+x+1)[/tex]
Therefore:
[tex](x^2+x+1)=0[/tex]
To find the complex roots, use the quadratic formula
[tex]x=\dfrac{-b \pm \sqrt{b^2-4ac} }{2a}\quad\textsf{when }\:ax^2+bx+c=0[/tex]
[tex]\implies x=\dfrac{-1 \pm \sqrt{1^2-4(1)(1)}}{2(1)}[/tex]
[tex]\implies x=\dfrac{-1 \pm \sqrt{-3}}{2}[/tex]
[tex]\implies x=\dfrac{-1 \pm \sqrt{3}\sqrt{-1}}{2}[/tex]
[tex]\implies x=\dfrac{-1 \pm \sqrt{3}i}{2}[/tex]
Therefore, all the roots of the function are:
[tex]x=1, \quad x=-3, \quad x=\dfrac{-1 + \sqrt{3}i}{2}, \quad x=\dfrac{-1 - \sqrt{3}i}{2}[/tex]
Part (d)
End behaviors:
As the degree of the function is even and the leading coefficient is positive, the end behaviors are:
[tex]\textsf{As } x \rightarrow - \infty, \:\: f(x) \rightarrow + \infty[/tex]
[tex]\textsf{As } x \rightarrow + \infty, \:\: f(x) \rightarrow + \infty[/tex]
y-intercept
Substitute x = 0 into the function:
[tex]f\:\!(0)=(0)^4+3(0)^3-(0)-3=-3[/tex]
Therefore, the y-intercept is (0, -3)
To find the approximate x-coordinates of the turning points (stationary points), differentiate the function, set it to zero, then solve for x:
[tex]f'\:(x)=4x^3+9x^2-1[/tex]
[tex]\implies 4x^3+9x^2-1=0[/tex]
Using a calculator, x ≈ -2.2, x ≈ -0.4, x ≈ 0.3
Therefore the approximate coordinates of the turning points are:
(-2.2, -9.3), (-0.4, -2.7) and (0.3, -3.2)
We don't need to plot these points, they merely help us with the approximate shape of the curve.
Related Questions
Robert climbed 125 steps in five minutes at this rate how many steps did he average per minute
Answers
The solution is 62. We have to multiply the mixed number by the whole number. 12 1/2 is 25/2 so we now multiply by flipping the 25/2 to 2/25 and when you multiply you get 1550/25, which is 62
Find the coordinates of the midpoint of the line segment with endpoints at ( 6, -7) and (-3,5)
Answers
Answer:
Coordinates of the midpoint are ( 9/2 , -6 )
If overhead is applied to individual jobs at a rate of 50% of direct labor costs incurred per job, and $50,000 in direct materials were used and $25,000 of direct labor was required, what was the total cost of the job?
a) $25,000
b) $50,000
c) $12,500
d) $87,500
Answers
The total cost of the Job in the question is; D: $87,500
How to find the total Cost?
We are given that;
Direct Labor = $25000
Cost of direct materials used = $50000
Overhead is 50% of direct labor costs. Thus;
Overhead Cost = 50% * $25000 = $12500
Total Cost of Job = Overhead Cost + Materials Cost + Labor Cost
Total Cost of Job = $87,500
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#88 will give brainliest to the best response!
Answers
Answer:
Mean: 80
Mode: 81
Median:81
Step-by-step explanation:
First, let's get the set of numbers from this Stem-and-Leaf plot:
[60,61,73,75,79,81,81,81,92,96,98]. From here we can solve for the mean by adding all the numbers up and dividing it by the number of terms that we have: [tex]\frac{60+61+73+75+79+81+81+81+92+96+98}{11} = 80[/tex] when rounded to the nearest whole number. The median is the middle number of the set when arranged from least to greatest, in this case, the median is 81.
Hope this helps! :)
subtract the quotient of 15 divided by 3 from the sum of 33 and 7
Answers
Answer:
(33+7)-(15/3) = 35
Step-by-step explanation:
35
Step-by-step explanation:
fancy way of saying (33+7) - (15/3)
33 + 7 = 40
15 / 3 = 5
40 - 5 = 35
please answer this!!!
Answers
Step-by-step explanation:
sin60 =opp/hyp
0.86=y/20
y=17.2
cos60=adj/hyp
0.5=x/20
x=10
sin57=y/15
0.83=y/15
y=12.45
Explanation: a right angle adds to 90 degrees, meaning you would Add 57 and 15, then subtract that from 90. You would get -18, but turn that into a positive. Your answer will be 18 for Y on the second problem.
Help meee pleaseeeeeeee
Answers
Answer:
The answer to this question is letter choice E. 25%.
Step-by-step explanation:
Since 3/12 circles are shaded it would be the same as 25% or 1/4 if simplified.
Alizeh plans on leaving her $10,000 invested for 20 years. Write an equation representing her investment’s value, y, after x years. What is the value of Alizeh’s investment after 20 years?
Answers
Answer:
$38696.84
Step-by-step explanation:
7% interest
so *1.07 per year
10,000*1.07^x=y
10,000*1.07^20=$38696.84
The equation that represent the value of Alizeh’s investment after 20 years is y = 1000(b)^20
Exponential equationThe increase in the amount of his money can be represented using the exponential function as shown:
y = ab^x
Given the following
a = 1000
x = 20 years
Substitute
y = 1000(b)^20
Hence the equation that represent the value of Alizeh’s investment after 20 years is y = 1000(b)^20
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11. The exponential function given by H(t)=80,030.83(1.0481)t, where t is the number of years after 2010, can be used to project the number of centenarians in a certain country. Use this function to project the centenarian population in this country in 2013 and in 2038.
Answers
A function assigns the value of each element of one set to the other specific element of another set. The number of centenarians in the country in 2013 is 9.4175×10⁴⁵.
What is a Function?A function assigns the value of each element of one set to the other specific element of another set.
The exponential function that can be used to project the number of centenarians in a certain country is [tex]H(t) = 80,030.83(1.0481)^t[/tex].
Now, the number of centenarians in the country in 2013 is,
[tex]H(t) = 80,030.83(1.0481)^t\\\\H(2013) = 80,030.83(1.0481)^{2013}\\\\H(2013) = 9.4175 \times 10^{45}[/tex]
Also, the number of centenarians in the country in 2038 is,
[tex]H(t) = 80,030.83(1.0481)^t\\\\H(2038) = 80,030.83(1.0481)^{2038}\\\\H(2038) = 3.048 \times 10^{46}[/tex]
Hence, the number of centenarians in the country in 2013 is 9.4175×10⁴⁵.
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PLSSS HELP ME WITH MATH
Answers
Step-by-step explanation:
always remember, equations and inequalities are like a scale with 2 pans (left and right).
a variable is like a placeholder, an empty spot on the pans. and the expressions on both sides are kind of the conditions for the things to put in these empty spots so that the status of the scale stays unchanged.
for an equation the scale has to stay balanced all the time.
and in an inequality one side is heavier that the other (it normally does not matter how much heavier) with "<" or ">" signs, or the scale can be also balanced with "<=" or ">=" signs. just the other side must never be heavier.
A
so, in our case here
2x - 5 = 3
yes, the solution (the ONLY solution, actually) is x = 4. no other value of x allows the scale to be balanced.
2x - 5 >= 3
well, since it is a ">=" sign, we can be lazy and treat it like the equation above, and x = 4 is therefore a valid solution for the inequality too.
but so is every other value of x that makes the left side "heavier". what about e.g. 5 ?
2×5 - 5 >= 3
10 - 5 >= 3
5 >= 3
true, great ! so, e.g. x = 5 is also a valid solution.
what else ?
let's simplify the inequality
2x - 5 >= 3
2x >= 8
x >= 4
so, really every value of x that is greater or equal to 4 is a valid solution for the inequality.
B
-2x - 5 = 3
yes, the solution (the ONLY solution, actually) is x = -4. no other value of x allows the scale to be balanced.
-2x - 5 >= 3
well, since it is a ">=" sign, we can be lazy and treat it like the equation above, and x = -4 is therefore a valid solution for the inequality too.
but so is every other value of x that makes the left side "heavier". what about e.g. -5 ?
-2×-5 - 5 >= 3
remember, -×- = +
10 - 5 >= 3
5 >= 3
true, great ! so, e.g. x = -5 is also a valid solution.
what else ?
let's simplify the inequality
-2x - 5 >= 3
-2x >= 8
x <= -4
a multiplication or division by a negative value flips the inequality sign, because such an operation makes a light weight heavy and a heavy weight light.
so, really every value of x that is less or equal to -4 is a valid solution for the inequality.
y
12
11
10
9
7
NWO0065
4
-12-11-10-9-8
-65
-3-2-1
1
2 3 4 5 6
8 9 10 11 12
-2
-9
-10
-11
-12
Answers
Answer:
y12111097
Step-by-step explanation:
because I know the answer
You asked students the survey question: How many hours did you do homework last week? The responses were 3, 10, 5, 7, 4, 8, 3, 5, 4, and 2 hours. Which statement best represents the data?
1. Most of the friends who responded did homework for 5 or fewer hours last week.
2. Most of the friends who responded did homework for more than 8 hours last week.
3. Most of the friends who responded did homework for 6 or more hours last week.
4. Exactly half of the friends who responded did homework for fewer than 6 hours last week.
Answers
If you write it out in order of smallest number to largest number, you can count that only 3 people out of the ten total did over 5 hours of homework
What is the range for the following set of measurements? 7.1 g, 9.8 g, 2.3 g, 8.5 g, 7.4 g, 5.7 g
Answers
Answer:
7.5
Step-by-step explanation:
you take the largest number (9.8) and subtract with the smallest number (2.3)
9.8-2.3= 7.5At Webster Elementary School, of
the fifth graders are enrolled in summer
camps. Of the students enrolled in
summer camps, 12 are enrolled in an
art camp. What fraction of the fifth
graders at Webster are enrolled in
art camp?
Answers
8/12 of fifth graders at Webster are enrolled in
art camp
Step-by-step explanation:
Question: At Webster Elementary School, 4/6 of the fifth graders are enrolled in summer camps. Of the students enrolled in summer camps, 8/12 are enrolled in an art camp. What fraction of the fifth graders at Webster are enrolled in art camp?
Given:
4/6 of the fifth graders are enrolled in summer camps.
8/12 are enrolled in an art camp.
fraction of the fifth graders enrolled in art camp:
find common denominator, 4/6 x 2/2 = 8/12
8/12 = 4/6 which means all went to art camp
I need help with part 2 on this question (Screenshot)
Answers
Hope this helps.
Determine why the triangles are congruent. Choose from SSS, SAS, ASA, AAS, and HL. If the
triangles are not congruent type NC. (Use all capitals in your answer)
Answers
Answer:
Step-by-step explanation:
Somebody please explain how to solve this by formula or theorem
Answers
[tex]\text{It is an equilateral triangle, so all angles are equal to}~ 60^{\circ}\\\\~~~~~~m\angle 2 = x+72\\\\\implies 60 = x+72\\\\\implies x = 60-72\\\\\implies x = -12[/tex]
What is the eighth term of the sequence below?
5, 9, 13, 17, …
A. 33
B. 30
C. 29
D. 34
Answers
Answer:
A. 33
Step-by-step explanation:
Add 4 to each number
1. 5 is your first number
2. 5+4= 9 is your second number
3. 9+4= 13 is your third number
4. 13+4= 17 ...
5. 17+4= 21 ...
6. 21+4= 25 ...
7.25+4= 29 ...
8.29+ 4= 33 is your eight number
Which number sentence is not true?
A. | -1 | > 1/2
B. | -1 | = -1
C. | 1/2 | < | -1 |
D. | -1/2 | < | -1 |
Answers
Answer:
Option B
Step-by-step explanation:
|-1| [tex]\neq[/tex] -1
|-1| = 1
Answer:b
Step-by-step explanation: bc it’s right
solve this using pythagorean theorem please and give me the right answer and i will give YOU brainliest and a like in return.
Answers
Answer:
[tex]They\ are\ 8.544\ feet\ apart[/tex]
Step-by-step explanation:
Step 1: Determine the important information
Math teacher 8 feet in front
Science teacher 3 feet to the right
Step 2: Determine how far apart the teachers are
Pythagorean theorem → [tex]a^2 + b^2 = c^2[/tex]
[tex](8)^2+(3)^2=c^2[/tex]
[tex]64 + 9 = c^2[/tex]
[tex]73 = c^2[/tex]
[tex]\sqrt{73} = \sqrt{c^2}[/tex]
[tex]8.544=c[/tex]
Answer: [tex]They\ are\ 8.544\ feet\ apart[/tex]
The surface area of a sphere is 1,017.36 square inches. What is the volume of the sphere? Use 3.14 for pi
Answers
Please help me its a math problem !! please see image
Answers
Answer:
y ≤ 4
Step-by-step explanation:
Solving :
Highest y-value is 4The other y-values are less than 4Hence, the range is : y ≤ 4A car travels a distance of D km at 80 km/h and arrives 20 min late. If the car travels at
120 km/h, the car will arrive 20 min early. Find the value of D.
Answers
Tell whether a triangle with the side lengths 65 feet, 72 feet, and 97 feet is a right triangle.
Answers
Answer:
when the sum of the squares of the sides are equal to the square of the hypotenuse
Step-by-step explanation:
we use Pythagoras, 97 is the hypotenuse, 65 and 72 the sides.
97² = 65² + 72²
97² = 97²
it is a right triangle
Identify the growth factor for the following exponential function: y= 2.3(1.32)*
A. 2.3
B. 1.32
C. X
D 32%
Answers
Triangles are stacked beside each other to form a prism. which statements are true regarding the prism? check all that apply. the prism has no vertices. the prism has 9 edges. the bases of the prism are triangles. the bases of the prism are rectangles. the prism has 5 faces.
Answers
The true statements based on the information are the bases of the prism are rectangles and the prism has 5 faces.
What is a pyramid?A pyramid is defines as a polyhedron formed by connecting a base to a point known as a vertex.
The pyramid formed from stacking triangles besides each other will have vertices, even number of edges and faces
The true statements based on the information are the bases of the prism are rectangles and the prism has 5 faces.
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Answer:
B C E , edge22
Step-by-step explanation
drink some water today and remember to take a break for yourself have a wonderful dayday
A water tank measures 1.4m by 8cm by 60cm. Find the capacity of the tank in litres
Answers
Answer:
672 litres.
Step-by-step explanation:
Convert the measurements to cm:
14 m = 1400 cm
The capacity = 1400 * 8 * 60
= 672,00 cm^3
There are 1000 cm^3 in 1 litre so the answer is
672 litres.
PLEASE HELP!!! NEED QUICKLY!!! WILL GIVE EXTRA POINTS AND WILL MARK BRAINLIEST!!! NEED ANSWERS TO ALL
Answers
Answer:
B and D have a horizontal line of symmetry ( their top and bottom parts match ) Some letters like X, H, and O, have vertical and horizontal lines of symmetry. some like P, R, and N have no lines of symmetry.
Step-by-step explanation:
A,. C,. D,. E,. H,. I,. M,. O,. T,. U,. V,. W,. X,. and Y have lines of reflective symmetry. Some of them have more than one.
'O' has an infinity of them.
There is no way for you to give extra points.
VW has a midpoint at M(12, 0.5). Point V is at (10, -18). Find the coordinates of point W. Write the coordinates as decimals or integers.
Answers
[tex]~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ V(\stackrel{x_1}{10}~,~\stackrel{y_1}{-18})\qquad W(\stackrel{x_2}{x}~,~\stackrel{y_2}{y}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{ x +10}{2}~~~ ,~~~ \cfrac{ y -18}{2} \right)~~ = ~~\stackrel{M}{(12~~,~~0.5)} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{ x +10}{2}=12\implies x+10=24\implies \boxed{x=14} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{ y -18}{2}=0.5\implies y-18=1\implies \boxed{y=19}[/tex]
Emmet set up a bubble station for his little sister's birthday party. He filled a bubble machine and a bottle for kids to blow their own bubbles. In the bubble machine, Emmet mixed 12 ounces of dish soap and 24 ounces of water. In the bottle, he mixed 5 ounces of dish soap and 9 ounces of water. Which container had the greater ratio of dish soap to water?
Answers
Answer:
It's 5912
Step-by-step explanation:
It's 5912 cause 5912 is the answer.