Answer:
There are four possible combinations of root types for a 7th degree polynomial:
(i) 7 rational roots. ([tex]a_{\mathbb{Q}} = 7[/tex], [tex]a_{\mathbb{I}} = 0[/tex], [tex]a_{\mathbb{C}} = 0[/tex])
(ii) 6 rational roots, 1 irrational root. ([tex]a_{\mathbb{Q}} = 6[/tex], [tex]a_{\mathbb{I}} = 1[/tex], [tex]a_{\mathbb{C}} = 0[/tex])
(iii) 2 rational roots, 1 irrational root, 4 complex roots. ([tex]a_{\mathbb{Q}} = 2[/tex], [tex]a_{\mathbb{I}} = 1[/tex], [tex]a_{\mathbb{C}} = 2[/tex])
(iv) 3 irrational roots, 4 complex roots. ([tex]a_{\mathbb{Q}} = 0[/tex], [tex]a_{\mathbb{I}} = 3[/tex], [tex]a_{\mathbb{C}} = 2[/tex])
Step-by-step explanation:
According to the Fundamental Theorem of Algebra, every polynomial with degree higher than zero has at least a solution and at best [tex]n[/tex] complex roots, where [tex]n[/tex] is the degree of the polynomial. In this case, we have 7 roots.
We can rewrite the 7th degree polynomial as a product of a 3rd degree polynomial and 4th degree polynomial. That is:
[tex]p_{7}(x) = p_{3}(x)\cdot p_{4}(x)[/tex] (Eq. 1)
From Algebra we know that both 3rd degree and 4th degree polynomials are solvable by analytical means:
1) All 4th degree polynomials are solvable by Ferrari's method, whose posible solutions are, respectively:
(i) 4 rational roots.
(ii) 4 irrational roots.
(iii) 2 rational roots, 2 irrational roots.
(iv) 3 rational roots, 1 irrational root.
(v) 1 rational roots, 3 irrational roots.
(vi) 2 rational roots, 2 complex roots.
(vii) 2 irrational roots, 2 complex roots.
(viii) 1 rational root, 1 irrational root, 2 complex roots.
(ix) 4 complex roots.
2) All 3rd degree polynomials are solvable by Cardano's method, whose possible solutions are, respectively:
(i) 3 rational roots.
(ii) 3 irrational roots.
(iii) 1 rational root, 2 complex roots.
(iv) 1 irrational root, 2 complex roots.
(v) 3 complex roots.
Please note that complex roots are always presented in conjugated pairs by Quadratic Formula.
The number of solutions of a polynomials are represented by the following expression:
[tex]n = a_{\mathbb{Q}}+a_{\mathbb{I}}+2\cdot a_{\mathbb{C}}[/tex] (Eq. 2)
Where:
[tex]a_{\mathbb{Q}}[/tex] - Number of rational roots, dimensionless.
[tex]a_{\mathbb{I}}[/tex] - Number of irrational roots, dimensionless.
[tex]a_{\mathbb{C}}[/tex] - Number of pairs of complex roots, dimensionless.
Based on this expression, we find the possible root types for a 7th degree polynomial:
(i) 7 rational roots. ([tex]a_{\mathbb{Q}} = 7[/tex], [tex]a_{\mathbb{I}} = 0[/tex], [tex]a_{\mathbb{C}} = 0[/tex])
(ii) 6 rational roots, 1 irrational root. ([tex]a_{\mathbb{Q}} = 6[/tex], [tex]a_{\mathbb{I}} = 1[/tex], [tex]a_{\mathbb{C}} = 0[/tex])
(iii) 2 rational roots, 1 irrational root, 4 complex roots. ([tex]a_{\mathbb{Q}} = 2[/tex], [tex]a_{\mathbb{I}} = 1[/tex], [tex]a_{\mathbb{C}} = 2[/tex])
(iv) 3 irrational roots, 4 complex roots. ([tex]a_{\mathbb{Q}} = 0[/tex], [tex]a_{\mathbb{I}} = 3[/tex], [tex]a_{\mathbb{C}} = 2[/tex])
Find the value of x.
(A) 16 (B) 41 (C) 5 (D) 25 (E) 66
Answer:
D. 25Step-by-step explanation:
D. 25 is the correct answer....Hopes it helps you....Hopes to get BRAINLIEST.. .Thank you ☺️☺️☺️
A line which has an undefined slope is _____.
A. zero
B. horizontal
C. vertical
D. undefined
Answer:
c, vertical
Step-by-step explanation:
blablablablablanlabla
The length of a rectangle is 4 feet less than twice its width, while the perimeter is 15.
Answer:
width: 3.83
length: 3.66
Step-by-step explanation:
The perimeter of a triangle is calculating by adding all the sides together (length + length + width + width). Since we are given the formula for the length we can simply substitute that formula and solve for the width which would be the only variable in the equation.
length + length + width + width = 15
(2w-4) + (2w-4) + w + w = 15 ... combine like terms
6w - 8 = 15 ... add 8 on both sides
6w = 15 ... divide both sides by 6
w = 3.83
Now that we know the value for the width we can plug it into the formula for the length to find its value
l = 2w - 4
l = 7.66 - 4
l = 3.66
Finally, we can see that the rectangle has a width of 3.83 and a length of 3.66
The units of the subway map below are in miles. Suppose the routes between stations are straight. Station G, not shown on the map, is 5 miles east and 7 miles south of Station A. What is the approximate distance between Station E and Station G when traveling by subway?
Answer:
8.60 miles
Step-by-step explanation:
If S=station G, not shown on the map, is 5 miles east and 7 miles south of Station A, the approximate distance between Station E and Station G when traveling by subway will be gotten using the Pythagoras theorem.
EG² = 5²+7²
EG² = 25+49
EG² = 74
EG = √74
EG = 8.60 miles
Hence the approximate distance between Station E and Station G when traveling by subway is 8.60miles
Describe the possible values of x.
Answer:
70x - 70 if it means 5 * 7 * (2x - 2)
Step-by-step explanation:
Explain how the graph of f(x) is obtained
Answer:
there is no picture
Step-by-step explanation:
sorry without a picture i cant :(
What is 0.66666667 as a fraction
Answer:
2/3
Step-by-step explanation:
Answer:
[tex]\frac{2}{3}[/tex]
I hope this helps!
I think of a number, add seven, then divide by 5
Answer:
[tex] \frac{x + 7}{5} [/tex]
Step-by-step explanation:
let take the number as X
[tex] \frac{x + 7}{5} [/tex]
Ruby says that the equation x-3 = x+3 has no solution. Is Ruby's comment correct? Explain your answer.
Answer:
yes
Step-by-step explanation:
x-3=x+3
combine like terms
x=x+6
subtract x from both sides
0 is not equal to 6
you must have a common denominator when ? fractions
Step-by-step explanation:
ADD::If the denominators are not the same, you must find the common denominator by finding the least common multiple of the 2.
SUBTRACT::If the denominators are not the same, you must find the common denominator by finding the least common multiple of the 2.
12.1.33 Question Help Suppose that 62% of families living in a certain country own an HDTV and 16% own a computer. The Addition Rule might suggest, then, that 78% of families own either an HDTV or a computer. What's wrong with that reasoning?
Answer:
Step-by-step explanation:
Suppose :
62% own an HDTV
16% own a computer
Using the addition rule : suggests that (62% + 16%) = 78% either own an HDTV or a computer.
The reasoning is wrong because ; since it is possible to own both an HDTV and also own a computer at the same time, the outcome of the addition rule is wrong as both events are independent. If both event cannot occur at the same time (mutually exclusive event) , then, the addition rule may be applied,. But since both can occur independently and each family can own an HDTV while also owning a computer, then the addition rule is wrong for the scenario described.
please help with one and two i have no clue and i’m really stressed, i’ll mark highest :(
Answer:
From Post Office to Store is 7.8 miles Henry would walk 2.2 hStep-by-step explanation:
From Post Office (P) to Library (L) is 6 miles, from Store (S) to Library (L) is 5 miles. And angle SLP is right angle (m∠SLP=90°).
So, from Pythagorean theorem:
[tex]SL^2+PL^2=SP^2\\\\5^2+6^2=SP^2\\\\SP^2=25+36=61\\\\SP=\sqrt{61}=7.810 249...\approx7.8[/tex]
If Henry walks 3.6 mph then:
[tex]3.5\qquad-\quad\quad1\,h \\7.8\qquad-\qquad x\\\\ x=\dfrac{7.8\cdot1\,h}{3.5}=\dfrac{78\,h}{35}=2,22857...\, h\approx2.2\,h[/tex]
If you subtract 12 from my number and multiply the difference by - 5, the result is -90." What is Sara's number?
Answer:
30
Step-by-step explanation:
Do the steps in reverse
-90 divided by -5 = 18
18 + 12 = 30
Your answer is 30
Hope I helped :)
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How do you turn a interval into an inequality?
convert 2 3/7 to an improper fraction
Answer:The mixed number 2 3/7 can be converted to the improper fraction 17/7. The easiest way to do this is to multiply the denominator of the fraction (7 in...
Step-by-step explanation:
Isabel is mailing back packages.each small package cost her $3.30 to send each large package cost her $4.10. How much will it cost her to send 7 small packages and 1 large package??l
Answer:
$27.20 is the answer. just multiply and then add
Answer:
It will cost her $23.1 to send the little packages and, $4.10 to send one big package.
Step-by-step explanation:
Its very easy multiply the cost which is $3.30 times the amount which is 7 and that your answer $23.1. Than do the same for the big package multiple $4.10 times 1 and you get $4.10.
Can someone help me solve this?
Answer:
2313
Step-by-step explanation:
Which of the following is a true statement about the triangles shown on the graph?
The slope of the smaller triangle is smaller than the slope of the larger triangle
The slope of the larger triangle is larger than the slope of the smaller triangle.
The triangles are congruent triangles.
The slopes of the two triangles are the same.
The correct statement about the triangles shown in the graph is given as follows:
The slopes of the two triangles are the same.
How to obtain the slope?Considering a graph, a slope is calculated as the division of the vertical change by the horizontal change.
For the smaller triangle, we have that:
The vertical change is of 2.The horizontal change is of 2.Hence the slope is of:
m = 2/2
m = 1.
For the larger triangle, we have that:
The vertical change is of 3.The horizontal change is of 3.Hence the slope is of:
m = 3/3.
m = 1.
Meaning that they have the same slope.
More can be learned about slope at https://brainly.com/question/24808124
#SPJ1
If a fuel efficient car can go 210 miles
on 4 gallons of fuel, how far can it go
on 12 gallons?
Answer:
The distance covered by the car in 12 gallons of fuel is 630 miles.
Step-by-step explanation:
Distance covered by car in 4 gallons of fuel = 210 miles.
Let us use unitary method to find distance covered by car in 1 gallon of fuel.
Distance covered by car in 1 gallon of fuel = 210/4 miles.
Let us use unitary method to find distance covered by car in 12 gallon of fuel.
Distance covered by car in 12 gallons of fuel = (210/4)*12 = 630 miles.
HELPPPP PLS ASAPPPPP EASY
Answer:
on what
Step-by-step explanation:
??????
Answer:
What do you need help with?
Step-by-step explanation:
Upload a picture
6 - 8x = 6x - 9x + 11
6-8x=-3x+11
-8x+3x=11-6
-5x=5
x=-1
The first step is to isolate the variable, then you solve. Hope this helps, brainliest if you can.
Answer:
hi
Step-by-step explanation:
hi i am a gamergirl add me in fortnite xxgamergilxx93_qt
Heeeeellllllllppppp lol
Answer: C
Step-by-step explanation: The sign wouldnt be greater than or equal to because he wants to spend less than 250.
Answer:
250
Step-by-step explanation:
Please help me ASAP!!
Answer:
bottom right
Step-by-step explanation:
Answer:
(2067+49) x 21
Step-by-step explanation:
Edpuzzle pls helppp
The answer is make a table
What is the Solution of
7x + 9 = 5x - 3 ?
Answer:
x = -6
Step-by-step explanation:
Original Equation:
7x + 9 = 5x - 3
Subtract 9 from both sides
7x = 5x - 12
Subtract 5x from both sides
2x = -12
Divide both sides by 2
x = -6
Hope I helped :)
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Each side of a square is (7 + 3x) units. Witch is the perimeter of the square?
Perimeter formula: Side × 4
Side is 7 + 3x
So, P = 4(7+3x) = 28 + 12x
Answer: 28 + 12x
math more math and more math i mark as brainly
Answer:
It's the first one
Its A
PLEASE MARK ME AS THE BRAINLIEST!!!!!!!!!!!!!!!!!!!!
hello I need help on it
42145357556574455247788555
12. According to a survey conducted by the American Pet Products Association, 68% of U.S. households own a pet. According to the Energy Information Administration, 31% of all U.S. households have trouble paying their energy bills. Suppose 17% of all U.S. households own a pet and have trouble paying their energy bills. If a U.S. household is randomly sampled, a. What is the probability that the household owns a pet or is having trouble paying energy bills
Answer:
0.82
Step-by-step explanation:
We are given;
P(households own a pet) = 68% = 0.68
P(have trouble paying their energy bills) = 31% = 0.31
P(households own a pet & have trouble paying energy bills) = 17% = 0.17
Now, to find the probability that a household either owns a pet or having trouble paying energy bills is gotten by the relationship;
P(households own a pet or have trouble paying energy bills) = P(households own a pet) + P(have trouble paying their energy bills) - P(households own a pet & have trouble paying energy bills)
This gives;
P(households own a pet or have trouble paying energy bills) = 0.68 + 0.31 - 0.17 = 0.82
The sum of three times a number and seven is 22
Answer:
3x+7=22
Step-by-step explanation:
three times a number is
[tex]3\times x=3x[/tex]
so the sum of 3x and 7 is 22
[tex]3x+7=22\\\\3x=15\\\\x=5[/tex]
you can leave it here
[tex]3x+7=22[/tex]
if you need the value of x
then
[tex]x=5[/tex]
