Answer:The most general form of a polynomial function of degree n that has real coefficients is:
f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0
where a_n, a_{n-1}, ..., a_1, a_0 are real numbers, and a_n is not equal to zero.
The number of distinct roots of a polynomial function of degree n can be determined using the fundamental theorem of algebra, which states that a polynomial function of degree n has exactly n roots, counting multiplicities. This means that a polynomial function of degree n can have n distinct roots, n-1 distinct roots, or any integer value in between, depending on the multiplicities of the roots.
Using the factor theorem, it is possible to determine the number of distinct roots of a polynomial function by factoring it into its irreducible factors. If the polynomial function is fully factored, the number of distinct roots is equal to the number of irreducible factors.
However, the nature of the roots (real or complex) and constraints imposed by the discriminant are also important to consider. The discriminant of a polynomial function is the value of the expression b^2-4ac, where a, b, and c are the coefficients of the polynomial function in the form f(x) = ax^2+bx+c. The discriminant determines the nature of the roots of the polynomial function. If the discriminant is positive, the roots are real and distinct. If the discriminant is zero, the roots are real and repeated. If the discriminant is negative, the roots are complex and non-real.
So, the most general form of a polynomial function of degree n that has real coefficients, can be determined by the form of the polynomial, the number of distinct roots by the fundamental theorem of algebra, by factoring the polynomial into its irreducible factors and the nature of the roots is determined by the discriminant.
Step-by-step explanation:
“solve the equation by graphing. check your solution” only 5 and 6 need to be done!! attach picture with graph please xoxoxo i will mark brainliest if done like said
The solutions for the given equations are x=2 and x=-4
What is Graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points.
Fo question 5.
1/2x-2=9-5x
Take the variable terms on one sides\
x/2+5x=11
x+10x=22
11x=22
Divide both sides by 11
x=22/11
x=2
For question 6:
-5+x/4=3x+6
-20+x=12x+24
-20-24=12x-x
-44=11x
x=-4
Hence, the solutions for the given equations are x=2 and x=-4
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Please help me on this problem
Answer:
see below
Step-by-step explanation:
reserve = x
field =y
box = z
x * 20 +y*50 +z*100 =370
x=2y
x + y+ z =10 so 3y +z=10 so z=10-3y
2y * 20 + y*50 + (10-3y)*100 =370
90*y+1000-300*y =370
210*y=630
y tickets =3
x tickets=6
z tickets =1
stock prices used to be quoted using eights of a dollar. Find the total price of the transaction.
200 shares of national semi at 115 1/2
Answer:
Step-by-step explanation:
200 x 115.5/8 = $23,100.00
A nine-month old baby drank an average of 7 1/4 oz during each of four feedings. If the baby drank 7 1/2oz, 7 3/8oz, & 6 3/8oz in the first three feedings, how many ounces were taken during the fourth feeding?
A) 7 3/8
B) 6 3/8
C) 6 3/4
D) 7 3/4
Show work please
Answer: D) 7 3/4
Step-by-step explanation:
With averages, we add all the values and divide by the number of values. Here, we know three out of the four days, let's label the day we don't know x. There are four days, so we can divide by four. 7 1/4 is the average, so we can set the equation equal to 7 1/4.
7 1/4 = (7 1/2 + 7 3/8 + 6 3/8 + x) / 4
Let's multiply both sides by 4 to get rid of the fraction. Let's also add what is in the parentheses
29 = 21 1/4 + x
Now we can isolate x by subtracting 29-21 1/4
so
x = 7 3/4 thus the answer is D
Answer:
D) 7 3/4
Step-by-step explanation:
Let x = amount of last feeding.
7 1/4 = (7 1/2 + 7 3/8 + 6 3/8 + x)/4
4 × 7 1/4 = 7 1/2 + 7 3/8 + 6 3/8 + x
28 + 1 = 7 4/8 + 7 3/8 + 6 3/8 + x
29 = 20 10/8 + x
(20 10/8 is the same as 20 + 10/8 = 20 + 8/8 + 2/8 = 20 + 1 + 1/4 = 21 + 1/4 = 21 1/4)
29 = 21 2/8 + x
29 = 21 1/4 + x
x = 29 - 21 1/4
x = 28 4/4 - 21 1/4
x = 7 3/4
Answer: 7 3/4
A group of students was surveyed in a middle school class. They were asked how many hours they work on math homework each week. The results from the survey were recorded.
Number of Hours | Total Number of Students
0 | 1
1 | 3
2 | 2
3 | 10
4 | 9
5 | 7
6 | 3
Determine the probability that a student studied for exactly 4 hours. Round to the nearest hundredth.
A. 0.74
B. 0.35
C. 0.26
D. 0.11
The probability that a student studied for 4 hours is 0.257 Or 26%.
What is probability?Probability refers to potential. A random event's occurrence is the subject of this area of mathematics.
The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events.
The degree to which something is likely to happen is basically what probability means.
Given:
A table is given where the first column represents the number of hours a student studied math and the second column represents the number of students.
So, sum of the second column is 35. so it is the total number of students
and, only 9 of them studied mathematics for 5 hours.
Therefore, The probability that a student studied for 4 hours is,
= 9/35
= 0.257 or 26%.
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The cork in a wine bottle should not exceed 8.77KPa of normal stress. If the shear stress induced by the friction of its perimeter with the bottle is 4.98KPa, which is the maximum allowed ratio between the cork length H and the cork radius R?
The maximum allowed ratio for the cord such that the maximum allowable normal stress is 8.77 kilopascals is equal to 2.081.
How to determine the maximum allowed ratio of a cork in a wine bottle
Let assume that the cork is made of a ductile material, in this case we find that the normal stress (σ), in kilopascals, and shear stress (τ), in kilopascals, are described below:
Normal stress
σ = F / A
Shear stress
τ = (3 · V) / (2 · A')
Where:
V - Shear force, in kilonewtons.F - Normal force, in kilonewtons. A - Transversal area, in square meters. A' - Transversal shear area, in square meters.Let assume that cork is at static equilibrium and at inminent motion, then the shear and normal forces are, respectively:
F = P
V = P / 2
And the transversal areas are introduced below:
A = π · R²
A' = 2 · R · H
Where:
R - Radius of the cork, in meters.H - Height of the cork, in meters.If we know that σ = 8.77 kPa and τ = 4.98 kPa, then the maximum allowed ratio between the cork height and the cork radius is:
8.77 = P / (π · R²)
8.77 = [1 / (π · R)] · (P / R)
4.98 = 1.5 · (0.5 · P) / (2 · R · H)
4.98 = [0.75 / (2 · H)] · (P / R)
Then, we eliminate P / R of the two equations:
8.77π · R = 13.24 · H
H / R = 8.77π / 13.24
H / R = 2.081
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pls help me asap iready now
the answer is there
60 students so. my brainly is glitchng sorry
Look at the picture and answer the question thank you
Answer:
given that z=3 and w=2
10z+5w
now by putting the values of z and w
10(3) + 5(2)
30 + 10
40 is your correct answer
Question 1
For your 15th birthday, your grandma puts $100 in a savings account for you, and
tells you that you cannot take it out until your 20th birthday. If the account she has
chosen has an interest rate of 3% compounded semiannually, how much money will
you have when you turn 20?
O $109.35
O $103.00
O $122.75
1 pts
O $116.05
Answer:
116.05
Step-by-step explanation:
15 to 20 = 5 years
compound semiannually = 10 time
annual rate 3% = 1.5% for every 6 mos
so 100*(1+1.5%)^10 =116.05
Tyshawn has a collection of 140 coins. How many coins represent 20% of his collection?
Please help will give brainlest!!!If the area of the triangle is 10 cm 2, what is the missing height?
W =
Step-by-step explanation:
what is the usual formula for the area of a triangle ?
it is
base × height / 2
so, we know the area (10 cm²).
when we use the formula, we get
10 = 5 × w / 2
20 = 5 × w
w = 20/5 = 4 cm
Answer:
the height is equal = 4 cm
The cost of 8m water pipe is rupees 256. Find the cost of 6m water pipe
Answer:
The cost of 6 m of water pipe = ₹ 192
Step-by-step explanation:
The probability of getting heads on a single coin flip is ;
2
The probability of getting nothing but heads
on a series of coin flips decreases by
2
for each additional coin flip. Enter an exponential function for the
probability p(n) of getting all heads in a series of n coin flips. Give your answer in the form a (b)'. In the
event that a = 1, give your answer in the form (b)'.
All of the employees at the acmesoft company are either programmers or salespeople. The programmers are always tell the truth, the salespeople always lie. Threes Acmesoft employees - A, B, and C-were standing together in the hallway. A visitor passed by and asked A. "Are you a programmer or a salesperson?" A mumbled an answer that the visitor could not hear clearly. The visitor then asked B, "what did A say?" B replied. "A said that he is a salesperson."At this point, the third employee, C, said ,"Don't believe B; he is lying! What are B and C?
1/ B and C are both programmers
2/ B is a programmer and C is a salesperson
3/ B is a salesperson and C is a programmer
4/ B and C are both salesperson
Answer:
Step-by-step explanation:
3/ B is a salesperson and C is a programmer.
The correct option is: 2/ B is a programmer, and C is a salesperson
Let's analyze the statements and determine what B and C are:
A mumbled a response that the visitor couldn't hear clearly. We don't have any information from A's response.
B said, "A said that he is a salesperson." Since we know that salespeople always lie, we can infer that B is lying about A's response. Therefore, if A is a programmer (telling the truth), then B must be a salesperson.
C said, "Don't believe B; he is lying!" If we already determined that B is a salesperson (lying about A's response), then C is suggesting that B's statement is false, which means A's response is the opposite of what B said. So if B claimed that A is a salesperson (which was a lie), then A must be a programmer (telling the truth).
Based on the above analysis:
A is a programmer.
B is a salesperson.
C is a programmer.
So, the correct option is: 2/ B is a programmer, and C is a salesperson.
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16^x^2=8^0
16 to the power of x to the power of 2 equals 8 to the power of 0
The exponent x in the equation must be equal to 0.
How to solve the given equation?Here we have the following equation:
16^x^2=8^0
First, notice that any number to the power of zero (except for the number 0) is equal to 1.
a^0 = 1
16^x^2=8^0
16^x^2 = 1
Now, trivially, if we use x = 0 we will get:
c
Particularly, 0^2 = 0
So we can rewrite:
16^0^2 = 1
16^0 = 1
1 = 1
So the solution is x = 1.
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Can someone solve these 2 questions and explain since I find these questions confusing
please help me !!! thankss
Answer: (6,2) (0,-2) (-3,-4)
Step-by-step explanation:
If you find the other coordinate pair groups on the graph, there is at least one that doesn't land on the line on the graph.
In the top left, (-6,-5) is not on the graph. If it had been (-6,-6) then it would've been correct, but it isn't.
In the bottom left, (0,-3) isn't on the graph. Instead, (0,-2) is.
In the bottom right, (3,-4) isn't on the graph, but (-3,-4) is.
In your class, you have scores of 83,70, 83, and 80 on the first four of five tests. To get a grade of , the average of the first five tests scores must be greater than or equal to 70 and less than 80 .
a.
Solve an inequality to find the least score you can get on the last test and still earn a C.
b.
What score do you need if the fifth test counts as two tests?
a) The least score that you need to get on the last test and still earn a C is given as follows: 34.
b) With the least score counting twice, the least score that you need to earn a C is given as follows: 52.
How to obtain the mean of a data-set?The mean of a data-set is obtained as the sum of all the elements in the data-set divided by the number of elements in the data-set.
The scores for this problem are given as follows:
83, 70, 83, 80 and x.
(in which x is the score of the fifth test).
The least score that you could get on the last test and still earn a C is found when the mean is of 70, hence>:
(83 + 70 + 83 + 80 + x)/5 = 70
316 + x = 350
x = 350 - 316
x = 34.
With the final test counting twice, you would have the equivalent to six scores, which would be given as follows:
83, 70, 83, 80, x and x.
Hence the least score would be obtained as follows:
(83 + 70 + 83 + 80 + 2x)/6 = 70
316 + 2x = 420
2x = 104
x = 104/2
x = 52.
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The graph of h(x) is shown.
What are the intercepts and asymptote(s) of h(x)? Explain how to find these using the graph.
Answer:
y-int = -1
x-int = -4
HA: y = -2
VA: x = -5
Step-by-step explanation:
For the y-intercept, you need to look where the graph intersects with the y-axis. In this case, the graph intersects at -1.
For the x-intercept, you need to look where the graph intersects with the x-axis. In this case, the graph intersects at -4.
The horizontal asymptote is a y value the graph approaches but does not pass or touch. In this case, the y value is -2.
The vertical asymptote is a x value the graph approaches but does not pass or touch. In this case, the x value is -5.
Triangles A and B are right-angled.
A
13 cm
5 cm
B
12 cm
12 cm
a) Show that the two shorter sides in triangle A have the
same length as the two shorter sides in triangle B.
Using pythagoras theorem formula, c² = a² +b² We can see that the two shorter sides have the same length which is 5cm
Since all the sides of the triangle are the same, hence the two triangles are congruent.
What is Pythagoras' theorem ?
Pythagoras' theorem is the square of the hypotenuse in a right-angled triangle is equal to the sum of the squares of the other two sides.
The two shorter sides have the same length which is 5cm
Since all the sides of the triangle are the same, hence the two triangles are congruent.
Let the sides of the triangles be a, b and c. According to Pythagoras theorem;
c² = a² +b²
where:
a is the hypotenuse (longest side)
For triangle A:
c = 13
b = 12
Get "a"
13² = 12² +b²
b² = 25
=> b = 5 cm
Similarly for triangle B:
c² = 5² +12² = 169
=> c=13 cm
We can see that the two shorter sides have the same length which is 5cm
Since all the sides of the triangle are the same, hence the two triangles are congruent.
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The shop floor manager in a dairy company believes that the milk packaging process unit for 1 liter packs is not working fine and needs calibration. If the Null Hypothesis is framed to be µ = 1 liter, what would be the alternate hypothesis?
Answer:
Step-by-step explanation:
Given :
Null Hypothesis :
Alterative Hypothesis
two tailed test
claim is Ha
Alterative Hypothesis: Ha: u ≠ 1
The alternative hypothesis that would be used to test the mean quantity is different from 1 liter is u ≠ 1.
What is alternative hypothesis?An assertion used in statistical inference experiments is known as the alternative hypothesis. It is contradictory to the null hypothesis and denoted by Ha or H1.
Given that the claim in the null us that u = 1.
Since, an alternative hypothesis is one in which the researchers anticipate a difference (or an effect) between two or more variables; thus, the observed pattern of the data is not the result of chance.
Therefore,
The alternative would be that u is not equal to 1.
Hence, the alternative hypothesis that would be used to test the mean quantity is different from 1 liter is u ≠ 1.
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PLS HELP
perform the indicated operation and simplify the answers (image attached )
Son solving the provided question we can say that - the fraction is (2 + 1/x ) / ( 7-1/x ) = 5/13
what is fraction?Any number of equal portions, or fractions, can be used to represent a whole. Fractions in standard English indicate how many units of a certain size there are. 8, 3/4. A whole includes fractions. The ratio of the numerator to the denominator is how numbers are expressed in mathematics. Each of these is an integer in simple fractions. In the numerator or denominator of a complex fraction is a fraction. True fractions have numerators that are less than their denominators. A fraction is a sum that constitutes a portion of a total. By breaking the entire up into smaller bits, you can evaluate it. Half of a full number or item, for instance, is represented as 12.
here, the fraction is
(2 + 1/x ) / ( 7-1/x )
2x+1/7x-1
x = 2
5/13
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11 pounds = how many kgs
Answer:
4.98952 kgs in 11 pounds
Step-by-step explanation:
First of all, how many kgs are in a pound?
Well 1 pound = 0.453592 kg.
So, to find how many kg are in 11 pounds you would multiply, 0.453592 by 11, which will be...
4.98952 kgs.
Hope this helps!
Prove that the limit x tends to 1 (2x^4-6x^3+x^2+3)÷(x-1)=-8
Answer:
full answer in below
Step-by-step explanation:
To prove that the limit of (2x^4-6x^3+x^2+3)÷(x-1) as x approaches 1 is equal to -8, we can use the definition of a limit and the fact that (x-1) is not equal to zero at x=1.
The definition of a limit states that:
If we have a function f(x) and a number L, then the limit of f(x) as x approaches a is L, if and only if, for every ε > 0, there exists a δ > 0 such that for every x:
|f(x) - L| < ε when 0 < |x - a| < δ
So, to prove that the limit of (2x^4-6x^3+x^2+3)÷(x-1) as x approaches 1 is -8, we need to show that:
|(2x^4-6x^3+x^2+3)÷(x-1) + 8| < ε when 0 < |x - 1| < δ
We know that (x-1) is not equal to zero at x=1, so we can safely divide both sides of the equation by (x-1) and simplify it to:
|2x^4-6x^3+x^2+11| < ε when 0 < |x - 1| < δ
Now, we can choose ε = 0.1 and δ = 0.1, so we have:
|2x^4-6x^3+x^2+11| < 0.1 when 0 < |x - 1| < 0.1
If we plug in x=1, we have:
|2(1)^4-6(1^3)+(1)^2+11| < 0.1
which simplifies to:
|11| < 0.1
and we see that this is true, since |11| = 11 is less than 0.1.
We can also plug in some values around x=1, for example x=0.99 and x=1.01:
|2(0.99)^4-6(0.99)^3+(0.99)^2+11| < 0.1
and
|2(1.01)^4-6(1.01)^3+(1.01)^2+11| < 0.1
In both cases, we can see that the inequality is true, and the absolute value of the expression on the left side is less than 0.1.
Therefore, we have shown that for any ε > 0, there exists a δ > 0 such that for every x:
|(2x^4-6x^3+x^2+3)÷(x-1) + 8| < ε when 0 < |x - 1| < δ
And thus, by the definition of a limit, we have proven that the limit of (2x^4-6x^3+x^2+3)÷(x-1) as x approaches 1 is -8.
Answer:
is that calculus?
am confuse in your question
can you try take derivative both numerator and denominator with respect to x
by using L - Hospitable rule
A recipe calls for the following spices: 1/8 teaspoon of turmeric, 1/4 teaspoon of ginger, and 1/2 teaspoon of cumin. What is the total quantity of these three spices?
The total quantity of the three spices is 7/8
How to calculate the total quantity of the three spices ?A recipe has three spices
1/8 teaspoon of turmeric
1/4 teaspoon of ginger
1/2 teaspoon of cumin
The total quantity of the three spices can be calculated as follows
1/8 + 1/4 + 1/2
= 7/8
Hence the total quantity of the three spices is 7/8
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Question 3(Multiple Choice Worth 6 points)
(02.06 MC)
Solve x² - 7x = -12.
Ox=2 and x = 6
Ox=6 and x = -2
Ox=-4 and x = -3
Ox=4 and x = 3
Answer:
x = 4 or x = 3
Step-by-step explanation:
x² - 7x = -12
x² - 7x + 12 = 0
(x - 4)(x - 3) = 0
x - 4 = 0 or x - 3 = 0
x = 4 or x = 3
Answer: The answer would be x=4 and x=3.
Step-by-step explanation:
Interest rate is 9 % and doubling time is 8 years. If you Invest $5,000.00, what will it grow to in 24 years?
Choose the best answer from the options below:
A $40.000.00
B $15,000.00
C $20,000.00
D$30,000.00
After 24 years, the amount will be $40,000
What is annual interest rate?
An annual percentage rate is expressed as an interest rate. It calculates what percentage of the principal you'll pay each year by taking things such as monthly payments into account.
It's the ratio of two integers stated as a fraction of a hundred parts. It is a metric for comparing two sets of data, and it is expressed as a percentage using the percent symbol.
Formula:
a(t) = p*2^(t/8)
given, t=24 , p =5000
a(24) = 5000*2^(24/8)
a(24) = 5000*2^3
a(24) = 5000*8
a(24) = $40,000
Thus, after 24 years, the amount will be $40,000
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Is (-1,5) a solution to the equation 2x + 5y = 0 Why or Why not? Is (-1,5) a solution to the equation 2x + 5y = 0 Why or Why not?
The point (-1,5) is not a solution to the equation 2x + 5y = 0 because this point does not satisfy the equation.
What is the equation?The equation is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are equal.
The equation is given in the question, as follows:
2x + 5y = 0
Substitute x = -1 and y = 5 into the equation and see if it holds true:
2(-1) + 5(5) = 0
-2 + 25 = 0
23 = 0
Since 23 ≠ 0, the point (-1,5) is not a solution to the equation 2x + 5y = 0.
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Determine whether a triangle with the given vertices is a right triangle.
So on solving the provided question we can say that from the given data - (a) and (b) are triangles only/
What is triangle?A triangle is a polygon since it has three sides and three vertices. It is one of the basic geometric shapes. The name given to a triangle containing the vertices A, B, and C is Triangle ABC. A unique plane and triangle in Euclidean geometry are discovered when the three points are not collinear. Three sides and three corners define a triangle as a polygon. The triangle's corners are defined as the locations where the three sides converge. 180 degrees is the result of multiplying three triangle angles.
(a) T(-2, 4), R(6, O), 1(-5, -2) - is triangle
(b) p(9, -3), Q(-l, 1), R(-5, -9)- is a triangle
(c) O), E(-6, 10), F(-IO, 4) - can not be determined
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what is the solution of √2x+4 - √x = 2
please show with work done!
Answer:
x=1
Step-by-step explanation:
To solve the equation, we'll need to get the square roots on one side and the non-square root terms on the other side.
First, we'll add √x to both sides to get:
√2x+4 - √x + √x = 2 + √x
√2x+4 = 2 + √x
Next, we'll square both sides to get rid of the square roots:
(√2x+4)^2 = (2 + √x)^2
2x+4 = 4 + 2√x + x
Then, we'll simplify the right side of the equation:
2x+4 = 4 + 2√x + x
2x+4 = 4 + x + 2√x
Now we'll subtract 4 from both sides
2x = x + 2√x
x = 2√x
Finally, we'll divide by x on both sides
1 = 2√x/x
x = √x
x = 1
Therefore, the solution of the equation is x=1
Answer:
To solve the equation √2x+4 - √x = 2, we can first square both sides of the equation to eliminate the square roots.
Squaring both sides gives: (√2x+4)^2 - (√x)^2 = 2^2
Simplifying the left side gives: 2x+4 - x = 4
Next, we can add x to both sides of the equation to get:
2x+4 = 4 + x
then we can subtract 4 from both sides of the equation to get:
2x = x
then we can divide both sides by x
2 = 1
However, it is not possible to divide by zero, so x = 0 is not a valid solution.
This equation is an extraneous solution because the assumption that x is not equal to 0 led to a valid solution, but when x = 0, the equation is not defined.
Therefore, the solution of √2x+4 - √x = 2 is x = any value except 0.
