According to the factor theorem, if "a" is any real integer and "f(x)" is a polynomial of degree n larger than or equal to 1, then (x - a) is a factor of f(x) if f(a) = 0. Finding the polynomials' n roots and factoring them are two of their principal applications.
What is the remainder and factor theorem's formula?When p(x) is divided by xc, the result is p if p(x) is a polynomial of degree 1 or higher and c is a real number (c). For some polynomial q, p(x)=(xc)q(x) if xc is a factor of polynomial p. The factor theorem in algebra connects a polynomial's components and zeros. The polynomial remainder theorem has a specific instance in this situation. According to the factor theorem, f(x) has a factor if and only if f=0.The remainder will be 0 if the polynomial (x h) is a factor. In contrast, (x h) is a factor if the remainder is zero.The factor theorem is mostly used to factor polynomials and determine their n roots. Factoring is helpful in real life for comparing costs, splitting any amount into equal parts, exchanging money, and comprehending time.To learn more about Factor theorem refer to:
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what's the leading term and constant of -.5x^5+1.5
We have the following polynomial:
[tex]-0.5x^5+1.5[/tex]And we have to determine which is the leading term, and the constant term of that polynomial.
1. To determine that we know that the leading term is that term in the polynomial that contains the highest power of the variable. In this case, the variable is x, and the term with the highest variable is:
[tex]-0.5x^5\rightarrow\text{ This is the leading term}[/tex]2. To determine the constant term, we have to remember that this term is not associated with the variable, that is, is not a coefficient of the variable. Therefore, the constant term is 1.5.
Hence, in summary, we have that:
[tex]\text{ Leading term: }-0.5x^5[/tex]And
[tex]\text{ Constant term: }1.5[/tex]Determine the value(s) of x at which the function is discontinuous. Describe the discontinuity as removale or non-removable.
Answer with explanation: To find the values of x where the f(x) is discontinuous, we have to set the denominator equal to zero, doing this gives:
[tex]\begin{gathered} f(x)=\frac{x^2+10+9}{x^2-81}\Rightarrow x^2-81=0 \\ x=\sqrt[]{81}=9 \\ x=9 \end{gathered}[/tex]The f(x) is discontinuous at x = 9, following graph confirms it:
In conclusion, discontinuity is non-removable.
In boot camp, a cadet must use a rope swing to cross an obstacle withoutfalling into the water hazard below. Unfortunately, they miss the platform onthe other side and swing back to where they started. If it takes the cadet 3.5seconds to swing from one side of the obstacle to the other and back, howlong is the rope swing? Use the formula:
Answer:
Choice C: 3.0 m
Explanation:
We are basically asked to solve for L using
While waiting for the school bus, Michiko records the colors, of all cars passing through an intersection. Thetable shows the results, Estimate the probability that the next car through the intersection will be red. Exgressyour answer as a percent. If necessary, round your anewer to the nearest tenth
Given the following question:
Estimate the probability that the next car will be red.
11, 24, 16, 9
[tex]\begin{gathered} 11\text{ + 24}=35 \\ 35\text{ + 16 = 51} \\ 51\text{ + 9 = }60 \\ 60=100\text{per} \end{gathered}[/tex][tex]p=\frac{11}{60}[/tex][tex]\frac{11}{60}\times100=18.333333[/tex][tex]\begin{gathered} 18.333333 \\ 3\text{ < 5} \\ 18.3 \end{gathered}[/tex]18.3% or the first option.
You might need:CalculatorHiro painted his room. After 3 hours of painting at a rate of 8 square meters per hour, he had 28 squaremeters left to paint.Let y represent the area (in square meters) left to paint after chours.Which of the following information about the graph of the relationship is given?
The graph is that of area painted against the number of hours. Given that he painted 8 square meters per hour, the slope is 8 square meters per hour because slope is known as unit rate.
After 3 hours of painting, he would have painted 8*3 = 24 square meters.
Since he has 28 meters left to paint, it means that the total rea of the room that
The same set of data has been fit using two different functions. The following images show the residual plots of each function.
We have the residuals of each function graphed.
They represent the distance, taking into account the sign, of each data point to the line of best fit.
A good fit will have residuals that are close to the x-axis. Also, the distribution for the residuals should not have too much spread, meaning that all the points should have approximately the same residual in ideal conditions.
In this case, we see that Function A has most residuals around the horizontal axis. Except for one of the points, that may be considered an outliert.
In the case of Function B there is a clear pattern (a quadratic relation between x and the residual) that shows that the degree of the best fit function is not the adequate (maybe two degrees lower than what should be).
This results in residuals that have a wide spread depending on the value of x.
Then, we can conclude that Function A has a better fit because the points are clustered around the x-axis.
Answer: Function A has a better fit because the points are clustered around the x-axis [Third option]
If a cell disruptor is purchased with a frequency of 60Hz, what is the wavelength traveling through human tissue? (1540 m/s).
The wavelength traveling through human tissue when the velocity is 1540 m/s and frequency is 60Hz will be 25.67 m.
According to the question,
We have the following information:
Frequency of a cell disruptor = 60 Hz
Velocity of the cell disruptor = 1540 m/s
We know that the following formula is used to find the wavelength:
Wavelength = Velocity/frequency
Wavelength = 1540/60 m
(Note that when velocity is given m/s and frequency is given in Hz then the unit of wavelength is m. Every physical quantity has to be expressed with its units.)
Wavelength = 25.67 m
Hence, the wavelength traveling through human tissue is 25.67 m.
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Use a sample mean to estimate a population mean with a certain specified OA certainty OB. accuracy OC. confidence OD. determination
SOLUTION
Given the question on the question tab;
Explanation:
When the sample mean is used as a point estimate of the population mean, some error can ... Lower levels of confidence lead to even more narrow intervals.
Final answer:
If the line y-7=3(x+5) is dilated with a center at the origin and a scale factor of 4, which of the following equations would describe its image?
When we dilate a line with a scale factor of "a", we change its equation by changing the slope.
The slope gets multiplied by "a".
In the equation given, the slope is "3". After a dilation with a scale factor of 4, the slope would become:
3 * 4 = 12
The equation, after dilation, would be:
[tex]y-7=12(x+5)[/tex]The graph shows the first four ordered pairs formed by the corresponding terms of two patterns. Which ordered pair would be the fifth point on this graph? (4,12) (12,4) (12,8) (10, 4) Q1 6 7 8 9 10 11 12
As shown in the graph:
There are four points:
(0,0) , (3, 1) , ( 6, 2) and ( 9, 3)
The points represent a proportion relation between x and y
The relation will be:
[tex]y=\frac{1}{3}x[/tex]So, the fifth point will be: ( 12, 4)
What is the slope of the line that passes through the points (2,8) and (12,20)?
The slope of the line with that passes through the coordinates (2,8) and (12,20) is 6/5.
What is the slope of the line with the given coordinates?Slope is simply expressed as change in y over the change in x.
Slope m = ( y₂ - y₁ )/( x₂ - x₁ )
Given the data in the question;
Point 1( 2,8 )
x₁ = 2y₁ = 8Point 2( 12,20 )
x₂ = 12y₂ = 20Slope m = ?
To find the slope m, plug the given x and y values into the slope formula and simplify.
Slope m = ( y₂ - y₁ )/( x₂ - x₁ )
Slope m = ( 20 - 8 )/( 12 - 2 )
Slope m = ( 12 )/( 10 )
Slope m = 12/10
Slope m = 6/5
Therefore, the slope of the line is 6/5.
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Statistics: a professor recorded 10 exam grades but one of the grades is not readable. if the mean score on the exam was 82 and the mean of the 9 readable scores is 84 what is the value of the unreadable score?
To mean of a set is given by the sum of all values in the data-set divided by the number of values.
We have that the mean of the whole set is 82.
The mean of the 9 readable scores is 84.
So:
[tex]\begin{gathered} \frac{x}{9}=84 \\ x=84\cdot9 \\ x=756 \end{gathered}[/tex]So, we 9 readable scores add up to 801. If we add 756 to a number, y, and divide by 10, we'll have the mean score of the exam, 82.
[tex]\begin{gathered} \frac{756+y}{10}=82 \\ 756+y=820 \\ y=820-756 \\ y=64 \end{gathered}[/tex]So, the grade of the unreadable score was 64.
Which data sets should be displayed on a stem display instead of a dot plot? Select all that apply. A) 11, 23, 9, 24, 34, 18, 15, 11, 8, 14, 16B) -14, -15, -17, -15, -15, -15, -12, -14.-14C) 5,3, 8, 3, 7,5,6,3, 7, 3, 7, 6,5,6D) 1.1, 1.2, 1.1, 1.3, 1.4, 1.1, 1.2, 1.4, 1.2, 1.1E) 42.7, 39.8, 41.1, 39.7, 40.1, 39.8.42.3
In order to determine which data sets should be displayed on a stem display, you consider that the stem display is usefull in the cases in which you have data which can be grouped easily. For instance, for data set in which there are differents number with the same first digit(s).
According with the previous definition you can notice that the options E) and A) are the best options, because there are different number that can be grouped, for example, according to the first number.
For other options you have other situations, for option D) there is no way to group the data. For option C) there is only one number on each data, so, there wouldn't be leafs in the diagram, and the same applies to option B), the first number is the same in all data, then, there is no way to group.
Tom said that the difference in length between the longest trail and the shortest trail is 2 5/8 miles. Does Tom's answer make sense? What mistake did he make? Answer in at least two complete sentences. Use the sentences below to get started: "Tom's answer (makes sense/does not make sense). His mistake was ________."
Solution:
Given:
From the trail lengths given,
[tex]\begin{gathered} The\text{ longest trail is }1\frac{7}{8} \\ The\text{ shortest trail is }\frac{3}{4} \end{gathered}[/tex]The difference in length between the longest trail and the shortest trail:
[tex]\begin{gathered} 1\frac{7}{8}-\frac{3}{4}=\frac{15}{8}-\frac{3}{4} \\ =\frac{15-6}{8} \\ =\frac{9}{8} \\ =1\frac{1}{8} \end{gathered}[/tex]
The sum of the longest trail and the shortest trail.
[tex]\begin{gathered} 1\frac{7}{8}+\frac{3}{4}=\frac{15}{8}+\frac{3}{4} \\ =\frac{15+6}{8} \\ =\frac{21}{8} \\ =2\frac{5}{8} \end{gathered}[/tex]From the calculations above, the conclusion can be reached that:
Tom's answer does not make sense. His mistake was he did the sum of the longest trail and the shortest trail.
What is period of the function, give the exact value
Solution
Step 1:
Find the midline
[tex]\begin{gathered} Midline\text{ = }\frac{maximum\text{ + minimum}}{2} \\ Maximum\text{ = 11.4} \\ minimum\text{ = -5.5} \\ midline\text{ = }\frac{11.4\text{ + \lparen-5.5\rparen}}{2} \\ midline\text{ = }\frac{5.9}{2} \\ midline\text{ = 2.95} \end{gathered}[/tex]Step 2:
Find the amplitude
[tex]\begin{gathered} Amplitude\text{ = }\frac{maximum\text{ - minimum}}{2} \\ Amplitude\text{ = }\frac{11.4\text{ - \lparen-5.5\rparen}}{2} \\ Amplitude\text{ = 8.45} \end{gathered}[/tex]Step 3:
Period:
To find the period, use the values of x.
[tex]\begin{gathered} Period\text{ = 2\lparen11.4 + 5.5\rparen} \\ Period\text{ = 2 }\times\text{ 16.9} \\ period\text{ = 33.8} \end{gathered}[/tex]Final answer
Period = 33.8
I dont know the steps to solve this expression, help.
5
1) Let's solve that expression step by step
[tex]\frac{35}{2^3-1}[/tex]2) As we have an exponent, let's firstly solve this
[tex]\frac{35}{8^{}-1}[/tex]Now proceeding with the subtraction, and then divide it:
[tex]\frac{35}{7}=5[/tex]3) Hence, the answer is 5
(1 point) A variable of a population has a mean of I = 250 and a standard deviation of o = 49.
Solution
Question 1a:
- The population mean and sample mean are approximately the same in theory. The only difference is that the distribution of the sample will be wider due to a larger uncertainty caused by having less data to work with.
- Thus, we have:
[tex]\begin{gathered} \text{ Sample Mean:} \\ 250 \\ \\ \text{ Standard Deviation:} \\ \frac{\sigma}{\sqrt{n}}=\frac{49}{\sqrt{49}}=\frac{49}{7}=7 \\ \end{gathered}[/tex]Question 1b:
- The assumption is that the distribution is a normal distribution (OPTION C)
Question 1c:
Yes, the sampling distribution of the sample mean is always normal (OPTION B). This is in accordance with the central limit theorem.
use the listing method to represent the following set. picture attached
The correct option is A
{3, 4, 5, 6, ...}
Explanation:The condition given states that x is greater or equal to 3.
The only option that corresponds to this condition is:
{3, 4, 5, 6, ...}
3. Convert the angle 3π/4 to degrees.
Answer:
135°
Step-by-step explanation:
To convert an angle from radians to degrees, multiply by [tex]180/\pi[/tex].
[tex]\frac{3\pi}{4} \cdot \frac{180}{\pi}=135^{\circ}[/tex]
The stem-and-leaf plot shows the number of hours per semester that students in a certain school club watchtelevision. What is the greatest number of hours that were watched?
Answer:
45
Explanation:
The number on the left of the line represent the tens and the numbers on the right of the line represent the units, so data for the stem and leaf plot is:
20, 22, 22, 22, 23, 23
30, 31, 33, 35
40, 42, 42, 43, 43, 43, 45
It means that the greatest number of hours is 45.
Then, the answer is 45.
A bicycle wheel is 63 centimeters from top to bottom . When the wheel goes all the way around one time , the bicycle travels 198 centimeters . How can this information be used to estimate the value of pi
Given :
A bicycle wheel is 63 centimeters from top to bottom .
So, the diameter of the wheel = 63 cm
When the wheel goes all the way around one time , the bicycle travels 198 centimeters .
So, the circumference of the circle = 198 cm
The circumference of the circle of diameter = d will be :
[tex]\pi\cdot d[/tex]So,
[tex]\begin{gathered} \pi\cdot63=198 \\ \\ \pi=\frac{198}{63}=\frac{22}{7} \end{gathered}[/tex]hello I don't know if you can help me with this but I no am doing something wrong. because at the bottom its not spelling right
8. The difference of three and a number means x+3 or x-3 because in both equations you have three units plus or minus the number X.
10. '"4 times the sum of a number and three" means
[tex]4\cdot(x+3)=4x+12[/tex]Letter D
Simplify the expression below. Show all steps and calculations to earn full credit. You may want to do this work by hand and upload an image of that written work rather than try to type it all out. \sqrt[]{\frac{ \sqrt[3]{64}+\sqrt[4]{256}}{\sqrt[]{64}+\sqrt[]{256}}}
We are given the expression:
[tex]\sqrt[]{\frac{ \sqrt[3]{64}+\sqrt[4]{256}}{\sqrt[]{64}+\sqrt[]{256}}}[/tex]We will simplify this as shown below:
[tex]\begin{gathered} \sqrt[]{\frac{ \sqrt[3]{64}+\sqrt[4]{256}}{\sqrt[]{64}+\sqrt[]{256}}} \\ \text{Let's consider \& solve the terms one after the order, we have:} \\ \sqrt[3]{64}\Rightarrow\sqrt[3]{4\times4\times4}\Rightarrow4 \\ \sqrt[4]{256}\Rightarrow\sqrt[4]{4\times4\times4\times4}\Rightarrow4 \\ \sqrt[]{64}\Rightarrow\sqrt[]{8\times8}\Rightarrow8 \\ \sqrt[]{256}\Rightarrow\sqrt[]{16\times16}\Rightarrow16 \\ We\text{ will substitute each of these expressions back into the parent expression, we have:} \\ \sqrt[]{\frac{4+4}{8+16}} \\ =\sqrt[]{\frac{8}{24}} \\ =\sqrt[]{\frac{1}{3}} \\ =\frac{\sqrt[]{3}}{\sqrt[]{3}\times\sqrt[]{3}} \\ =\frac{\sqrt[]{3}}{3} \\ \Rightarrow\sqrt[]{\frac{\sqrt[3]{64}+\sqrt[4]{256}}{\sqrt[]{64}+\sqrt[]{256}}}=\frac{\sqrt[]{3}}{3} \\ \\ \therefore\frac{\sqrt[]{3}}{3} \end{gathered}[/tex]y= -2x - 7x - y = -8
Given the system of equations:
[tex]\begin{gathered} y=-2x-7\rightarrow(1) \\ x-y=-8\rightarrow(2) \end{gathered}[/tex]we will find the solution to the system by graphing
To draw the lines, we need to know two points on each line
so, substitute with two values of x and calculate the corresponding value of y
For line (1): y = -2x - 7
[tex]\begin{gathered} x=0\rightarrow y=-2\cdot0-7=-7 \\ x=1\rightarrow y=-2\cdot1-7=-9 \end{gathered}[/tex]so, line (1) passes through the points ( 0, -7) and ( 1, -9)
For line (2): x - y = -8
y = x + 8
[tex]\begin{gathered} x=0\rightarrow y=8 \\ x=1\rightarrow y=1+8=9 \end{gathered}[/tex]So, line 2 passes through the points ( 0, 8) and ( 1, 9)
The graph of the line will be as shown in the following picture
As shown in the figure:
Line (1) is the blue line
Line (2) is the red line
The point of intersection = ( -5, 3)
So, the solution is point ( -5, 3)
2) A humane society claims that 30% of U.S. households own a cat. In a random sample of 210 U.S. households, 80 say they own a cat. Is there enough evidence to show this percent has changed? Use a level of significance of 0.05.
ANSWER:
There is enough evidence to reject the humane society claims
STEP-BY-STEP EXPLANATION:
Given:
p = 0.3
q = 1 - p = 1 - 0.3 = 0.7
n = 210
x = 80
Therefore:
[tex]\hat{p}=\frac{x}{n}=\frac{80}{210}=0.381[/tex]The critical values are:
[tex]Z_0=\pm1.96\text{ due }\alpha=0.05[/tex]The test statistic is:
[tex]\begin{gathered} Z=\frac{\hat{p}-p}{\sqrt[]{\frac{pq}{n}}} \\ \text{ replacing:} \\ Z=\frac{0.381-0.3}{\sqrt{\frac{0.3\cdot0.7}{210}}} \\ Z=2.56 \end{gathered}[/tex]Observe that
Z < 1.96
Therefore, reject the null hypothesis
There is enough evidence to reject the humane society claims
ANSWER:
There is enough evidence to reject the humane society claims
STEP-BY-STEP EXPLANATION:
Given:
p = 0.3
q = 1 - p = 1 - 0.3 = 0.7
n = 210
x = 80
Therefore:
[tex]\hat{p}=\frac{x}{n}=\frac{80}{210}=0.381[/tex]The critical values are:
[tex]Z_0=\pm1.96\text{ due }\alpha=0.05[/tex]The test statistic is:
[tex]\begin{gathered} Z=\frac{\hat{p}-p}{\sqrt[]{\frac{pq}{n}}} \\ \text{ replacing:} \\ Z=\frac{0.381-0.3}{\sqrt{\frac{0.3\cdot0.7}{210}}} \\ Z=2.56 \end{gathered}[/tex]Observe that
Z < 1.96
Therefore, reject the null hypothesis
There is enough evidence to reject the humane society claims
A gamer spinner, circle O, is divided into 3 regions as shown. RP is a diameter. what is the area of the shaded sector ROS if RP=8 in a m
we get that the radius is 4 and the angle is 135° which is radians 3/4 pi
so the area is
[tex]A=\frac{\pi}{2}\cdot4^2-\frac{\pi}{8}\cdot4^2=8\pi-2\pi=6\pi\approx18.85[/tex]I’m a parent and I’m not sure I’m understanding this question the practice test question says “How would you take apart 14 to solve 28 - 14? The choices are 7 and 710 and 4 12 and 220 and 8. I said 7 and 7
There are several ways of taking apart the number 14:
1 and 13
2 and 12
3 and 11
4 and 10
5 and 9
6 and 8
7 and 7
Nevertheless, as we can see by the note below that exercise,
"Have your child take apart 16 to make a ten to find 87 - 16",
we can conclude that the exercise is asking how to take apart 14 to make a ten. By doing so, the subtraction operation (28 - 14) gets simpler since you could subtract 4, and then subtract the ten:
14 = 10 + 4 (1 ten and 4 units)
So, when we do 28 - 14, we can do that in two steps:
• 28 - 4 = ,24
,• 24, - 10 = 14
Therefore, based on the note below exercise 6, the expected answer is
10 and 4
Given the following confidence interval for a population mean compute the margin of error E
Given that the Confidence Interval for a population mean:
[tex]11.81<\mu<13.21[/tex]In this case, you can set up these two equations:
[tex]\bar{x}+E=13.21\text{ \lparen Equation 1\rparen}[/tex][tex]\bar{x}-E=11.81\text{ \lparen Equation 2\rparen}[/tex]Because by definition:
[tex]\bar{x}-E<\mu<\bar{x}+E[/tex]Where "ME" is the margin of error and this is the mean:
[tex]\bar{x}[/tex]In this case, in order to find the "ME", you need to follow these steps:
1. Add Equation 1 and Equation 2:
[tex]\begin{gathered} \bar{x}+E=13.21 \\ \bar{x}-E=11.81 \\ -------- \\ 2\bar{x}=25.02 \end{gathered}[/tex]2. Solve for the mean:
[tex]\begin{gathered} \bar{x}=\frac{25.02}{2} \\ \\ \bar{x}=12.51 \end{gathered}[/tex]3. Substitute the mean into Equation 1 and solve for "ME":
[tex]12.51+E=13.21[/tex][tex]\begin{gathered} E=13.21-12.51 \\ E=0.7 \end{gathered}[/tex]Hence, the answer is:
[tex]E=0.7[/tex]General Mills is testing 12 new cereals for possible production. They are testing 3 oat cereals, 5 wheat cereals, and 4 rice cereals. If each of the 12 cereals has the same chance of being produced,
and 4 new cereals will be produced, determine the probability that of the 4 new cereals that will be produced, 2 are oat cereals, 1 is a wheat cereal, and 1 is a rice cereal.
The probabilis
(Type an integer or a simplified fraction.)
Using the combination formula, the probability that of the 4 new cereals that will be produced, 2 are oat cereals, 1 is a wheat cereal, and 1 is a rice cereal is of 4/33.
Combination Formula[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula, involving factorials.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
This formula is used when the order in which the objects are chosen is not important, as is the case in this problem.
A probability is given by the number of desired outcomes divided by the number of total outcomes.
For the total outcomes, 4 cereals are taken from a set of 12, hence:
[tex]T = C_{12,4} = \frac{12!}{4!8!} = 495[/tex]
For the desired outcomes, we have that:
2 oat are taken from a set of 3.1 wheat is taken from a set of 5.1 rice is taken from a set of 4.Hence the number is:
[tex]D = C_{3,2}C_{5,1}C_{4,1} = \frac{3!}{2!1!} \times \frac{5!}{1!4!} \times \frac{4!}{1!2!} = 3 \times 5 \times 4 = 60[/tex]
Hence the probability is:
p = 60/495.
Both numbers can be simplified by 5, hence:
p = 12/99.
They can also be simplified by 3, hence:
p = 4/33.
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5-74.The number of girls at Middle SchoolCyber Summer Camp was six morethan twice the number of boys. Therewere a total of 156 middle schoolstudents at the camp. Use the 5-DProcess to find the number of boysand the number of girls at camp.
We have the following:
Describe/Draw
The statement tells us that the number of girls was 6 plus twice the number of boys and that in total there are 156 students.
Define
Number of girls: y, y = 6 + 2x
Number of boys: x
Number of students: 154
Do
[tex]\begin{gathered} x+y=156 \\ x+6+2x=156 \\ 3x=156-6 \\ x=\frac{150}{3} \\ x=50 \end{gathered}[/tex]for y:
[tex]y=6+2\cdot50=6+100=106[/tex]Decide
The answer is correct because the sum of both is equal to 156 students
Declare
In total there are 50 boys and 106 girls