Answer:
3,4,5
Step-by-step explanation:
Because, it is the only pair that fulfill the pythagorean theorem/formula
Pythagorean formula: [tex]c^2 = \sqrt{a^2 + b^2}[/tex]
Where c is the longest side of the triangle, or the hypothenuse, and a and b is the two perpendicular side.
A sampling distribution of the sample means of many large samples of a probability distribution is normally distributed. Which statement is true according to the Central Limit Theorem?
A the original PD can have any shape
B the original PD must also be normal
C the original PD must be uniform
D the original PD must be symmetric
The statement that is true according to the Central Limit Theorem is B. the original PD must also be normal.
Which is the Central Limit Theorem?According to the central limit theorem, if you take sufficiently large samples from a population, the means of the samples will be normally distributed even if the population is not normally distributed. The central limit theorem (CLT) states that as sample size increases, the distribution of sample means approaches a normal distribution, regardless of the distribution of the population.
The central limit theorem says that the sampling distribution of the mean will always be normally distributed, as long as the sample size is large enough.
In conclusion, the correct option is B.
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You can use induction. Here's a very very rusty sketch of the proof.
Given -
Union of two regular languages is regular.
Let f(n) be a function representing the union of n regular languages.
Question Is f(n) a regular language?
Lk+1 is the k+1th regular language, for example. The presented hypothesis states that f(n) = f(k+1) is regular since f(k) and Lk+1 are both regular.
What is the purpose of the inductive hypothesis?The induction hypothesis's function The assumption made at the beginning of the induction phase is known as the induction hypothesis, and it is the situation where n = k for the proposition we are trying to prove ("P(k)"). If you don't bring this hypothesis into play at any point throughout the proof of the induction phase, something is amiss.
Given:
It is normal for two regular languages to be combined.
Give the function f(n) the name "union of n regular languages."
Question Is the language f(n) regular?
The union of a single regular language is regular
if n = 1. Base Case
The provided hypothesis tells us that f(2) is regular
if n = 2.
Assume that f(n) is regular for every n = k.
This is the inductive hypothesis.
Let n equal k+1 in the inductive step. The inductive premise tells us that f(k) is a regular language.
Consequently, f(n) = f(k+1) = Lk+1 U f(k) (k)
Lk+1 is the k+1th regular language,
for example. The presented hypothesis states that f(n) = f(k+1) is regular since f(k) and Lk+1 are both regular.
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the mass of a radioactive substance follows a continuous exponential decay model, with a decay rate parameter of per day. a sample of this radioactive substance was taken six days ago. if the sample has a mass of today, find the initial mass of the sample. round your answer to two decimal places.
As per the concept of exponential decay, the initial mass of the sample is 5,254.87 kg
What is meant by Exponential decay:
In math, the term exponential decay is a very common process especially when we are talking about radioactive materials.
The standard formula for this type of behavior which is written below:
A = Pe^-rt
where
A refers the amount left after time t
P refers the initial amount at t=0
r refers the rate
Here we have given that the mass of a radioactive substance follows a continuous exponential decay model, with a decay rate parameter of per day. a sample of this radioactive substance was taken six days ago.
And we need to find if the sample has a mass of today, then the initial mass of the sample.
Now, we have to substituting the values,
Then we get the equation as,
=> A = (9575 kg)(e^-0.12*5)
When we simplify this one, then we get,
=> A = 5,254.87 kg
Therefore, the initial mass of the sample is 5,254.87 kg
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Player First
Round Final
Round
Golfer 11 72 72
Golfer 12 72 70
Golfer 13 70 73
Golfer 14 70 77
Golfer 15 68 70
Golfer 16 68 66
Golfer 17 71 70
Golfer 18 70 68
Golfer 19 69 68
Golfer 20 67 71
Suppose you would like to determine if the mean score for the first round of a golf tournament event is significantly different than the mean score for the fourth and final round. Does the pressure of playing in the final round cause scores to go up? Or does the increased player concentration cause scores to come down?
(a) Use α = 0.10 to test for a statistically significantly difference between the population means for first- and fourth-round scores. State the null and alternative hypotheses. (Use μd = mean score first round − mean score fourth round.
Calculate the value of the test statistic. (Round your answer to three decimal places.)
Calculate the p-value. (Round your answer to four decimal places.)
p-value =
(b) What is the point estimate of the difference between the two population means? (Use mean score first round − mean score fourth round.)
The difference between the two population means is estimated at -1.05.
What is meant by Null Hypothesis?Null Hypothesis: The null hypothesis is a typical statistical theory which suggests that no statistical relationship and significance exists in a set of given single observed variable, between two sets of observed data and measured phenomena.
The information presented here represents the golfers' first and final round scores.
a. α= 0.10 is the significance level.
The following hypothesis test can be created:
Hypothesis of nullity H₀ μₐ= 0
This suggests that the population means of the first round and final round are identical.
Alternative formulation: Hₐ μₐ≠0
This suggests that the population means of the first round and final round are different.
The test statistic is now:
t=(d-meanₐ)[tex]\sqrt{n}[/tex]/sₐ
Excel can be used to determine the test statistic.
This will result in: 0.10<P<0.20.
b. The 90% confidence level is the primary margin of error. This indicates that there is no discernible difference, and the result is -1.05.
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As part of her science project , Shay is making a model of a wind farm.She wants to put 24 turbines in her model.What arrays can Shay make using 24 turbines
Answer: In Shay's wind farm their can be 4 arrays
Search up the meaning of a Array if you are confused
Step-by-step explanation:
24 has 4 pairs of factors.
1,242,123,84,6The map of a walking trail is drawn on a coordinate grid with three points of interest. The trail starts at R(−3, 2) and goes to S(2, 2) and continues to T(2, −5). The total length of the walking trail is
units. (Input whole numbers only.) (3 points)
Answer:
12
Step-by-step explanation:
Some teachers use a weighted mean to calculate grades. Each score is assigned a weight based on its importance. To find a weighted mean, multiply each Lab Repor score by its weight and add the results. For example, a student's final chemistry grade is based on four sources: 30% from lab reports, 10% from quizzes, 25% from the midterm exam, and 35% from the final exam. What is the student's weighted mean given the scores shown in the table? Lab reports:81 Quizzes: 94 Midterm Exam: 75 Final Exam: 85
The students weighted score is__
Select all the equations that have g=5 as the solution.
Select all that apply.
A. g-1=4
B. 34+g=50
C. g-1=15
D. g+5=15
A
5-1=4
34+5=39
5-1=4
5+5=10
Answer:
[tex]A)g-1=4\\g=4+1\\g=5[/tex]
Step-by-step explanation:
[tex]A)g-1=4\\g=4+1\\g=5\\\\B)34+g=50\\g=50-34\\g=16\\\\C)g-1=15\\g=15+1\\g=16\\\\\\D)g+5=15\\g=15-5\\g=10\\So the only Option remaining is A) which the correct answer[/tex]
Use the given data to find the minimum sample size required to estimate the population proportion. Margin of error: 0.04; confidence level: 99%; from a prior study, p is estimated by 0.12.
The value of the minimum sample size required to estimate the population proportion is 438.
What is the margin of error?Your results will vary from the actual population value by how many percentage points, as indicated by the margin of error. A 95% confidence interval with a 4% margin of error, for instance, indicates that your statistic will, 95% of the time, be within 4% of the true population figure.
Given:
The margin of error, ME = 0.04,
The value of p = 0.12,
The confidence level = 99%,
Calculate the minimum sample size as shown below,
ME = z √[p(1 - p) / n]
Calculate the value of z from the table attached below considering the confidence level, z = 2.576,
0.04 = 2.576 √[0.12(1 - 0.12) / n]
Squaring both sides,
0.0016 = 6.636 × 0.12 × 0.88 / n
n = 0.7 / 0.0016
n = 437.9 or 438
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In Mr.Hernandez class 7 out of 20 students have an A what percent of the students in the class have an A
To figure this out, you set up the fraction
[tex]\dfrac{\text{number that meet the criteria}}{\text{total possible number}}[/tex]
In this case, it's [tex]\dfrac{7 \text{ A's}}{20 \text{ total students}}[/tex].
You then divide 7/20 = 0.35 and convert that decimal into a percent: 35%
A greengrocer has sorted their apples into three
categories A weighs 12g, B weighs 16g and C
weighs 24g
A customer buys 1 apple A, 3 apples B and 1 apple
C
If A, B and C were to represent different isotopes of
apple what would be the relative abundance of
each type
We need to use the concept of Relative Abundance. Relative Abundance of A,B and C are 14.28,57.14 and 28.57% respectively.
What is Relative Abundance?The percentage of an organism of a specific sort in relation to the overall number of organisms in the area is known as relative abundance. Relative species abundances frequently follow particular patterns, some of the most well-known and extensively researched patterns in macroecology. Over a large variety of biological ecosystems, relative species abundances exhibit relatively similar patterns. Most species are rare (represented by a single individual in a community sample), while only a small number of species are prolific, according to a histogram of the number of species represented by 1, 2, 3,..., n individuals.
(M1)(x) + (M2)(1-x) = M(E) is the formula for Relative Abundance.
CalculationA weighs 12g, B weighs 16g and C weighs 24g.
A customer buys 1 apple A, 3 apples B and 1 apple
So assume A,B and C as different isotopes of apples.
Relative Abundance of A=12/84 ×100⇒14.28%
Relative Abundance of B=48/84 ×100⇒57.14%
Relative Abundance of C=24/84 ×100⇒28.57%
We need to use the concept of Relative Abundance. Relative Abundance of A,B and C are 14.28,57.14 and 28.57% respectively.
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test for an association between handeness and career for these 5 professions. state the null and alternative hypothesis
Null Hypothesis: There is no association between handedness and career for Accountants. And Alternative Hypothesis: There is an association between handedness and career for Accountants.
Profession 1: Lawyer
Null Hypothesis: There is no association between handedness and career for Lawyers.
Alternative Hypothesis: There is an association between handedness and career for Lawyers.
Profession 2: Pilot
Null Hypothesis: There is no association between handedness and career for Pilots.
Alternative Hypothesis: There is an association between handedness and career for Pilots.
Profession 3: Accountant
Null Hypothesis: There is no association between handedness and career for Accountants.
Alternative Hypothesis: There is an association between handedness and career for Accountants.
Profession 4: Nurse
Null Hypothesis: There is no association between handedness and career for Nurses.
Alternative Hypothesis: There is an association between handedness and career for Nurses.
Profession 5: Teacher
Null Hypothesis: There is no association between handedness and career for Teachers.
Alternative Hypothesis: There is an association between handedness and career for Teachers.
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Suppose that the length of a confidence interval is 0.06 when the sample size is 400. Determine how the sample size must change to decrease the length of the confidence interval to 0.03 (Use decimal notation. Give your answer as a whole number:) new sample size:
We use Z-Distribution for large sample sizes, The reduced sample size is 200.
What do you mean by probability distribution?A mathematical function called a probability distribution explains the likelihood of many alternative values for a variable. Graphs or probability tables are frequently used to represent probability distributions.
What do you mean by Z-distribution?A unique type of normal distribution with a mean of 0 and a standard deviation of 1 is known as the standard normal distribution, or z-distribution.
confidence interval = 0.06
Sample size= 400
new interval of confidence = 0.03
new sample size = 400/0.06 * 0.03
=200
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Fill in the blank.
​_____________ is the distribution of all values of the statistic when all possible samples of the same size n are taken from the same population.
The sampling distribution of statistics
The sampling distribution of statistics is the allocation of all matters of the statistic when all viable examples of the exact size n are taken from the exact people.
What is sampling distribution mean?
The group of averages drawn from all feasible random samples of a certain size (n) drawn from a particular population is referred to as the distribution of sample means.
A statistical probability distribution that results from selecting random samples from a specific population is referred to as a sampling distribution.
It depicts the distribution of frequencies on how dispersed various outcomes will be for a particular population and is also referred to as a finite-sample distribution.
To determine the sample distribution, you must know the population's standard deviation.
Divide the total number of observations in the sample by the sum of all the observations.
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Which of the following is true about the t-distribution? Assume a reference (random) sample of size 100 with sample mean 10 and sample standard deviation of 1 with unknown population mean and variance.
a. The t-distribution will look similar to the standard normal distribution as the sample size increases which results in the degrees of freedom increasing.
b. The t-distribution will look similar to the standard normal distribution as the sample size increases which results in the degrees of freedom decreasing.
c. The t-distribution will look similar to the normal distribution with mean 10 and standard deviation 1 as the sample size increases which results in the degrees of freedom decreasing.
d. The t-distribution will look similar to the normal distribution with mean 10 and standard deviation 1 as the sample size increases which results in the degrees of freedom increasing.
e. None of these
1000 times were added to the n=30 sample. Even with n=30, the sample variance distribution is still very biased.
What is probability ?Probability refers to possibility.
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.
You will understand the potential outcomes for a random experiment using this fundamental theory of probability, which is also applied to the probability distribution.
Knowing the total number of outcomes is necessary before we can calculate the likelihood that a specific event will occur.
According to our question-
The sample variance's variance (SD) is still very large. The range is from 20 to 30%.
Because we are utilizing estimated population variance rather than the genuine population variance, which has some degree of uncertainty, the e t distribution has a longer tail when degrees of freedom are limited. Given that the estimated population variance is nearer to the actual population variance when sample size = 30, the t distribution will be closer to the z distribution.
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a teacher figures that final grades in the statistics department are distributed as: a, 25%; b, 25%; c, 40%; d, 5%; f, 5%. at the end of a randomly selected semester, the following number of grades were recorded. find only the test statistics to test the claim that the grade distribution for the department is different than expected. be sure to set up the equation. do not conduct the hypothesis testing. grade a b c d f number 42 36 60 8 14
5.25 option c is the test statistics for the given data.
Given,
The observed values;-
A: 36
B: 42
C: 60
D: 14
E: 8
Total =160
To verify the following claim, a chi square test must be performed:
H0: The proportions for the final grades are the same.
H1: The final grade proportions varies from one another.
The level of significance assumed for this case is α = 0.01
The statistic to check the hypothesis is given by:
X² = ∑ⁿi = 1 (Oi - Ei)²/Ei
Now all that remains is to use the following formula to determine the predicted values, Ei = % × total
And the calculations are given by:
EA = 0.25 × 160 = 40
EB = 0.25 × 160 = 40
EC = 0.4 × 160 = 64
ED = 0.05 × 160 = 8
EF = 0.05 × 160 = 8
Here, the statistics;-
X² = (36 - 40)²/40 + (42 - 40)²/40 + (60 - 64)²/64 + (14 - 8)²/8 + (8 -8)²/8
= 16/40 + 4/40 + 16/64 + 36/8 + 0/8
= 5.25
Therefore,
The test statistics for the given data is option c 5.25
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Solve each equation by completing the square:
0.1²-0.6x+0.9=0
2x²-12x+18=0
Answer:
-3
3
Step-by-step explanation:
0.1x^2-0.6x+0.9=0
(1/10)(x^2+6x+9)=0
(1/10)(x+3)^2=0
x=-3
2x^2-12x+18=0
2(x^2-6x+9)=0
2(x-3)^2=0
x=3
The solution of the equation 0.1x²-0.6x+0.9=0 is -3 and 2x²-12x+18=0 is 3.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
We are given the system of equation as;
0.1x²-0.6x+0.9=0
2x²-12x+18=0
Solving by completing the square:
0.1x²-0.6x+0.9=0
(1/10)(x²+6x+9)=0
(1/10)(x+3)²=0
x=-3
2x²-12x+18=0
2(x²-6x+9)=0
2(x-3)²=0
x=3
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What is the y intercept of the equation y - 6 = -2(x + 5)
Answer:
Step-by-step explanation:
To find the y-intercept of the equation y - 6 = -2(x + 5), we can set x equal to 0, since the y-intercept is the point where the graph of the equation crosses the y-axis.
Substituting 0 for x, we get:
y - 6 = -2(0 + 5)
y - 6 = -2(5)
y - 6 = -10
y = -4
Therefore, the y-intercept of the equation y - 6 = -2(x + 5) is (0, -4). This means that the graph of the equation crosses the y-axis at the point (0, -4).
If angle EBC is congruent to angle ECB, then ? is congruent to ?
By using the property of angle, it can be calculated that [tex]\angle EBA \cong \angle ECD[/tex]
What are angles?
Angles are a way of measuring the amount of turn between two lines or three or more planes. An angle is formed when two lines intersect or when two planes meet. It is measured in degrees, with a full circle being 360 degrees. The angle is measured in terms of the amount of turn between the two lines or planes. Angles are typically used to describe the orientation of objects in space, such as the corners of a triangle, the angle of a flight path, or the angle of the sun’s rays. Angles are also used to measure the size of an angle and the distance between two points. Angles are important in mathematics because they help to analyze the size and shape of objects, and also to help solve problems.
Here,
It is given that
[tex]\angle EBC \cong \angle ECB[/tex]
Now,
[tex]180-\angle EBC \cong 180-\angle ECB[/tex] [Angle on a straight line]
[tex]\angle EBA \cong \angle ECD[/tex]
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Can someone please explain what formula to use when solving this problem ?
Answer:
Step-by-step explanation:
10 months=10/12=5/6 years.
Amount=Principal+ Interest
A=P+I
A=P+P×r×t
A=P(1+rt)
18500=P(1+4%×5/6)
18500=P(1+20/600)
18500=P(1+1/30)
18500=P(31/30)
P=18500×30/31≈17903.2258≈17903.23
A food company distributes its tomato soup in two cans of different sizes. For the larger can, both the diameter and the height have been increased by 10%.
By what percentage does the volume of the can increase from the smaller can to the larger can? Round your answer to the nearest percent.
Answer:
Step-by-step explanation:
If the diameter increases by 10%,
new diameter = d+⅒ = 1.1d
also new radius = 0.55r
Height also increasesby 10%,
New height = h+⅒h = 1.1h
hence diameter of larger can =
π*(0.55)²*1.1= 1.0453
hence percentage change in volume
is 4.53% which can be rounded off to
givena triangle with side ollenghts of 5 and 8 which iniquality describes all possible lenghs of the thhird side
given a triangle with side of lengths of 5 and 8 then c^2 =89 inequality describes all possible lengths of the third side
In the right triangle of sides a, b, and c
a^2 + b^2 = c^2
Since a = 5 and c = 8
Substitute them in the given rule
we get
25+64= c^2
89= c^2
Take a square root for both sides
c= sqrt 89
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what part of 72 is 36a
Answer: 2
Step-by-step explanation: hope it helps
the data below are the final exam scores of 10 randomly selected statistics students and the number of hours they studied for the exam. find the equation of the regression line for the given data. (points : 4) hours x, 3 5 2 8 2 4 4 5 6 3 scores y, 65 80 60 88 66 78 85 90 90 71
Dependent variable: A dependent variable is a variable that changes or is impacted by other factors. independent variable: An independent variable is a variable that does not change or are impacted by other factors.
What is Regression?Regression is a method for figuring out the relationship between two or more variables. In other words, it is determined by how the predictor variable changes and how the dependent variable changes. Additionally, when more than one independent variable is included in a regression analysis, the change in the dependent is examined while maintaining the same values for the other independent variables.
The best fit for the data is linear regression if the data set is bivariate. The data with two variables are best represented by the straight line known as the least squares regression line.
When the explanatory variable has a value of zero, the response variable's projected value is shown by the regression line's y-intercept (though be wary of extrapolation in interpreting the intercept or other values outside the original data range).
The best fit for the data is linear regression if the data set is bivariate. The data with two variables are best represented by the straight line known as the least squares regression line. The line's equation is provided by,
y = a+bx
where,
a—Intercept
b—Regression coefficient
y—Predicted value of the dependent variable
x —Independent or predictor variable
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(a) plot a scatter diagram. does it appear that a simple linear regression will be a suitable model?
On a scatter diagram, you can see outliers if there is a regression line. The point or points that are furthest distant from the regression line are considered outliers in a scatter diagram.
How is a scatter plot related to simple linear regression?In a scatter plot, dots are used to show the values of two separate numerical variables. To see how different variables relate to one another, utilize scatter plots. The link between an independent and dependent variable is shown as a straight line in a linear regression.
How do you know if a linear model is appropriate for a scatter plot?If a linear model is adequate, the residuals scatterplot should display random scatter and the histogram should appear about normal. The linear model is inappropriate if we see a curved connection in the residual plot. The residuals are displayed against the explanatory variable in a different form of residual plot.
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point(s) possible
K
Write the standard form of the equation and the general form of the equation of the circle of radius r and center (h,k).
Graph the circle.
r=√√3; (h,k) = (1,-5)
Write the equation for the circle in general form
The standard equation of a circle is [tex](x-1)^{2} + (y+5)^{2}= (\sqrt{3})^{2}[/tex]. The equation of the circle in general form is (x-1)² + (x-1)² = 3.
What is the standard equation of a circle?Standard form of circle equation is (x – a)² + (y – b)² = r²
What is the general form of the equation of a circle?General form of Equation of a Circle.
x² + y² + 2gx + 2fy + c = 0, for all values of g, f and c.
The equation of a circle is
(x-h)²+ (y-k)² = r²
So, it's given that radius,
r = √3
h=1 and k=-5
So, by substituting all values to the standard equation of a circle is :
[tex](x-1)^{2} + (y+5)^{2}= (\sqrt{3})^{2}[/tex]
The equation of the circle in the general form is written as :
(x-1)² + (x-1)² = 3
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Algebra HELP PLS
The ceiling of Victoria's living room is a square that is 15√2 ft long on each side. To decorate for a party, she plans to hang crepe paper around the perimeter of the ceiling and then from each corner to the opposite corner. Victoria can buy rolls that each contain 30 ft of crepe paper. what is the minimum number of rolls she should buy?
pls show your work
Answer:
Minimum 5 rolls---------------------------------------
Total length we need is the sum of the perimeter and two diagonals of the square.
Perimeter:
P = 4s P = 4(15√2) = 60√2 ≈ 85 ft (rounded up to the whole number)Diagonals:
2d = 2s√2 = 2(15√2)(√2) = 30*2 = 60 ftTotal length:
85 + 60 = 145 ftNumber of rolls:
145/30 ≈ 5 (rounded up to the whole number)The weight of a small car when rounded to the nearest 2 decimal place is 1.24 tons. Find the greatest and least possible weight of the car in 3 decimal place before it is rounded to the nearest hundred. Explain how u arrived at ur answer
Answer:
Below
Step-by-step explanation:
1.235 <= Weight < 1.245 before rounding
ASTU = AXYZ, #27 = 28°, m2U = (4x + y)²
#2X=130%, #2Y = (&r=6p)
The equations are -2-
and y
-[
Answer:
[tex]x=5, y=2[/tex]
Step-by-step explanation:
[tex]4x+y=22 \implies 24x+6y=132 \\ \\ 8x-6y=28 \\ \\ 32x=160 \\ \\ x=5 \\ \\ \implies 4(5)+y=22 \implies y=2[/tex]
Write a quadratic function with the given zero. 3 + i
Answer:
[tex]x^2 - 6x + 10[/tex] or in general: [tex]a(x^2 - 6x + 10)[/tex] where "a" is any constant real number except zero.
Step-by-step explanation:
Complex Conjugate Root Theorem:If we're given a complex root of a polynomial to be: [tex](a+bi)\text{ or }(a-bi)[/tex], then we know that its conjugate is also a root of the polynomial: [tex](a-bi)\text{ or }(a+bi)[/tex].
So since we're given the root: [tex](3+i)[/tex], then we know that its conjugate: [tex](3-i)[/tex] is also a root. Thus we have two roots so far.
Fundamental Theorem of Algebra:The Fundamental Theorem of Algebra simply states that any polynomial, with degree "n", will have "n" roots, which can consist of both real and imaginary solutions.
So let's say I have a polynomial: [tex]x^6+3x+2=0[/tex], this has 6 roots, according to the Fundamental Theorem of Algebra, since the degree of the polynomial is 6 (remember the degree is simply the variable with the highest degree)
Since we already have two roots, and a quadratic function has a degree of two, we know that these are all the roots the quadratic has, since the Fundamental Theorem of Algebra states that a quadratic will only have 2 roots.
Factored Form:We can generally express any polynomial in factored form, that is: [tex]a(x-b)(x-c)(x-d)...=y[/tex]
where "b", "c", and "d", and so on... are zeros of the function. This aligns with the behavior of zeros, since if we plug any of them into the function, then one of the factors will become zero, and since zero times anything is zero, well the entire thing is zero.
For example if I plug in "b", which is zero we get: [tex]a(b-b)(b-c)(b-d)...=y\implies a(0)(b-c)(b-d)=y\implies 0=y[/tex]
this is since plugging in zero, makes the factor: [tex]x-b[/tex] equal to zero, making the entire thing zero.
One last thing to mention is the "a" value in front which just generally determines the stretch/compression of the polynomial.
So with this information, we can write a polynomial:
[tex]a(x-(3+i))(x-(3-i))[/tex]
From here let's distribute the negative sign:
[tex]a(x-3-i)(x-3+i)[/tex]
Now this may look a bit tough to distribute, but you may notice a pattern here, we have a difference of squares. It may not be obvious, but if we make one substitution, let's say: [tex]u=x-3[/tex], then we get the equation
[tex]a(u-i)(u+i)[/tex]
The difference of squares is expressed as: [tex]a^2 - b^2 = (a-b)(a+b)[/tex], so our expression will expand out into:
[tex]a(u^2 - i^2)[/tex]
The substitution was just to make things a bit more clear, but let's just substitute x-3 back into the equation.
[tex]a((x-3)^2 - i^2)[/tex]
From here, we know that i^2 is simply -1, since: [tex]\sqrt{-1} = i, \text{ so }i^2 = -1[/tex]
We can also easily expand out the polynomial using FOIL
[tex]a((x^2 - 6x + 9) -(-1))\\a(x^2 - 6x + 9 + 1)\\a(x^2 - 6x + 10)[/tex]
You may notice the "a value, and that's just a constant, which determines the stretch/compression of the polynomial, and can be any real number, except for zero since that will make the entire thing zero regardless of what x is. For the sake of simplicity, let's just write this to be 1.