Answers
Answer:
\(\huge\color{purple}\boxed{\colorbox{black}{Answer♡}}\)
\( \frac{2}{ \sqrt{7} - 1} \\ = > \frac{2}{ \sqrt{7} - 1} \times \frac{ \sqrt{7} + 1}{ \sqrt{7} + 1} \\ = > \frac{2( \sqrt{7} + 1)}{ \sqrt{7} {}^{2} - 1 {}^{2} } \\ = > \frac{2 \sqrt{7} + 2}{7 - 1} \\ = > \frac{2( \sqrt{7} + 1)}{6} \\ = > \frac{ \sqrt{7} + 1}{3} \)
———————————————\(\color{red}\boxed{\colorbox{skyblue}{Additional Information~}}\)
Here..Step 3 contains the algebraic identity( a + b ) × ( a - b ) = a² - b²
Also.. in the very last step ,we earlier had 2 multiplied to rest of the numerator and 6 in the denominator..
we simply divided both the numerator as well as denominator by 2 and hence got the result obtained.
hope helpful~
~Be Brainly!~Exhausted Zombie
Related Questions
Problem # 1. Consider the roll of a pair of fair dice. Let Ak denote the event that the number of dots facing Up is k; for k = 2, 12_ (There are 1l such events ) Let Bk denote the event that this number is greater O equal to k. Let E and denote the events that the number is even or odd, respectively. Find the probabilities: P[Ak]; and P[Bk], for k = 2, b) P[OIBs] P[Ag U An|Bs] P[Bslo] P[BslBz] f) P[E 0 BolBs] 12 The probability that the two dice show different outcomes
Answers
a) P[Ak] is the probability that the sum of the two dice is k, which can be calculated by counting the number of ways that the dice can add up to k and dividing by the total number of possible outcomes (36).
For example, P[A2] is the probability that both dice show 1, which is 1/36. Similarly, P[A7] is the probability that the sum is 7, which is 6/36 or 1/6.
b) P[Bk] is the probability that the sum of the two dice is greater than or equal to k. This can be calculated by counting the number of ways that the dice can add up to a number greater than or equal to k and dividing by the total number of possible outcomes (36). For example, P[B2] is 1 because the sum of the two dice is always greater than or equal to 2. P[B7] is 6/36 or 1/6 because there are six ways to roll a sum of 7.
c) P[OIBs] is the probability that the sum of the two dice is either odd or greater than or equal to 6. This can be calculated by adding the probabilities of the two events and subtracting the probability of their intersection. P[O] is the probability that the sum is odd, which is 18/36 or 1/2. P[B6] is the probability that the sum is greater than or equal to 6, which is 21/36 or 7/12. P[OIB6] is the probability that the sum is both odd and greater than or equal to 6, which is 9/36 or 1/4. Therefore, P[OIBs] is (1/2) + (7/12) - (1/4) = 5/6.
d) P[Ag U An|Bs] is the conditional probability that the sum is either even or odd, given that the sum is greater than or equal to 7. This can be calculated using Bayes' theorem: P[Ag U An|Bs] = P[Bs|Ag U An] P[Ag U An] / P[Bs]. P[Bs] is the probability that the sum is greater than or equal to 7, which is 20/36 or 5/9. P[Ag U An] is the probability that the sum is either even or odd, which is 1. P[Bs|Ag U An] is the probability that the sum is greater than or equal to 7, given that it is even or odd. This is the same as P[B7], which we calculated earlier. Therefore, P[Ag U An|Bs] = (1/6) * (1) / (5/9) = 3/10.
e) P[Bslo] is the probability that the sum is greater than or equal to 10, which is 3/36 or 1/12. P[BslBz] is the probability that the sum is greater than or equal to 11, which is 2/36 or 1/18.
f) P[E 0 BolBs] is the probability that the sum is even and greater than 4, given that it is odd. This can be calculated using Bayes' theorem: P[E 0 BolBs] = P[BolBs|E] P[E] / P[BolBs]. P[E] is the probability that the sum is even, which is 18/36 or 1/2. P[BolBs] is the probability that the sum is odd, which is also
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Handmade socks, knitted using pure cashmere wool, are very expensive to buy. Rowena buys cashmere wool in 20g balls. Each ball of cashmere wool costs her £1.42 . She pays her sister £8 to knit each pair of socks. 135g of cashmere wool is used to knit each pair of socks. Rowena sells 40 pairs of cashmere socks at £18.95 per pair. What is her percentage profit? Give your answer correct to 2 significant figures.
Answers
The percentage profit for the so is that were made is 40.32%.
How to calculate the profit?From the information, Rowena buys cashmere wool in 20g balls. Each ball of cashmere wool costs her £1.42 and she pays her sister £8 to knit each pair of socks. The cost will be:
= (£1.42 × (20 + 135) + (40 × £8)
= £220.1 + £320
= £540.1
The selling price will be:
= £18.95 × 40
= £758
Therefore, the percentage profit will be:
= Profit / Cost price × 100
= (758 - 540.1) / 540.1 × 100
= 217.9 / 540.1 × 100
= 40.32%
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Number 10 is the question
Answers
B is your answer
every answer equals 0.4 except B
(Chapter 12) If u * v = 0 and u X v = 0, then u or v = 0
Answers
Therefore, in either partial derivatives, we have u = 0 or v = 0.
The given information implies that two vectors u and v satisfy:
u * v = 0, where * denotes the dot product between vectors.
u X v = 0, where X denotes the cross product between vectors.
From the first equation, we know that the angle between u and v is either 90 degrees or 270 degrees. That is, u and v are orthogonal (perpendicular) to each other.
From the second equation, we know that the magnitude of the cross product u X v is equal to the product of the magnitudes of u and v multiplied by the sine of the angle between them. Since u and v are orthogonal, the angle between them is either 90 degrees or 270 degrees, which means that the sine of the angle is either 1 or -1. Therefore, we have:
|u X v| = |u| * |v| * sin(θ)
= 0
Since the magnitudes of u and v are non-negative, it follows that sin(θ) must be zero. This can only happen if the angle between u and v is either 0 degrees (i.e., u and v are parallel) or 180 degrees (i.e., u and v are anti-parallel).
In the case where u and v are parallel, we have:
u * v = |u| * |v| * cos(θ)
= |u|²
= 0
This implies that |u| = 0, which means that u = 0.
In the case where u and v are anti-parallel, we have:
u * v = |u| * |v| * cos(θ)
= -|u|²
= 0
This again implies that |u| = 0, which means that u = 0.
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A total of 50,000 contestants are chosen for a survey. A
computer randomly eliminates 20% of the participants
each day.
Which function models the survey?
Answers
Answer:
you are correct do not say any other answer
Answer:
The answer is either
y=50000 x 0.8^x
or
y=50000(20%)^x but this just seems wrong
Step-by-step explanation:
Given that csc(θ) < 0 and cot(θ) < 0, in which quadrant does θ lie? Select the correct answer below:
Quadrant I
Quadrant II
Quadrant III
Quadrant IV
Answers
The angle θ lies in Quadrant II.
To determine the quadrant in which θ lies, we need to analyze the signs of the trigonometric functions csc(θ) and cot(θ). The cosecant function (csc) is defined as the reciprocal of the sine function (sin), and the cotangent function (cot) is defined as the reciprocal of the tangent function (tan).
Since csc(θ) is negative, it means that the sine function (sin) is negative in Quadrant II and Quadrant III. However, since cot(θ) is also negative, it implies that the tangent function (tan) is negative in Quadrant II only.
In Quadrant II, both sine (sin) and tangent (tan) are negative, which satisfies the given conditions. In contrast, in Quadrant III, the sine function (sin) is negative but the tangent function (tan) is positive. Therefore, the angle θ must lie in Quadrant II, where both csc(θ) and cot(θ) are negative.
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which graph shows the solution to the system of linear equations?
y=-1/3x+1
y=-2x-3
Answers
y = -1/3x + 1
y = -2x - 3
We can compare the equations to the graphs and see which graph represents the intersection point of the two equations.
The first equation, y = -1/3x + 1, has a negative slope (-1/3) and a y-intercept of 1.
The second equation, y = -2x - 3, also has a negative slope (-2) and a y-intercept of -3.
Based on the slopes and y-intercepts, we can identify the correct graph by finding the point where the two lines intersect.
Unfortunately, since the graphs are not provided, I am unable to determine which specific graph shows the solution to the system of linear equations. I recommend referring to the graph representation of the equations and identifying the intersection point to determine the correct graph.
There are 42 runners in a race. How many different ways can the runners finish first, second, and third?
Answers
Answer:
There are 68,640 different ways the runners can finish first, second, and third in the race.
Concept of Permutations
The number of different ways the runners can finish first, second, and third in a race can be calculated using the concept of permutations.
Brief Overview
Since there are 42 runners competing for the top three positions, we have 42 choices for the first-place finisher. Once the first-place finisher is determined, there are 41 remaining runners to choose from for the second-place finisher. Similarly, once the first two positions are determined, there are 40 runners left to choose from for the third-place finisher.
Calculations
To calculate the total number of different ways, we multiply the number of choices for each position:
42 choices for the first-place finisher × 41 choices for the second-place finisher × 40 choices for the third-place finisher = 68,640 different ways.
Concluding Sentence
Therefore, there are 68,640 different ways the runners can finish first, second, and third in the race.
If f(1) = 0, what are all the roots of the function ? use the remainder theorem.
Answers
By using synthetic division, we can find all the roots of the function f(x) = x³ + 3x² - x - 3 and determine that the roots are 1, -1, and -3.
A root of a function is a value for x that makes the function equal to zero.
In this case, we have a function f(x) = x³ + 3x² - x - 3 and we know that f(1) = 0. To find all the roots of this function, we can use the remainder theorem.
The remainder theorem states that if we divide a polynomial by a linear expression (such as x - a), the remainder will be equal to the value of the polynomial at that value of x (a). In other words, if we write f(x) = (x - a)q(x) + r, where r is the remainder, then f(a) = r.
So, if we divide our function f(x) by x - 1, the remainder will be equal to f(1), which we know is zero. Therefore, (x - 1) is a factor of f(x), and the function has a root at x = 1.
Next, we can divide f(x) by (x - 1) to get a quotient, q(x), and a new remainder, r. We can repeat this process, dividing q(x) by another linear expression until we have no more remainders.
The roots of f(x) will be the values of x that we used as factors in each step of the division process.
This process of dividing a polynomial by a linear expression to find its roots is called synthetic division.
Complete Question
If f(1) = 0, what are all the roots of the function f(x) = x³ + 3x² - x - 3. Use the remainder theorem.
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Each box holds 48 pencils. The chart below shows the number of pencils in 1 box and 5 boxes.
Help ASAP please
Answers
Answer: No, 6 boxes will give you 288 penicils.
Step-by-step explanation: 1 box is 48 penicils, so if use this to the other boxes (4x48) we get 192. Now with six boxes (6x48) we get 288.
Rationalizing the denominator
Answers
Answer:
Fraction form: \(\frac{10\sqrt{14} }{7}\)
Decimal form: 5.34522483
I believe it is 5.35 if rounded
In how many ways can 7 people line up for play tickets?A. 40,320B. 5,040C. 823,543D. 7
Answers
Given the question: In how many ways can 7 people line up for play tickets?
The number of ways for the first = 7
The number of ways for the second = 6
The number of ways for the third = 5
The number of ways for fourth = 4
The number of ways for the fifth = 3
The number of ways for the sixth = 2
The number of ways for the seventh = 1
So, the total number of ways = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5,040
So, the answer will be option B. 5,040
The sum of 4 and Janelle's height is 88
Answers
Answer:
So her height would be 84
Step-by-step explanation:
Samuel invested money in his bank account. He had a principal, P, of $100 in his account at the beginning of the period, which increased at a rate, r, of 0.15 per year. At the end of the period, he had interest, I, of $105 in his account. Use the simple interest formula I = Prt, to solve for the time, t, in months that it took to earn this amount.
Answers
Hey there! I'm happy to help!
Let's see what we have for each variable.
I = 105
P = 100
r = 0.15
Now, let's plug it in and solve for our t (time)
105=100(0.15)(t)
Swap sides so t is on the left.
100(0.15)(t)=105
Divide both sides by 100.
0.15t=1.05
Divide both sides by 0.15
t=700
So, we see that it takes 700 months to earn this amount.
Have a wonderful day and keep on learning! :D
The number of months that it took to earn this amount will be 84 months.
What is simple interest?Simple interest is the concept that is used in many companies such as banking, finance, automobile, and so on.
I = (PRT)/100
Where P is the principal, R is the rate of interest in percentage, and T is the time.
Samuel invested money in his bank account.
He had a principal, P, of $100 in his account at the beginning of the period, which increased at a rate, R, of 0.15 per year.
At the end of the period, he had interest, I, of $105 in his account.
105 = 100 x 0.15 x t
t = 1.05 / 0.15
t = 7 years
The number of months that it took to earn this amount will be
⇒ 7 x 12
⇒ 84 months
The number of months that it took to earn this amount will be 84 months.
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Explain why public opinion polls use sample sizes of more than 1000 people instead of using a smaller sample size.
Answers
Public opinion polls use sample sizes of more than 1000 people, rather than smaller sample sizes in order to obtain more accurate and reliable results.
What are public opinion polls?Public opinion polls are surveys conducted to gather data and insights on the opinions, attitudes, and preferences of a particular population or a specific group of individuals.
Public opinion polls use sample sizes of more than 1000 people for the following reasons:
A larger sample size increases the statistical power of the pollThe margin of error decreases as the sample size increases.A larger sample size increases the likelihood that the sample is representative of the entire population.A larger sample size gives higher confidence levels.Learn more about public opinion polls at: https://brainly.com/question/10404069
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Using the GCF you found in Part b re write 35 +63 factor one factor is the GCF and the other is the sum of the two numbers that do not have a common factor show your work
Answers
Answer:
7(9 + 14)
Step-by-step explanation:
The Greatest Common Factor of 35 and 63 is 7; both 35 and 63 are evenly divided by 7. Thus, 35 + 63 can be rewritten as 7(5) + 7(9), or 7(14)
So 35 + 63 = 7(9 + 14)
Answer:
= 98
Step-by-step explanation:
which object is more likely to be 5kg?
a lamp
a car
a dollar bill
????
Answers
Answer:
A dollar bill is to light and the car is two heavy so the car
Step-by-step explanation:
Hope this helps!
Pretty sure a lamp.
A car would be 1000kg, a dollar bill less than 1kg, so most likely a lamp.
Prove that ∑i=1[infinity]2i1=1.
Answers
After using the formula for the sum of an infinite geometric series, we conclude that the given infinite series does not converge to 1.
To prove that the infinite series ∑(i=1 to ∞) 2^(i-1) equals 1, we can use the formula for the sum of an infinite geometric series.
The sum of an infinite geometric series with a common ratio r (|r| < 1) is given by the formula:
S = a / (1 - r)
where 'a' is the first term of the series.
In this case, our series is ∑(i=1 to ∞) 2^(i-1), and the first term (a) is 2^0 = 1. The common ratio (r) is 2.
Applying the formula, we have:
S = 1 / (1 - 2)
Simplifying, we get:
S = 1 / (-1)
S = -1
However, we know that the sum of a geometric series should be a positive number when the common ratio is between -1 and 1. Therefore, our result of -1 does not make sense in this context.
Hence, we conclude that the given infinite series does not converge to 1.
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only 18 percent of the peices in the box were unpainted. if 738 oieces were painted, how many pieces were in the box?
Answers
If 18% of the pieces in the box were unpainted, that means 82% of the pieces in the box were painted.
And since 738 pieces were painted, that number comprises 82% of the total number of pieces.
Therefore, the number of pieces in the box was : 738 : 82% = 900 (pieces)
how many solutions are there to square root x =9
Answers
Answer:
There are 2 solutions to square root x = 9
They are 3, and -3
Step-by-step explanation:
The square root of x=9 has 2 solutions,
The square root means, for a given number, (in our case 9) what number times itself equals the given number,
Or, squaring (i.e multiplying with itself) what number would give the given number,
so, we have to find the solutions to \(\sqrt{9}\)
since we know that,
\((3)(3) = 9\\and,\\(-3)(-3) = 9\)
hence if we square either 3 or -3, we get 9
Hence the solutions are 3, and -3
Hi can anyone help me with question 1 and 3
Answers
Answer:
292 and 2092
Step-by-step explanation:
Surface area:
Wall 1: 40 x 3 = 120m^2
Wall2: 40 x 3 = 120m^2
Wall 3: 15 x 3 = 45m^2
Wall 4: 15 x 3 = 45m^2
Ceiling: 15 x 40 = 600m^2
Total : 240 + 600 + 90 = 730
So, cost of painting = 730/25 x 10 = 29.2 x 10 = 292
Total: 292 + 1800 = 2092
The total cost of paint would be 2092 dollars.
The area of the cuboid is the sum of product of the length, breadth of the given prism.
Surface area:
Wall 1: 40 x 3 = 120m^2
Wall2: 40 x 3 = 120m^2
Wall 3: 15 x 3 = 45m^2
Wall 4: 15 x 3 = 45m^2
Ceiling: 15 x 40 = 600m^2
The Total area: 240 + 600 + 90 = 730
So, The cost of painting = 730/25 x 10
= 29.2 x 10
= 292
Total: 292 + 1800 = 2092
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Complete the square for the expression.
x2 − 11x + ____
−121/2
121/2
121/4
−121/4
Answers
To complete the square for the expression x2 − 11x + ____, we need to add a constant term that will make the expression a perfect square trinomial.
We can do this by taking half of the coefficient of the x-term (-11/2) and squaring it (121/4). Thus, x2 − 11x + 121/4 can be written as (x - 11/2)2.
We can check this by expanding (x - 11/2)2:
(x - 11/2)2 = x2 - 11/2 x - 11/2 x + (121/4)
= x2 - 11x + 121/4
Therefore, the complete expression is (x - 11/2)2, which is equivalent to x2 − 11x + 121/4.
In summary, to complete the square for x2 − 11x + ____, we add the constant term 121/4. This turns the expression into a perfect square trinomial, which can be written as (x - 11/2)2.
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A cylindrical object is 3.13 cm in diameter and 8.94 cm long and
weighs 60.0 g. What is its density in g/cm^3
Answers
A cylindrical object is 3.13 cm in diameter and 8.94 cm long and weighs 60.0 g. The density of the cylindrical object is 0.849 g/cm^3.
To calculate the density, we first need to find the volume of the cylindrical object. The volume of a cylinder can be calculated using the formula V = πr^2h, where r is the radius (half of the diameter) and h is the height (length) of the cylinder.
Given that the diameter is 3.13 cm, the radius is half of that, which is 3.13/2 = 1.565 cm. The length of the cylinder is 8.94 cm.
Using the values obtained, we can calculate the volume: V = π * (1.565 cm)^2 * 8.94 cm = 70.672 cm^3.
The density is calculated by dividing the weight (mass) of the object by its volume. In this case, the weight is given as 60.0 g. Therefore, the density is: Density = 60.0 g / 70.672 cm^3 = 0.849 g/cm^3.
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Use the Binomial Theorem to expand the binomial. SHOW STEPS
(d-4b)^2
Some one please help!!!
Answers
The binomial theorem says
\(\displaystyle (a + b)^n = \sum_{k=0}^n \binom nk a^{n-k} b^k \\\\(a+b)^n= \binom n0 a^n + \binom n1 a^{n-1}b + \cdots + \binom n{n-1} ab^{n-1} + \binom nn b^n\)
where
\(\dbinom nk = \dfrac{n!}{k!(n-k)!}\)
is the so-called binomial coefficient. The binomial coefficients follow a neat pattern called Pascal's triangle:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
where the n-th row, starting with n = 0, lists the coefficient of the k-th term in the sum (with 0 ≤ k ≤ n). For example,
\(\displaystyle n=0 \implies (a + b)^0 = \sum_{k=0}^0 \binom nk a^{0-k} b^k = \binom00 a^{0-0} b^0 = \underline{1}\)
\(\displaystyle n=1 \implies (a + b)^1 = \sum_{k=0}^1 \binom 1k a^{1-k} b^k = \binom10 a^{1-0} b^0 + \binom11 a^{1-1}b^1 = \underline{1}a + \underline{1}b\)
\(\displaystyle n=2 \implies (a + b)^2 = \sum_{k=0}^2 \binom 2k a^{2-k} b^k = \binom20a^{2-0}b^0+\binom21a^{2-1}b^1+\binom22a^{2-2}b^2 = \underline{1}a^2 + \underline{2}ab+\underline{1}b^2\)
Each row starts and ends with 1, and every coefficient in between is obtained by adding the coefficients directly above and to the left. The next row, for instance, would be
1 1 + 4 4 + 6 6 + 4 4 + 1 1
or
1 5 10 10 5 1
which is to say,
\(\displaystyle (a + b)^5 = \underline{1}a^5+\underline{5}a^4b+\underline{10}a^3b^2+\underline{10}a^2b^3+\underline{5}ab^4+\underline{1}b^5\)
In your case, we have (d - 4b)², so we take a, b, and n above with d, -4b, and 2, respectively:
\(\displaystyle (d - 4b)^2 = \sum_{k=0}^2 \binom 2k d^{2-k} (-4b)^k\)
\(\displaystyle (d - 4b)^2 = \sum_{k=0}^2 \binom 2k (-4)^k d^{2-k} b^k\)
\(\displaystyle (d - 4b)^2 = \binom20 (-4)^0 d^{2-0} b^0 + \binom21 (-4)^1 d^{2-1} b^1 + \binom22 (-4)^2 d^{2-2} b^2\)
\(\displaystyle (d - 4b)^2 = 1\cdot1\cdot d^2\cdot1 + 2 \cdot (-4) d \cdot b + 1 \cdot 16 \cdot 1 \cdot b^2\)
\(\boxed{(d - 4b)^2 = d^2 - 8bd + 16b^2}\)
Which theorem shows that AABC A DEF?
A The triangles are not congruent.
B. SAS Triangle Congruence Theorem
C. Isosceles Triangle Theorem
D. SSS Triangle Congruence Theorem
Answers
The graphs below have the same shape. What is the equation of the blue
graph?
F(X) = x2
5
G(X) = ?
5
G(X) =
Answers
Answer:
\(g(x) = (x - 2)^2 + 1\)
Step-by-step explanation:
Given
\(f(x) = x^2\)
Required
Determine g(x)
f(x) represents the parent function. And from the graph, we have:
\(f(x) = x^2\)
f(x) is first shifted 2 units right.
The rule is:
\(f'(x) = f(x - 2)\)
So, we have:
\(f'(x) = (x - 2)^2\)
Next, f'(x) is shifted 1 unit up to give g(x), the blue graph.
The rule to this is:
\(g(x) = f'(x) + 1\)
\(g(x) = (x - 2)^2 + 1\)
five yellow m&m for every 3 green. How many green m&m if there are 95 yellow ?
PLEASE HELP
Answers
Which of the following statements best describe an invertible function? Select all that apply.
Answers
Statement (B) "an invertible function is a function that has an inverse that is also a function" best describes an invertible function.
What is an invertible function?The inverse function of a function f in mathematics is a function that reverses the operation of f. When a function has an inverse, it is said to be invertible. It is symbolized by f-1. A function must be one-to-one, Onto, or just bijective in order to have a clearly defined inverse. For instance, since it is one-to-one and Onto or Bijective, f(x)=2x+11 is invertible. Invertible, which as its name implies means "inverse," refers to a function's inverse. In the broadest sense, inverse functions are functions that "reverse" one another. For instance, if f goes from a to b, then f-1 must go from b to a.So, we now know that an invertible function is a function that has an inverse which is also a function.
Therefore, statement (B) "an invertible function is a function that has an inverse that is also a function" best describes an invertible function.
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Solve for x. Round to the nearest tenth, if necessary.
Answers
So the answer is 1.3 after rounding to 10.
Which equation is best to use to determine the zeros of the graph of y = 4x^2 – 8x - 5?
(2x + 1)(2x - 5) = 0
(2x - 1)(2x - 5) = 0
(4x - 1)(x - 5) = 0
(4x + 1)(x - 5) = 0
Answers
Answer:
(a) (2x +1)(2x -5) = 0
Step-by-step explanation:
The best equation to use to find the zeros is the one that represents the correct factored form of the given equation. The only factoring that is correct is ...
(2x +1)(2x -5) = 0
__
Additional comment
This tells you the zeros are x = -1/2 and x = 5/2.
__
The factoring can be checked by checking each of the terms of the product. The leading term is the product of leading terms, 4x² in every case, so that is uninformative.
The constant is the product of the constants, so only (1)(-5) gives the correct -5 constant (eliminates B and C).
The linear term is (4)(-5) +(1)(1) = -19 for the last choice, so that is incorrect. It is (2)(-5) +(1)(2) = -8 for the first choice, so that is the correct factorization.