Answers
a) We are given that f(x) = x^3 - 10x^2 + 19x + 30 and x - 6 is a factor of f(x). Using long division or synthetic division, we can divide f(x) by x - 6 to get the other factor(s):
6 │ x^3 - 10x^2 + 19x + 30
x^3 - 6x^2
-------------
-4x^2 + 19x
-(-4x^2 + 24x)
---------------
5x + 30
5x - 30
-------
0
Therefore, we can write:
f(x) = (x - 6)(x^2 - 4x + 5)
The quadratic factor x^2 - 4x + 5 cannot be factored further using real numbers.
b) We are given that f(x) = x^3 + 6x^2 + 5x - 12 and x + 4 is a factor of f(x). Using long division or synthetic division, we can divide f(x) by x + 4 to get the other factor(s):
-4 │ x^3 + 6x^2 + 5x - 12
-x^3 - 4x^2
--------------
2x^2 + 5x
2x^2 + 8x
------------
-3x - 12
-3x - 12
--------
0
Therefore, we can write:
f(x) = (x + 4)(x^2 + 2x - 3)
The quadratic factor x^2 + 2x - 3 can be factored further using the product-sum method:
x^2 + 2x - 3 = (x + 3)(x - 1)
Therefore, we have:
f(x) = (x + 4)(x + 3)(x - 1)
Related Questions
Find the value of x.
Answers
The value of x in the chord is 3 units.
How to find line segment when chord intersect?The chord intersection theorem states that the products of the lengths of the line segments on each chord are equal.
Therefore,
6(x + 5) = 4(2x + 6)
Open the brackets
6x + 30 = 8x + 24
subtract 8x from both sides of the equation
6x - 8x + 30 = 24
-2x + 30 = 24
subtract 30 from both sides of the equation'
-2x + 30 - 30 = 24 - 30
-2x = -6
divide both sides by -2
x = -6 / 2
x = 3
Therefore,
x = 3
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What’s the correct answer for problem 16???
Answers
The values of x and y of the given transverse lines are:
x = 44.25° and y = 7.25°
How to find the missing angle on the line?We know that there are different classification of angles such as:
Corresponding angles
Alternate angles
Opposite angles
Supplementary angles
Complementary angles
Now, we also know that sum of angles on a straight line is 180 degrees. Thus:
5x - 9y - 16 = 140
5x - 9y = 156 -----(1)
We also know that alternate angles are defined as two angles, that are formed when a line crosses two other lines, that lie on opposite sides of the transversal line and on the opposite relative sides of the other lines. If the two lines crossed are parallel, then the alternate angles are equal.
Thus:
3x + y = 5x - 9y - 16
2x - 10y - 16 = 0
2x - 10y = 16 ----(2)
Solving simultaneously gives us:
x = 44.25° and y = 7.25°
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PLEASE HELP I have 30 minutes! What is the end behavior of this equation and how did you get it?
Answers
The limit of the function will be:
lim f(x) = ∞ as x → ∞
lim f(x) = 0 as x → -∞
How to explain the functionThe limit of f(x) as x goes towards infinity is infinite while the limit of f(x) approaches 0 when x tends to a negative value. Mathematically speaking,
lim f(x) = ∞ as x → ∞
lim f(x) = 0 as x → -∞
We can comprehend such behavior because the function's base is greater than 1 which explains how its growth accelerates as x increases and diminishes quickly as x declines towards zero.
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A leading magazine (like Barron's) reported at one time that the average number of weeks an individual is unemployed is 32 weeks. Assume that for the population of all unemployed individuals is normally distributed and the population mean length of unemployment is 32 weeks and that the population standard deviation is 3.9 weeks. Suppose you would like to select a random sample of 66 unemployed individuals for a follow-up study.
Answer the following, rounding all answers to three decimal places.
Find the probability that a single randomly selected value is greater than 31.9.
P(X > 31.9) = ???
Find the probability that a sample of size
is randomly selected with a mean greater than 31.9.
P(M > 31.9) = ???
Answers
The probability that a single randomly selected value is greater than 31.9 is 0.511.
The probability that a sample of size 66 is randomly selected with a mean greater than 31.9 is 0.641.
How to calculate the probabilityProbability that a single randomly selected value is greater than 31.9:
z = (31.9 - 32) / 3.9 = -0.026
We can find the probability:
P(X > 31.9) = P(Z > -0.026) = 0.511
Also, μ = 32, σ = 3.9, n = 66
z = (31.9 - 32) / (3.9 / √66) = -0.363
Using a standard normal distribution table or calculator, we can find the probability:
P(M > 31.9) = P(Z > -0.363) = 0.641
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A man drove 14 miles directly east from his home, made a left turn at an intersection, and then traveled 7 miles north to his place of work. If a road was made directly from his home to his place of work, what would its distance be to the nearest tenth of a mile?
Answers
The distance traveled east and the distance traveled north form the two legs of a right triangle, and the distance we want to find is the hypotenuse.
Using the Pythagorean theorem, we have:
distance^2 = (14 miles)^2 + (7 miles)^2
distance^2 = 196 + 49
distance^2 = 245
distance = sqrt(245)
distance = 15.62 miles (rounded to the nearest tenth)
Therefore, the distance between the man's home and his place of work, if a road was made directly between them, would be approximately 15.6 miles.
In how many distinct ways can the digits in the number 2,563,183,083 be arranged?
(If there are any zero digits then assume they can be placed in any position)
Answers
However, we need to account for the fact that three of the digits are 3s and two of the digits are 8s. To do this, we need to divide out the number of arrangements of those duplicates.
There are 3! = 6 ways to arrange the three 3s among themselves, and 2! = 2 ways to arrange the two 8s among themselves. So, we divide the total number of arrangements by these permutations of the duplicates:
10! / (3! * 2!) = 10 x 9 x 8 x 7 x 6 x 5 x 4 = 604,800
Therefore, there are 604,800 distinct ways to arrange the digits in the number 2,563,183,083, assuming that any zero digits can be placed in any position.
A company produces two products, A and B. At
least 30 units of product A and at least 10 units of
product B must be produced. The maximum
number of units that can be produced per day is
80. Product A yields a profit of $15 and product B
yields a profit of $8. Let a = the number of units of
product A and b = the number of units of product
B.
What objective function can be used to maximize
the profit?
P=
DONE✔
a+
b
Answers
The objective function that can be used to maximize the profit is Profit = 15a + 8b
Let a is the number of units of product A and b is the number of units of product B.
Profit = 15a + 8b
This function represents the total profit earned by producing a units of product A and b units of product B
Given that the profit per unit of product A is $15 and the profit per unit of product B is $8.
To maximize the profit, we would need to find the values of a and b that satisfy the constraints given in the problem and maximize the value of the objective function
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-3k - 7 ≤ 17 help!!!!!!!!!!!!!!!
Answers
k is less than or equal to -8
work in the pic i added but i’ll explain it too :)
you move the 7 to the right side of the equation by adding it to both sides.
now you have
-3k < 24
then you isolate the k by dividing both sides by -3
then you get
k is less than or equal to -8
What is 15% of 100
Please help
Answers
Answer:
Step-by-step explanation:15
Best way to think about it 1% of 100 is 1. So 2% is 2 and so on so forth.
How do I help my student understand this?
Answers
Cell 1 is in the "7th grade" row and "Pets" column. Have your student circle these labels or highlight them. Then it should be clear why choice E is the final answer.
Another way to rephrase it: the 35 people in cell 1 represent the number of 7th graders with pets.
What is the probability that a random sample of 20 second grade students from the city results in a mean reading rate of more than 96 words per minute?
Answers
Answer:
Step-by-step explanation:
We can calculate the t-statistic and the p-value as follows:
If we assume that the null hypothesis is true, then the t-statistic will follow a t-distribution with 19 degrees of freedom. We can compare the calculated t-statistic with the critical t-value at the 0.05 level of significance using a t-distribution table or statistical software.
If the calculated t-statistic is greater than the critical t-value, then we can reject the null hypothesis and conclude that there is sufficient evidence to support the alternative hypothesis that the population mean reading rate is greater than 96 words per minute.
The p-value is the probability of obtaining a t-statistic as extreme or more extreme than the calculated t-statistic, assuming the null hypothesis is true. We can use the t-distribution table or statistical software to find the corresponding p-value.
If the p-value is less than the significance level of 0.05, then we can reject the null hypothesis at the 0.05 level of significance and conclude that the probability of obtaining a random sample of 20 second grade students with a mean reading rate of more than 96 words per minute is statistically significant.
How to write Two times the sum of y and 5 is 24
Then solve
Answers
Answer:
Step-by-step explanation:
2(y+5)=24
Solved answer: y=7
You’d first have to distribute the 2 into the parenthesis.
Then you’d have to isolate the 2y by moving the 10 on the other side of the equal sign.
Finally, divide by 2 on both sides to get rid of the 2y, making it just y.
Hope this helps!
Your friend has a bag of yellow and purple candy. He wants to use them to play a game with you.
Purple is worth 2 points and yellow is worth three points. Pick 9 candies from the bag
Create a system of equations where the combination of the candies gives you a total of 22 points. Make sure you label and define your variables.
Answers
The system of equations where the combination of the candies gives you a total of 22 points are:
x + y = 9
2x + 3y = 22
How can we create system of equations?To create the equations, we shall first define the variables:
x = the number of purple candies
y = the number of yellow candies
Next, we shall use the given information that purple candies are worth 2 points and yellow candies are worth 3 points.
Then, we would make sure that the total number of candies selected is 9.
The first equation represents the total number of candies selected:
x + y = 9
The second equation represents the total number of points obtained:
2x + 3y = 22
Therefore, the system of equations is:
x + y = 9
2x + 3y = 22
This is a system of linear equations.
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Out of 600 people sampled, 42 had kids. Based on this, construct a 95% confidence interval for the true population proportion of people with kids.
Use GeoGebra to calculate! Give your answers as decimals, to three places
Answers
Answer:
So we can be 95% confident that the true proportion of people with kids in the population is between 0.044 and 0.096.
Step-by-step explanation:
To construct a confidence interval, we need to use the formula:
CI = p ± zsqrt((p(1-p))/n)
Where:
CI = confidence interval
p = sample proportion (in this case, 42/600 = 0.07)
z = the z-score corresponding to the desired level of confidence (in this case, 1.96 for a 95% confidence interval)
n = sample size (in this case, 600)
Plugging in the numbers, we get:
CI = 0.07 ± 1.96sqrt((0.07(1-0.07))/600)
Simplifying this, we get:
CI = 0.07 ± 0.026
Therefore, the 95% confidence interval for the true population proportion of people with kids is:
0.044 ≤ p ≤ 0.096
A straw is placed inside a rectangular box that is g inches by 6 inches by 7 inches, as shown. If the straw fits exactly into the box diagonally from the bottom left corner to the top right back corner, how long is the straw? Leave your answer in simplest radical form.
Answers
Answer:
The length of the straw will be 5.91607 inches long in radical form.
What is Length of diagonal in cuboid?
If a cuboid has length = l units, breadth = b units, and height = h units, then we can evaluate the length of the diagonal of a cuboid using the formula
As per the question the length of the straw is equal to length of the diagonal of Rectangular box.
l= 3 inch , b= 5 inch , h= 1 inch
Let us consider the length of the straw be 'x'.
As, Diagonal of Cuboid is,
X= 5.91607 (in radical form)
Step-by-step explanation:
Hence, the length of the straw will be 5.91607 inches.
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Prove the following this : Let H be a subgroup of a group G . The left cosets of H constitute a partition of G ; the right cosets of H constitute a partition of G ?
Answers
To prove that the left cosets of H constitute a partition of G, we need to show that they satisfy two conditions:
1) Each element of G belongs to some left coset of H.
2) Any two distinct left cosets of H are disjoint.
Let g be any element of G. Then, by definition of a left coset, there exists an element h of H such that g = h·x for some x in G. Therefore, g belongs to the left coset H·x. Since this holds for any element g of G, we have shown that the left cosets of H cover all of G, satisfying the first condition.
Now, let H·x and H·y be two distinct left cosets of H. Suppose there exists an element z in both H·x and H·y.
Then, z = [tex]h_{1}[/tex]·x = [tex]h_{2}[/tex]·y for some [tex]h_{1}[/tex], [tex]h_{2}[/tex] in H.
Solving for x,
x = [tex]h_{1}^{-1}[/tex]· [tex]h_{2}[/tex] · y.
Since H is a subgroup, [tex]h_{1}^{-1}[/tex]· [tex]h_{2}[/tex] is also in H, and therefore x belongs to H·y.
Thus, H·x is contained entirely in H·y, and the two cosets cannot intersect.
This satisfies the second condition, and hence the left cosets of H constitute a partition of G.
A similar argument can be used to show that the right cosets of H also constitute a partition of G. This completes the proof.
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Express 1:0.2:0.75 in their simple form
Answers
Answer:
20 : 4 : 15
Step-by-step explanation:
1 : 0.2 : 0.75
1 : 2/10 : 75/100
1 : 20/100 : 75/100
(×100)
100 : 20 : 75
(÷5)
20 : 4 : 15
Select all the correct answers.
If the measure of angle is which statements are true?
sin (0) = -
The measure of the reference angle is 30°.
cos (0) = √3
The measure of the reference angle is 45°.
tan (0) = -√3
The measure of the reference angle is 60°.
Answers
The measures of the trigonometric relations and reference angles are solved
Given data ,
Let the measure of the angle be θ = 2π/3
Now , from the trigonometric relation ,
The tangent of the function tan ( 2π/3 ) = -√3
And , the reference angle of the θ = 2π/3 is given by A = π/3
So , A = 60°
Hence , the reference angle is 60°
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Please if you know the answer put the steps on thank you.
Answers
Answer:
1. # of people who predicted they would pass = 30
2. # of people who predicted they would fail = 20
3. # of people who predicted they would pass and actually passed = 27
4. # of people who predicted they would pass and actually failed = 3
5. # of people who predicted they would fail and actually passed = 11
6. # of people who predicted they would fail and actually failed = 9
Step-by-step explanation:
1. # of people who predicted they would
The total number of people who took the test is 50.The number of people who predicted they would pass is 30.The number of people who predicted they would fail is 20 (since 50 - 30 = 20)Let x be the number of people who predicted fail and actually passed the test. Since three times as many people who passed predicted pass than predicted fail, we know that 3x is the number of people who predicted pass and actually passed the test. Therefore, the total number of people who passed the test is x + 3x = 4x, and we know that 36 people passed the test, so 4x = 36, and x = 9.Since x is the number of people who predicted fail and actually passed the test, then the number of people who predicted fail and actually failed the test is 20 - x = 20 - 9 = 11.The number of people who predicted pass and actually passed the test is 3x = 3(9) = 27.The number of people who predicted pass and actually failed the test is 30 - 27 = 3.I also filled in the frequency table by extracting it from Brainly and drawing on it to show how the math works and fits in the table.
Emilio puts $4,000.00 into an account to use for school expenses. The account earns 15% interest, compounded annually. How much will be in the account after 6 years?
Round your answer to the nearest cent.
Answers
Answer:
$10,359.73.
Step-by-step explanation:
We can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = the amount of money after the specified time
P = the principal amount (the initial amount of money)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the time (in years)
In this case, we have:
P = $4,000.00
r = 15% = 0.15
n = 1 (compounded annually)
t = 6 years
Substituting into the formula, we get:
A = 4000(1 + 0.15/1)^(1*6)
A = 4000(1.15)^6
A ≈ $10,359.73
Therefore, the amount in the account after 6 years, rounded to the nearest cent, is $10,359.73.
A = P(1 + r/n)^(nt)
where:
A = the amount of money in the account after t years
P = the initial principal (or starting amount), which is $4,000.00 in this case
r = the annual interest rate as a decimal, which is 0.15 (since the interest rate is 15%)
n = the number of times the interest is compounded per year, which is 1 since the interest is compounded annually
t = the number of years, which is 6 in this case
Plugging in the values, we get:
A = 4000(1 + 0.15/1)^(1*6)
A = 4000(1.15)^6
A ≈ $9,432.44
Therefore, the amount in the account after 6 years is approximately $9,432.44.
In tetrahedron $ABCO,$ $\angle AOB = \angle AOC = \angle BOC = 90^\circ.$ A cube is inscribed in the tetrahedron so that one of its vertices is at $O,$ and the opposite vertex lies on face $ABC.$ Let $a = OA,$ $b = OB,$ and $c = OC.$ Show that the side length of the cube is
\[\frac{abc}{ab + ac + bc}.\]
[asy]
import three;
size(180);
currentprojection = orthographic(6,3,2);
real a, b, c, s;
triple A, B, C, O;
a = 6;
b = 3;
c = 2;
s = a*b*c/(a*b + a*c + b*c);
A = (a,0,0);
B = (0,b,0);
C = (0,0,c);
O = (0,0,0);
draw(O--A,dashed);
draw(O--B,dashed);
draw(O--C,dashed);
draw(A--B--C--cycle);
draw((0,0,s)--(s,0,s)--(s,0,0)--(s,s,0)--(0,s,0)--(0,s,s)--cycle,dashed);
draw((s,s,0)--(s,s,s),dashed);
draw((s,0,s)--(s,s,s),dashed);
draw((0,s,s)--(s,s,s),dashed);
label("$A$", A, SW);
label("$B$", B, E);
label("$C$", C, N);
dot("$O$", O, NW);
dot((s,s,s));
[/asy]
pls help me
Answers
Answer:
Step-by-step explanation:To solve this problem, we can use similar triangles. Let the side length of the cube be s, and let the midpoints of AB, AC, and BC be P, Q, and R, respectively. Then OP, OQ, and OR are the diagonals of the faces of the cube, and so they are equal to .Consider triangle AOB. By the Pythagorean theorem, we have , so . Since angle AOB is 90 degrees, we can use similarity to see that triangle OAP is similar to triangle OAB, so . Solving, we find that .Using similar triangles, we can find that . Since the sum of the squares of the lengths of the diagonals of a cube is equal to three times the square of the side length, we have
$$(0P2 + 0Q? + OR?) = 35388 Substituting the values we found for $OP$, $OQ$, and $OR$, we haveSimplifying and solving for , we find that . Thus, the side length of the cube inscribed in the tetrahedron is .
Can you help me with this question?
Answers
Answer:
C
Step-by-step explanation:
It mentions that they charge 1.20 per kg of lettuce
What’s the answer? Will give brainliest if correct
Explain reasoning
Answers
The statement that is not true is: C. The translation of a line is a pair of parallel lines.
What is transformation?In geometry, a transformation is a procedure that modifies a figure's position, size, or shape. Translation, rotation, reflection, and dilation are the four main categories of transformations.
Every point in a figure is translated by the same amount and in the same direction. A rotation revolves a figure around the centre of rotation, a fixed point. A figure is reversed over a line known as the line of reflection in a reflection.
In relation to a fixed point known as the centre of dilation, a dilation expands or contracts a figure by a specific scale factor.
The statement that is not true is: C. The translation of a line is a pair of parallel lines as translation is a transformation that moves every point of a figure the same distance in the same direction.
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(1/2 + 1/3) + [1/4 + (2/3)] ÷ 2/5 -4 x 5/6 ÷ 3/7
Answers
Answer:
Step-by-step explanation:
first, take 1/2+1/3 + 1/4+2/3
1/2+1/3=5/6 , 1/4+2/3=11/12
5/6+11/12=22/12
2/5-4=-18/5
-18/5*5/6=-3
22/12/3=22*3/12
=11/2
11/2/3/7=11/2*7/3
=77/6=12.8
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Javier and Naida are discussing the elevation of Lima, Peru.
200
150 Lima
100
50+
0+Sea level
-50-
-100+
-150
-200
The elevation of Lima, Peru is 154 meters.
Javier: Lima is 154] meters from sea level.
Naida: Lima is 154 meters above sea level.
Whose statement is true?
Choose 1 answer:
Answers
Answer:
Naida's statement is true
Step-by-step explanation:
Since the elevation of Lima, Peru is a positive number, you can automatically tell that it is above sea level since sea level is zero
Can someone help me with this please
Answers
The measure of x is 36.
We have,
The angle opposite to equal sides is equal.
And,
The sum of the triangle.
27 + 90 + (x + 27)= 180
27 + 90 + x + 27 = 180
x = 180 - 144
x = 36
Thus,
The measure of x is 36.
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Please help will give a lot of points
Answers
Answer:
how many points will you give?
Step-by-step explanation:
Math question sb please do ts giving branliest
Answers
Answer:
All lines are parallel
Step-by-step explanation:
Get each equation in Slope-Intercept form:
1. Divide both sides by 3: [tex]y=-\frac{4}{3}x+\frac{7}{3}[/tex]
2. No change
3. Subtract 8x and divide by 6 on both sides: [tex]y=-\frac{4}{3}x-\frac{2}{3}[/tex]
Notice:
a. All slopes are -4/3, AND
b. All y-intercepts are different
A sixth-grade class collected data on the number of letters in the first names (name lengths) of all the students in class. Here is the dot plot of the data they collected:
How many students are in the class?
Answers
Based on the data displayed in the dot plot, it can be concluded there are 25 students in this class.
How to know the number of students based on the dot plot?Dot plots represent data and trends but using dots. In these types of graphs, each individual is represented with a dot. For example, in the number 3, we can see there are 3 dots, which means three students have a first name made of three letters.
Therefore, you can get the total of students by counting all the dots. Based on this, there are 25 students in this class.
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Calculate, to the nearest cent, the Present Value of an investment that will be worth $1,000 at the stated interest rate after the stated amount of time.
3 years, at 1.3% per year, compounded weekly (52 times per year).
PV=?
Answers
[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\dotfill & \$ 1000\\ P=\textit{original amount deposited}\\ r=rate\to 1.3\%\to \frac{1.3}{100}\dotfill &0.013\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{weekly, thus fifty two} \end{array}\dotfill &52\\ t=years\dotfill &3 \end{cases}[/tex]
[tex]1000 = P\left(1+\frac{0.013}{52}\right)^{52\cdot 3} \implies 1000=P(1.00025)^{156} \\\\\\ \cfrac{1000}{(1.00025)^{156}}=P\implies 961.76\approx P[/tex]