10. The remainder, when the polynomial is divided P(r) = ax³ + 3x² + bx + 15 is divided by (x - 1) is -12. When it is divided by (x - 2), the remainder is -21. Find (a) the value of the constants a and b. (b) Factorize P(x) completely.
The value of the constants a and b is 2 and -32 respectively.
The complete factorization of the polynomial is (x - 3)(x + 5) (2x - 1).
What are the values of the constants?The value of the constants is calculated as follows;
P(r) = ax³ + 3x² + bx + 15
If x - 1 is factor with a remainder of -12, then x = 1
P(1) = a1³ + 3(1²) + b(1) + 15 = -12
a + 3 + b + 15 = -12
a + b = -30
b = -30 - a
also, if x - 2 is a factor with a remainder of -21, then x = 2;
P(2) = a(2)³ + 3(2²) + b(2) + 15 = -21
8a + 12 + 2b + 15 = -21
8a + 2b = -48
4a + b = - 24
Substitute the value of b into the equation;
4a - 30 - a = - 24
3a = 6
a = 2
b = -30 - a
b = -30 - 2
b = -32
The quadratic equation will become;
P(r) = 2x³ + 3x² -32x + 15
We will factorize the equation as follows;
x - 3 is a factor;
2x² + 9x - 5
----------------------------------
x - 3 √ (2x³ + 3x² -32x + 15)
- (2x³ - 6x²)
----------------------------------
9x² - 32x + 15
- (9x² - 27x )
----------------------------
-5x + 15
- ( -5x + 15)
-----------------------------
0
We will factorize 2x² + 9x - 5
= 2x² - x + 10x - 5
= x(2x - 1) + 5(2x - 1)
= (x + 5)(2x - 1)
The complete factor = (x - 3)(x + 5) (2x - 1)
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a net for the gift box.
14 m
15.65 m
15.65 m
I
I
7m
15 m
1
14 m
what do u need help with
Pincushion circumference=8
If circumference of circle is 8 units then the radius of circle is 1.27 units
The circumference of the circle is 8 units.
We have to find the radius of the circle.
The formula for circumference of cirlce is two times of pi and radius.
Circumference = 2πr.
Where r is the radius of the circle.
The value of radius we have to find.
The circumference is 8 units.
Plug in the values of circumference to find r:
8=2×3.14 r
When 2 is multiplied with three point one four we get six point two eight.
8=6.28 r
Divide both sides by 6.28:
r = 8/6.28
When eight is divided by six point two eight we get one point two seven.
r=1.27 units
Hence, the radius of circle is 1.27 units when circumference is 8 units.
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Complete question
If circumference of circle is 8 units then find the radius of circle?
what is the lower quartile of the numbers 1 5 6 6 7 7 8 8 8 8 9 9 18 20 20 20 20 20 24 32 50 50 68 100
To find the lower quartile of a set of numbers, we need to determine the value that separates the lowest 25% of the data from the rest. In the given set of numbers, the lower quartile can be found by arranging the data in ascending order and identifying the value at the 25th percentile.
First, let's arrange the data in ascending order: 1 5 6 6 7 7 8 8 8 8 9 9 18 20 20 20 20 20 24 32 50 50 68 100.
To find the lower quartile, we need to determine the position of the 25th percentile. Since there are 23 data points, the 25th percentile corresponds to the value at the (25/100) * 23 = 5.75th position.
Since the position is not a whole number, we need to find the value between the fifth and sixth data points. The fifth data point is 7, and the sixth data point is also 7. Therefore, the lower quartile is 7.
The lower quartile represents the value below which 25% of the data lies. In this case, 25% of the data points are less than or equal to 7. It indicates that a quarter of the values in the dataset are smaller than or equal to 7, while the remaining 75% are larger.
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help please. thank you bro.
The area of the small and large sectors are about 11,760 feet² and 23,520 feet² respectively.
We have to given that;
A circle is shown in image.
Here, We have;
For small sector,
Angle = 120 deg
Radius = 14 feet
Since, Area of sector of circle is,
⇒ A = 1/2 (θr²)
Hence, For small sector,
Area = 1/2 (120 × 14²)
Area = 11,760 feet²
For large sector is,
Area = 1/2 (θr²)
Area = 1/2 (360 - 120) × 14²
Area = 1/2 × 240 × 14²
Area = 23,520 feet²
Thus, The area of the small and large sectors are about 11,760 feet² and 23,520 feet² respectively.
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suppose we want to select a set of 4 coins from a box containing 6 different pennies and 8 different dimes. how many of the possible sets contain 2 pennies and 2 dimes?
Answer:
420 possible sets------------------------
Use a combination formula.
We want to select 2 pennies from a set of 6 and 2 dimes from a set of 8.
The number of ways to do this is:
C(6,2) * C(8,2) = (6!/(2!4!)) * (8!/(2!6!)) = 15 * 28 = 420Therefore, there are 420 possible sets of 4 coins that contain 2 pennies and 2 dimes.
Cher has a motorcycle worth $11,357. She has $605 in cash, and
$1,811 in her savings account. Her only debt is the ken she took out
to buy the motorcycle. If Cher's net worth is $3,934, what is the
Amount of the loan?
The amount of the loan Cher took out to buy the motorcycle is $9839.
Here, we have,
To find the amount of the loan, we need to calculate Cher's total assets and then subtract her total liabilities (debts).
Given:
Value of motorcycle = $11,357
Cash = $605
Savings account = $1,811
Net worth =$3,934
Total assets = Value of motorcycle + Cash + Savings account
Total assets = $11,357 + $605 + $1,811
Total assets = $13,773
To find the amount of the loan, we subtract the total assets from the net worth:
Loan amount = Total assets - Net worth
Loan amount = $13,773 -$3,934
Loan amount = $9839
Hence, The amount of the loan Cher took out to buy the motorcycle is $9839.
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sherry buys a 5-ounce cup of ice cream. the summer heat melts the ice cream before she can eat any. what describes the weight of the melted ice cream?
The weight of the melted ice cream would be the same as the weight of the original ice cream, which is 5 ounces.
a quantity or thing weighing a fixed and usually specified amount. : a heavy object (such as a metal ball) thrown, put, or lifted as an athletic exercise or contest. 3. : a unit of weight or mass see Metric System Table.
When the ice cream melts, it undergoes a change in state from solid to liquid, but the total mass or weight remains unchanged. Therefore, the weight of the melted ice cream is still 5 ounces.
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A survey was given to a random sample of 85 voters in the United States to ask about
their preference for a presidential candidate. Of those surveyed, 68 respondents said
that they preferred Candidate A. Determine a 95% confidence interval for the
proportion of people who prefer Candidate A, rounding values to the nearest
thousandth.
The 95% confidence interval for the proportion of people who prefer Candidate A is 0.8 ± 0.080.
How do we calculate?Confidence Interval = Sample Proportion ± Margin of Error
We first determine the values for each term:
Sample Proportion = Number / Total sample size
Sample Proportion = 68/85
Sample Proportion = 0.8
Standard Error = √((Sample Proportion * (1 -Sample Proportion )) / n)
n = sample size
Standard Error = √((0.8 * (1 - 0.8)) / 85)
Standard Error = 0.041
We then calculate the Margin of Error = Critical Value * Standard Error Margin of Error = 1.96 * 0.041
Margin of Error = 0.080
We substitute all these values and calculate for :
Confidence Interval = Sample Proportion ± Margin of Error
Confidence Interval = 0.8 ± 0.080
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Finding an angle measure given a triangle and parallel lines
Answer:
We have congruent alternate interior angles.
39° + 96° + x° = 180°
135° + x° = 180°
x = 45
Rey is checking the distance between two rooms on a blueprint. The
blueprint uses a scale in which 3 inches equals 4 feet. If the distance
on the blueprint is 11 inches, how far is it between the rooms, in feet?
Your answer can be exact or rounded to two decimal places.
i don't get it pls i need help pls pls pls
Answer:
IDONTKNOW but your answer might be 35 .
Step-by-step explanation:
i think the answer is
5 x 7
So basically you have 4 and 3 so you want to multiply that and idontrealyknow the rest. eh sorry idontknow i tried.
a box of baseballs cost $66.12. what equation represents the cost,c, of n boxes
Hello !
c = the cost
n boxes = 66,12n
c = 66,12n
Perform the following area application.
Compute the number of vinyl tiles, measuring 8 in. each side, needed to tile a kitchen measuring 24 ft. by 18 ft.
__________tiles
To find the number of vinyl tiles needed to tile a kitchen measuring 24 ft. by 18 ft. with tiles measuring 8 in. each side, we need to convert all measurements to the same units.
First, we need to convert the kitchen measurements from feet to inches:
- 24 ft. = 24 x 12 = 288 in.
- 18 ft. = 18 x 12 = 216 in.
Next, we need to divide the length and width of the kitchen by the length of each tile to find the number of tiles needed in each direction:
- Length: 288 in. ÷ 8 in. = 36 tiles
- Width: 216 in. ÷ 8 in. = 27 tiles
Finally, we multiply the number of tiles needed in each direction to find the total number of tiles needed:
- Total tiles: 36 tiles x 27 tiles = 972 tiles
To tile a kitchen, it is important to accurately calculate the number of tiles needed to ensure that there is enough material to complete the job. In this case, we converted all measurements to the same units and then used division and multiplication to find the total number of tiles needed.
To tile a kitchen measuring 24 ft. by 18 ft. with vinyl tiles measuring 8 in. each side, we need a total of 972 tiles.
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In a scatter plot, if all data points were aligned along a 45° angle, your correlation would be:
A) Of medium strength
B) Of weak strength
C) Perfect
D) 0
E) None of the above
Answer:
C) Perfect correlation
Kai uses an entire can of paint on a square backdrop for the school play. The label on the can states that one can covers 27 m2 of wall surface. Estimate the backdrop’s side length, to one decimal place.
can you help me with math
Answer:
B
Step-by-step explanation:
203.95 is equal to 4 of the tickets plus 11.95
You can write this as
203.95=4(price of ticket)+11.95
4(price of ticket)=192
Divide by 4
The price of a ticket is 48.
The equation would be 11.95+48t because its 48 times the amount of ticket plus the service fee.
Alice drops a rock off a building that is 64 feet tall. If the equation for height as a function of time is: y = -16+2 + 400 where t is time in seconds and y is height in feet, how many seconds will it take for the drone to hit the ground? Now, set the equation equal to zero and solve.
By using Algebraic methods It will take 5 seconds for the rock to hit the ground when dropped from a building that is 64 feet tall, given the equation y = -16t^2 + 400.
The equation is y = -16t^2 + 400, where y represents the height of the rock in feet and t represents the time in seconds.
To find out how many seconds it will take for the rock to hit the ground, we need to set the equation equal to zero because the ground is at a height of zero.
0 = -16t^2 + 400
Now, we can solve for t using Algebraic methods.
First, we can subtract 400 from both sides of the equation:
-400 = -16t^2
Next, we can divide both sides by -16:
25 = t^2
Finally, we can take the square root of both sides:
t = ±5
Since we can't have negative time, we know that it will take 5 seconds for the rock to hit the ground.
So, to summarize, it will take 5 seconds for the rock to hit the ground when dropped from a building that is 64 feet tall, given the equation y = -16t^2 + 400.
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You spin a spinner that is equally divided into 9 parts. 3 parts are white, 3 parts are black, and 3 parts are yellow.
What is the probability of the spinner not stopping at a black section? Write your answer as a percent rounded to the nearest whole number.
(Thank you!!)
Answer:
6/9 67%
Step-by-step explanation:
Answer:
2/3
Step-by-step explanation:
66,6% are 2/3 -> 67%
From the 9 parts are 6 left - > 9/6 = 2/3
analysis is a form of horizontal analysis that can reveal patterns in data across periods. it is computed by taking the (analysis period amount/base period amount) x 100.multiple choice question.ratiotrendmodified horizontal
The type of analysis described in the statement is ratio analysis.
Ratio analysis is a tool used to evaluate the financial performance of a company by comparing different financial ratios over time or between companies. It involves calculating different ratios such as liquidity ratios, profitability ratios, and solvency ratios and comparing them across different periods to identify trends or patterns in the data. The ratio analysis formula mentioned in the statement is used to calculate the percentage change in a ratio between two periods, with the base period amount serving as the denominator.
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Which one of the following statements is true? A. If 4 3 h = 12 , then h = 16 . B. If − 2 5 r = − 20 , then r = − 50 . C. If 1 2 = − 3 7 q , then q = − 1 1 6 . D. If 3 8 = 24 c , then c = 9 .
The only True statement is B: If −2/5r = −20, then r = −50
We can solve each equation to determine which statement is true.
A. If 4/3h = 12, then h = 16.
To solve for h, we can multiply both sides of the equation by 3/4:
4/3h * 3/4 = 12 * 3/4
h = 9
Therefore, statement A is false.
B. If −2/5r = −20, then r = −50.
To solve for r, we can multiply both sides of the equation by -5/2:
-2/5r * -5/2 = -20 * -5/2
r = 50
Therefore, statement B is true.
C. If 1/2 = −3/7q, then q = −116.
To solve for q, we can multiply both sides of the equation by -7/3:
1/2 * -7/3 = -3/7q * -7/3
q = 21/3 = 7
Therefore, statement C is false.
D. If 3/8 = 24c, then c = 9.
To solve for c, we can divide both sides of the equation by 24 * 8/3:
3/8 * 3/24 * 8/3 = 24c * 3/24 * 8/3
c = 1/3 * 8
c = 8/3
Therefore, statement D is false.
Therefore, the only true statement is B: If −2/5r = −20, then r = −50.
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Find the volume of the triangular prism with a height of 2.8, width of 4.2, and a length of 8.
Answer:
Step-by-step explanation:
The formula for this is:
V = (1/2) x b x h x l
b= base
h=height
l=length
V= (1/2) x 4.2 x 2.8 x 8
so...I think the volume could be 47.04m^3?
PRO TIP: If you want to solve anything with volume, just write down every single formula they have for volume or area. It is good to memorize formulas like that.
The volume of the triangular prism with a height of 2.8, width of 4.2, and a length of 8 is 46.96 cubic unit.
The base area of the triangular cross-section must be multiplied by the prism's height to determine its volume. The triangle cross-section in this instance has a base of 4.2 and a height of 8.
The following is the formula to determine a triangle's area:
(Base * Height) / 2 = Area
When the values are plugged in, we get:
Base = 4.2
Size = 8
Area = (4.2 * 8) / 2 = 33.6 / 2 = 16.8
Next, increase the prism's base area by its height:
Volume = Base Area * Height = 16.8*2.8 = 46.96
The triangular prism therefore has a 46.96 cubic unit volume.
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find the length of the curve y=ln(cosx) from 0 to pi/3
To find the length of the curve y = ln(cos(x)) from 0 to π/3, we can use the arc length formula for a curve defined by a function y = f(x):
L = ∫[a to b] √(1 + (f'(x))²) dx.
In this case, the function is y = ln(cos(x)). Let's first find the derivative:
y' = d/dx ln(cos(x)).
Using the chain rule, we have:
y' = -tan(x).
Now we can calculate the arc length:
L = ∫[0 to π/3] √(1 + (-tan(x))²) dx.
Simplifying the integrand, we have:
L = ∫[0 to π/3] √(1 + tan²(x)) dx.
Using the trigonometric identity tan²(x) + 1 = sec²(x), we can rewrite the integrand as:
L = ∫[0 to π/3] √(sec²(x)) dx.
Taking the square root of sec²(x), we have:
L = ∫[0 to π/3] sec(x) dx.
Integrating sec(x), we get:
L = ln|sec(x) + tan(x)| [0 to π/3].
Evaluating the integral at the upper and lower limits, we have:
L = ln|sec(π/3) + tan(π/3)| - ln|sec(0) + tan(0)|.
Simplifying further, we have:
L = ln|2 + √3| - ln|1 + 0|.
Since ln|1| = 0, the second term on the right-hand side becomes 0, and we are left with:
L = ln|2 + √3|.
Therefore, the length of the curve y = ln(cos(x)) from 0 to π/3 is ln|2 + √3|.
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Subtract the polynomials
(3x² + 2x − 5) – (4x² – 3x − 4)
A:-7x²+5x-9
B:7x²-x-9
C:x²-x-9
D:-x²+5x-1
please helpppp asap
Answer:
-x² + 5x - 1
Step-by-step explanation:
separate the x², the x, and the constants (numbers without any x).
3x² - 4x² = -x²
2x - -3x = 2x + 3x = 5x
-5 - -4 = -5 + 4 = -1.
we have -x² + 5x - 1
Alang and his children went into a restaurant where they sell hamburgers for $7.50 each and drinks for $2.50 each. Alang has $95 to spend and must buy a minimum of 18 hamburgers and drinks altogether. If Alang decided to buy 15 drinks, determine the maximum number of hamburgers that he could buy
Alang can buy at most 10 hamburgers if he buys 15 drinks.
How to solve for the maximum number that can be boughtLet's say Alang buys x hamburgers. Then, he must buy (18 - x) drinks to meet the requirement of buying a minimum of 18 hamburgers and drinks altogether.
The cost of x hamburgers is 7.5x dollars.
The cost of (18 - x) drinks is 2.5(18 - x) = 45 - 2.5x dollars.
The total cost must be less than or equal to $95:
7.5x + (45 - 2.5x) ≤ 95
Simplifying and solving for x, we get:
5x ≤ 50
x ≤ 10
So Alang can buy at most 10 hamburgers if he buys 15 drinks.
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Use the formula to find the surface area of the figure. Show your work.
Formula Used:-
[tex] \qquad \hookrightarrow{ \underline{ \overline{ \boxed{ \sf{Surface \: Area =3\pi {r}^{2} }}}}}\\[/tex]
SolutioN:-[tex] \sf \longrightarrow \: Surface \: Area =3\pi {r}^{2} \: ) \\ [/tex]
[tex]\sf \longrightarrow \: Surface \: Area =3 \times \frac{22}{7} \times {r}^{2} \: \\[/tex]
[tex]\sf \longrightarrow \: Surface \: Area =3 \times \frac{22}{7} \times {11}^{2} \: \\[/tex]
[tex]\sf \longrightarrow \: Surface \: Area =3 \times \frac{22}{7} \times 11 \times 11 \: \\[/tex]
[tex]\sf \longrightarrow \: Surface \: Area = \frac{3 \times 22 \times 11 \times 11}{7} \: \\[/tex]
[tex]\sf \longrightarrow \: Surface \: Area = \frac{66 \times 11 \times 11}{7} \: \\[/tex]
[tex]\sf \longrightarrow \: Surface \: Area = \frac{66 \times 121}{7} \: \\[/tex]
[tex]\sf \longrightarrow \: Surface \: Area = \frac{7986}{7} \: \\[/tex]
[tex]\sf \longrightarrow \: Surface \: Area = 1140.8 \: {ft}^{2} \: \\[/tex]
______________________________________________
Therefore, Surface Area of Hemisphere is 1140.8 ft²
Answer:
1140.8 ft²Step-by-step explanation:
We know that,
[tex]\large {\boxed {\boxed{\bf\leadsto \: surface \: area \: of \: hemisphere \: = 3\pi {r}^{2} }}}[/tex]
Here,
Radius = 11ftNow
Putting the given value[tex]\large \sf\leadsto3 \times \frac{22}{7} \times 11 {}^{2} [/tex]
[tex]\large \sf\leadsto3 \times \frac{22}{7} \times 121[/tex]
[tex]\large \sf\leadsto \: \frac{66}{7} \times 121[/tex]
[tex]\large \sf\leadsto \frac{7986}{7} [/tex]
[tex] \boxed{\large \bf\leadsto1140.8 \: {ft}^{2} }[/tex]
Function g can be thought of as a translated (shifted) version of f(x)=x^2
write the equation of g(x)
The equation of the red graph, g(x) is g(x) = (x + 3)² - 4
How to calculate the equation of the red graphFrom the question, we have the following parameters that can be used in our computation:
The functions f(x) and g(x)
In the graph, we can see that
The blue graph passes through the vertex (0, 0)The red graph passes through the vertex (-3, -4)This means that
g(x) = f(x + 3) - 4
Recall that
f(x) = x²
This means that
g(x) = (x + 3)² - 4
This means that the equation of the red graph is g(x) = (x + 3)² - 4
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doris needs to renew her real estate license for the first time, and she's already completed 20.5 hours of continuing education. how many additional hours of ce does she need to complete?
Doris needs to complete a total of 30 hours of continuing education to renew her real estate license for the first time. As she has already completed 20.5 hours of CE, she needs to complete an additional 9.5 hours of CE.
It's important to note that continuing education requirements vary by state and licensing board, so it's always a good idea to check with the relevant authorities to ensure you meet all the requirements for license renewal. Real estate agents are typically required to complete a certain number of continuing education hours within a specific time frame in order to maintain their license and stay up-to-date with industry developments and best practices. This is important to ensure they are equipped with the necessary knowledge and skills to provide quality services to their clients.
In summary, Doris needs to complete an additional 9.5 hours of CE to meet the 30-hour requirement for renewing her real estate license for the first time.
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suppose the largest possible cutout length is 3.5 inches. over what interval of x does the volume of the box decrease as the cutout length gets larger?
The volume of the box decreases as the cutout length increases over the interval (0, 3.5/2) and increases over the interval (3.5/2, infinity).
Let the dimensions of the rectangular box be x, y, and z. The volume of the box is given by V = xyz. Suppose that the cutout length is along the x-axis, and let it be denoted by w. Then, we have:
V = (x - w)yz
Taking the derivative of V with respect to w, we get:
dV/dw = -yz
The volume of the box will decrease as the cutout length increases if dV/dw < 0, and it will increase if dV/dw > 0. Since y and z are positive, dV/dw will be negative if w is less than x/2 and positive if w is greater than x/2. Therefore, the volume of the box decreases as the cutout length increases over the interval (0, x/2) and increases over the interval (x/2, infinity).
In our case, the largest possible cutout length is 3.5 inches, so the interval over which the volume of the box decreases as the cutout length gets larger is (0, 3.5/2), which is (0, 1.75). The interval over which the volume of the box increases is (3.5/2, infinity), which is (1.75, infinity).
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Ms. Beck has invested $1000 that will gain 2% annual interest, which can be calculated using the expression \large 1000\left(1.02\right)^t, where`t` is the number of years.
1. Determine how much money Ms. Beck would have after one year.
t =
Final Amount =
2. Determine how much money Ms. Beck would have after one half of a year.
t =
Final Amount =
Answer: t=1
Final Answer=$1,020
t=1/2
Final Answer=$1,009.95
Step-by-step explanation: This is correct
its cammy, im telling that your cheating (and i definitely wasnt looking up this same question)
