In the polynomial expression f(x) = x³ + px² + qx +r, the values of p is 1/5, q is -66/5 and r is 12.
Polynomial:
Polynomial means an expression which is composed of variables, constants and exponents, that are combined using mathematical operations such as addition, subtraction, multiplication and division.
Given,
Here we have the polynomial f(x) = x³ + px² + qx +r
And it is divisible by (x+2),(x-1),and (x-3) the remainders are respectively -48,0 and 2.
Here we need to find the value of p, q and r.
Let us consider that the equations,
x + 2 = 0 then x = -2
x - 1 = 0 then x = 1
x - 3 = 0 then x = 3.
Now, apply these values on the polynomial expression then we get,
f(-2) = (-2)³ + p(-2)² + q(-2) +r = -48
f(-2) = -8 + 4p - 2q +r = -48
f(-2) = 4p - 2q +r = -48 + 8
f(-2) = 4p - 2q + r = -40 ----------------------(1)
Apply the value of x as 1, then we get
f(1) = (1)³ + p(1)² + q(1) +r = 0
f(1) = 1 + p + q + r = 0
f(1) = p + q + r = -1 ----------------------(2)
Apply the value of x as 3, then we get
f(3) = (3)³ + p(3)² + q(3) + r = 2
f(3) = 27 + 9p + 3q + r = 2
f(3) = 9p + 3q + r = -25 ----------------------(3)
Now we have to solve the equation (2) and (1), then we get,
=> (4p -p) + (-2q - q) + (r - r) = -40 - (-1)
=> 3p -3p = -39
Divide it by 3, then we get
=> p - q = -13 ----------------------(4)
Similarly, solve equations (2) and (3), then we get,
=> (9p - p) + (3q - q) + (r - r) = -25 - (-1)
=> 8p + 2q = -24
Divide it by 2, then we get,
=> 4p + q = -12 ---------------------------(5)
Solve the equations (4) and (5), then we get,
=> (4p + p) + (q - q) = -12 - (-13)
=> 5p = 1
=> p = 1/5
Apply the value of p on equation (4), then we get the value of q as,
=> (1/5) - q = -13
=> -q = -13 -1/5
=> -q = -66/5
Apply the value of p and q on equation (2), then we get,
=> 1/5 + (-66/5) + r = -1
=> -65/5 + r = -1
=> -13 + r = -1
=> r = 12
Therefore, the value of p is 1/5, q is -66/5 and r is 12.
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Tickets for an Amtrak train cost $21 for children and $49 for adults. Josie paid $3,332 for a total of 88 tickets. How many children tickets and how many adult tickets did Josie buy?
SOLUTION:
Case: System of equation word problem
Required: Find the number of adults and children tickets Josie bought.
First we create a system of equation. One for the total number of tickets, the other for the total cost of tickets.
Assumption: Let the number of adult tickets be a and the number of children tickets c.
Sytems of equations:
[tex]\begin{gathered} a\text{ + c= 88. (ticket equation)} \\ 49a\text{ + 21c = 3332. (Cost equation)} \end{gathered}[/tex]Solving the system of equation, using substitution method
[tex]\begin{gathered} \text{Make c the isolated variable from ticket equation} \\ c=\text{ 88-a} \\ \text{Substitutute in Cost equation.} \\ 49a\text{ + 21c = 3332. } \\ 49a\text{ + 21(88-a) = 3332. } \\ 49a\text{ + 1848 - 21a = 3332. } \\ \text{Taking like terms} \\ 49a\text{ - 21a = 3332 - 1848} \\ 28a\text{ = }1484 \\ Divide\text{ both sides by 28} \\ a=53 \end{gathered}[/tex]Next we find the value of c
[tex]\begin{gathered} We\text{ plug in the value }a\text{ in the isolated variable equation} \\ c=\text{ 88 - a} \\ c=\text{ 88 - 53} \\ c=\text{ 35} \end{gathered}[/tex]Final answers:
The total number of children tickets bought were 35 WHILE
The total number of adult tickets bought were 53.
Solve the equation 6 = -3(s+2)
Answer:
-4 = s
Step-by-step explanation:
6=-3(s+2) multiply -3 into the parenthesis.
6=-3s-6 add 6 to the other side
12=-3s divide by -3 on both sides.
-4=s
The owner of a restaurant wants to move the restaurant from Braddock Avenue to a new location onKnox Avenue that has more foot traffic. The owner would also like to double the perimeter of therestaurant due to the expected increase in business but keep the overall shape exactly the same, asshown in the coordinate plane.
We want to find a series of transformations that change the polygon ABCD into the polygon A'B'C'D'.
We see that a counter-clockwise rotation must be necessary, as:
By doing a traslation of 4 units to the right, and a dilation by a scale factor of 2, we obtain the desired figure. This is, the answer is B.
Explain why the statement line LR ≅ line RL and ∠CBA ≅ ∠ABC are both true by the Reflexive Property of Congruence.
The statement line LR ≅ line RL and ∠CBA ≅ ∠ABC are both true by the Reflexive Property of Congruence due to being a reflection of itself
What is reflexive property of congruence?The reflexive property of congruence says that the considered geometric quantity, whether it be angle, line segment, or shape etc, is congruent to itself.
WE have been given that the statement line LR ≅ line RL and ∠CBA ≅ ∠ABC are both true by the Reflexive Property of Congruence.
The reflexive property of congruence says that any geometric figure is congruent to itself.
Any figure being a reflection of itself and that is why they are congruent. For example, the triangle ∠CBA is congruent to ∠ABC.
The statement line LR ≅ line RL and ∠CBA ≅ ∠ABC are both true by the Reflexive Property of Congruence due to being a reflection of itself
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b) Solve x^2 + 5x + 6 = 0
Answer: (x+2)(x+3)
Step-by-step explanation:
1) Try to simplify with the line method
List the multiplicative numbers for a and c.
x^2 + 5x + 6 = 0
1x-----------> 3
1x-----------> 2
3 + 2 = 5
Make sure the two numbers multiply-add to b
Afterward, flip the top and bottom numbers.
1x--------------> 2
1x--------------> 3
Put them into an equation.
(x+2)(x+3)
What is x in the table below.
Green
2
3
5
6
10
Red
6
9
15
18
X
Answer
3.333
30
3
.333
Answer:
(b) 30
Step-by-step explanation:
Given a set of ordered pairs (green, red) = {(2, 6), (3, 9), (5, 15), (6, 18), (10, x)}, you want to know the value of x.
ProportionThe ratio of the red value to the green value is 6/2 = 3 in each case. To keep that ratio, we must have ...
x/10 = 3
x = 30 . . . . . . multiply by 10
Name two transformation sequences that can be done in any order without changing the outcome and explain how you know.
Answer:
There are two different categories of transformations: The rigid transformation, which does not change the shape or size of the preimage. The non-rigid transformation, which will change the size but not the shape of the preimage.
Step-by-step explanation:
yep sorry if you get it wrong it is
B
it has been determined with 95% confidence that the proportion of on-line students at spc who live in pinellas county is between 0.73 and 0.77. determine the margin of error associated with this confidence interval.
The margin of error associated with the confidence interval is 0.02.
The 95% confidence interval for the population proportion is defined as:
p ± z(0.05 / 2) × √[p ( 1 − p )/ n]
We have,
p − z(0.05 / 2) × √[ p ( 1 − p )/ n ] = 0.73 ----------- ( 1 )
p + z(0.05 / 2) × √[ p ( 1 − p )/ n ] = 0.77 ----------- ( 2 )
Adding the above two equations,
p − z(0.05 / 2) × √[ p ( 1 − p )/ n ] + p + z(0.05 / 2) × √[ p ( 1 − p )/ n ] = 0.73 + 0.77
2p = 0.73 + 0.77
2p = 1.5
p = 0.75
Now, Subtracting the equations ( 1 ) and (2),
We get,
p − z(0.05 / 2) × √[ p ( 1 − p )/ n ] - p - z(0.05 / 2) × √[ p ( 1 − p )/ n ] = 0.73 - 0.77
−2 × z(0.05 / 2) × √[ p ( 1 − p )/ n ] = -0.44
z( 0.10 / 2) × √[ p ( 1 − p )/ n] = 0.02
The margin of error is defined by the formula:
M E = z(0.05 / 2) × √[ p ( 1 − p )/ n]
Therefore, The margin of error will be:
M E = 0.02
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Find the Highest Common factor of 60 and 96
Answer: hi someone else answered this but you should report it its not an answer anyway here is the answer. 12
Step-by-step explanation:
write out the factors of both numbers:
60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96.
then find the highest number in both, 12.
and thats the hcf. hope this helped but report that person they are not helping. :)
I need to show a picture of the problem
x and y intercepts are the points where lines cut off the x and the y axis respectively.
In this case, the line cuts the x axis at x=-8. It means that the x intercept of the line is (-8,0).
The line also cuts the y axis at y=-6, which means that the y intercept of the line is (0,-6).
4/6= ? x 1/6 what number makes the equation
The first step here is to write out this question as a proper algebraic expression. That means you assign a letter to the question mark and you now have;
4/6 = y * 1/6
(Note that the symbol * represents the multiplication sign)
y * 1/6 can be re-written as y/6 since it is actually y/1 * 1/6
Therefore you now have;
4/6 = y/6
Cross multiply and you now have,
4 * 6 = y * 6
24 = 6y
Divide both sides of the equation by 6 (to eliminate the 6 on the right hand side)
4 = y
Therefore the number that should be in the question mark is 4.
Write an equation that represents your total earnings based on the price of your item and how many you sell.
The price of the item is $3, and I sell 40 items.
Answer:
y = 3(40)
Step-by-step explanation:
Let's break it down:
Total earnings = y
Price of item = 3
Amount of items sold = x (in this case 40)
what unit would you use to measure the length of a parking lot
A yard is a unit of measurement of length equal to 3 feet or approximately 0.914 meter. Hence, to measure the length of a parking lot we use unit yard
What is a Unit ?
A unit, like a building block, is a single, complete portion of something. Before moving on to another unit on geometry in math class, you might complete a unit on algebra. There are also other types of units of measurement, such inches and miles.
Since we know that
A Parking lot is a big area
So we should take a unit that is used to measure bigger area
Thus we use yard for measuring the Parking lot
A yard is a unit of measurement of length equal to 3 feet or approximately 0.914 meter
Hence, to measure the length of a parking lot we use unit yard
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The county fair charges $4 for 8 rides and $10 to get
in. If you have $32, write and solve an equation to
find the number of rides (r) you can go on.
For A Brainlist !!Please help its Urgent I need Help I think its C i dont know
Answer:
It would be 4/(-1) , which is -4
Step-by-step explanation:
You take f(1) and g(1) and then divide them to get f/g of (1)
Hello!
Let's solve:
[tex]\dfrac{f}{g} (1)=\dfrac{f(1)}{g(1)} \\\\\\\\\text{Plug in the value for f(1) and g(1) given by the table}\\\\\dfrac{f(1)}{g(1)}=\dfrac{4}{-1} =-4[/tex]
Answer: -4
Note:
To find value of f(1) and g(1) on the table
Start on the top bar of the chart (value of x)Move horizontally to the right until you reach the number 1. You have just now reached x when it is equal to 1Now move down one step into the box below the '1'. You have just found f(1)Now move down one more step into the box below f(1)'s value. You have just found g(1)Hope that helps!
My daughter needs help with her homework I’m confused she’s supposed to fill in the missing digits
Answer:
A is 5
B is 3
C is 2
D is 4
E is 3
F is 5
Solve this inequality:
5w-4>59-2w
Please show working out
Step-by-step explanation:
5w - 4 > 59 - 2w
add 4 to both sides:
5w - 4 + 4 > 59 - 2w + 4
5w > 63 - 2w
subtract 2w from both sides:
5w - 2w > 63 - 2w - 2w
3w > 63
divide both sides by 3:
3w/3 > 63/3
w > 21
in internal notation: (21, ∞)
what are three fractions between 1/8 and 3/4
Answer:
5/32, 3/16, 1/4
Step-by-step explanation:
Make an common denominator: multiply 1/8 by 4/4 to get 4/32
Multiply 3/4 by 8/8 to get 24/32
all possible values: 4/32 through 24/32
5/32
6/32 (Divide by 2 => 3/16 )
8/32 (Divide by 8 to get 1/4)
Use the Distributive
nt Property to find an equivalent
expression:
5(3x-7)
Approximate pi plus the square root of forty to the nearest hundredth.
To the nearest hundredth, pi + the square root of forty roughly equals 9.47. The best choice is B.
What does it mean to round a number to a particular place?Rounding a number to a particular value reduces the complexity of the number while increasing readability and accessibility.
When you round to a certain point, the exact digits are maintained on the left side of that point but are rounded or somewhat trimmed from the right.
The limit is set at five.
All right-sided values are made zero if the right-side digit to the digit where we must round off is less than or equal to five; otherwise, it is increased by one and the remaining right-side digits are made zero. If the digit to the right of the decimal point is 5, and there are just zeros or no values after it, we round up (raise by 1 as in 347500 to 34800). (Well there are many detailed rules of rounding)
The following can be used to represent the provided expression, "pi plus the square root of forty to the closest hundredth,"
π + √40
= 3.1416 + 6.3245
= 9.4661 ≈ 9.47
Value is therefore 9.47.
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an urn contains 18 red marbles and 15 blue marbles. 12 marbles are chosen. in how many ways can 7 red marbles be chosen? a) 792 b) 95,567,472 c) 34,827 d) 31,824 e) 126 f) none of the above.
The total number of ways 7 red marbles can be chosen is 95,567,472 i.e., option b is correct.
12 marbles are chosen in which 7 are red marbles and remaining marbles, which are 5, are blue marbles.
In order to find in how many ways 7 red marbles can be chosen we will follow the steps mentioned below:
Out of 18 red marbles 7 are chosen and out of 15 blue marbles 5 are chosen. So, we can write
Total number of ways 7 red marbles can be chosen= [tex]18C_{7}[/tex] × [tex]15C_{5}[/tex]
Total number of ways 7 red marbles can be chosen= [tex](\frac{18!}{7! (18-7)!} )[/tex][tex](\frac{15!}{5! (15-5)!})[/tex]
Total number of ways 7 red marbles can be chosen= 31824 × 3003
Total number of ways 7 red marbles can be chosen= 95,567,472
Thus, we can conclude that in 95,567,472 number of ways 7 red marbles can be chosen.
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two non-decreasing sequences of nonnegative integers have different first terms. each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is . what is the smallest possible value of ?
The smallest possible value of the seventh term of the sequence is 104 .
A sequence is an ordered group of items in mathematics where repetitions are permitted and order is important.
Similar to a set, it has members (also called elements, or terms). The size of the series is the amount of elements (potentially infinite). In contrast to a set, the same items might appear more than once in a sequence at various points, and unlike a set, the order is important.Let the first two terms of the first sequence be a and b and the first two of the second sequence be c and d.
Computing the seventh term,
we see that 5a + 8b = 5c + 8d.
Note that this means that a and c must have the same value of |8|.
To minimize, let one of them be 0;
We assume that a= 0.
Thus, the smallest possible value of c is 8; and since the sequences are non-decreasing we get d > 8.
To minimize, let d = 8. Thus, 5c + 8d = 40 + 64 = 104.
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I’ll give brainliest!! What is the standard form of the equation for a line that passes through (1, 2) and that has a slope of -4?
PLEASE SHOW WORK OR EXPLAIN
Answer:
[tex]y=-4x+6[/tex]
Step-by-step explanation:
Point Slope Form = Y-y1=m(X-x1)
m=-4, (x1,y1) = (1,2)
y-2=-4(x-1)
y-2=-4x+4
y=-4x+6
:]
(Lol please give Brainliest if I'm correct :] )
Math
I need an explanation too btw
The transformation from the graph of f to the graph of g is reflection in the x-axis.
We are given the function f(x) = -4x + 8
Also it says that g(x) = -f(x)
= 4x - 8
Since f(x) is a linear function, the graph of f(x) and hence the graph of g(x), both are straight lines. But the graph of g(x) will be a transformed image of the graph of f(x).
Here, if f(x) = y, then the transformation from f(x) to g(x) can be represented as, (x, y) -------> (x, -y)
This transformation corresponds to the reflection in the x-axis. When reflected along x-axis, the x- co-ordinate does not change, but the y- co-ordinate changes its sign. This is exactly what happens when the function f(x) is transformed to g(x) = -f(x).
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Susan is unpacking a full box after moving. She removes a 9 in. x 18 in. x 20 in. computer from the box below. How much volume do the rest of the items in the box take up (in cubic inches)?
The volume the rest of the items in the box take up (in cubic inches) is (x-3240) cu in.
What is volume?
Volume is a mathematical term that describes how much three-dimensional space an object or closed surface occupies. Volume is measured in cubic units like m^3, cm^3, in3, etc.Volume is sometimes referred to as capacity. For instance, a cylindrical jar's volume can be used to calculate how much water it can hold.The area that any three-dimensional solid occupies is known as its volume. These solids can take the form of a cube, cuboid, cone, cylinder, or sphere.Various forms have various volumes.Let the volume of the full box be = x cu in
The volume of the computer box = 9 in. x 18 in. x 20 in. = 3240 cu in
Volume the rest of the items in the box take up (in cubic inches)
= x cu in - 3240 cu in = (x-3240) cu in
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Action movies make up 80% of Devons movie collection. If she has 40 movies, how many action movies are in Devons collection
The number of action movies are in Devon's collection are, 32.
What is percentage?
A figure or ratio stated as a fraction of 100 is called a percentage. Although the abbreviations "pct.", "pct.", and occasionally "pc" are also used, the percent symbol, "%," is frequently used to indicate it. A % is a number without dimensions and without a standard measurement.
Given: Action movies make up 80% of Devon's movie collection. She has 40 movies.
We can find the number of Devon's movie collection by multiplying its percentage by the number of Action movies. That is;
Number of Devon's movie collection = [tex]\frac{80}{100} (40) = 32[/tex]
Therefore, Therefore he has 32 movies that are in Devon's collection.
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help please!!!!!!!!!!!!
In the estimation, the missing values are 8 and 1 respectively.
How to work out the missing valuesApproximation is the process of using rounded values when presenting numerical data and making rough calculations.
The digits 1,2,3 and are rounded down and digits 5,6,7,8 and 9 are rounded up.
Since the number after the digit 8 in 8.41 is 4, we will round down 8.41 to 8
Also, the number after the digit 0 in 0.61 is 6, so we will round up 0.62 to 1
Therefore, the missing values in the estimation are 8 and 1 respectively.
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identify the incorrect statement describing the sample sizes and time frames for conducting the various stages of human clinical trials: group of answer choices stage 3: 500-50,000 individuals - usually 4-5 years stage 4: millions of individuals - between 1-2 years stage 1: 5-50 individuals (healthy, young); few weeks to months stage 2: 50-500 individuals (cross section, more cohorts) - months to years
The incorrect statement is 50-500 individuals (cross section, more cohorts) - months to years
The answer provided above has been developed in a clear step by step manner.
Stage 1: 5-50 individuals (healthy, young); few weeks to months is correct considering the hypothesis
Second option is incorrect
Stage 2: 50-500 individuals (cross section, more cohorts) - months to years
Stage 3: 500-50,000 individuals - usually 4-5 years is also correct
Stage 4: millions of individuals - between 1-2 years is also correct
Appropriate sample size generally depends on the specified statistical hypotheses and a few study design parameters. These include the minimal meaningful detectable difference (effect size), estimated measurement variability, desired statistical power, and significance level
Therefore,50-500 individuals (cross section, more cohorts) - months to years can't be used to describe the sample sizes and time frames for conducting the various stages of human clinical trials
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recall the birthday paradox problem: take a random sample of size k with replacement from the set {1, 2,⋯, 365}. let pk be the probability that there is repetition in the sample (there are at least two people who share a common birthday).
the probability of at least two person have same birthday would be: 100%-99.72%= 0.28%
In this question, you are asked the probability for any of the 2 person to have the same birthday. To answer this it will be easier to calculate how much the probability for no one has same birthday. Let say the first person birthday is 1. Then the next person birthday should be other than 1, which mean 364 possible days out of 365 days.
Then the calculation for 2 people would be:
(365!/365-2!)/(365^(2)= (365!/363!)/ 365^2= 0.9972=99.72%
Then the probability of at least two person have same birthday would be: 100%-99.72%= 0.28%
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Suppose that 11 inches of wire costs 66 cents.
At the same rate, how much (in cents) will 21 inches of wire cost?
Answer:
8 inches can be purchased for 32 cents.