The expression for the nth term of the sequence is, Tn = 3n^2 - 7n - 1
To find an expression for the nth term of a quadratic sequence, we need to find a quadratic function that describes the sequence.
Let the nth term of the sequence be denoted by Tn. We can use the method of finite differences to determine the degree of the quadratic function that describes the sequence.
The first differences between the terms are 6, 10, 14, 18, 22. The second differences between these first differences are all equal to 4. This tells us that the sequence is quadratic, since the second differences are constant.
To find the quadratic function that describes the sequence, we can use the formula for the nth term of a quadratic sequence,
Tn = an^2 + bn + c
where a, b, and c are constants to be determined.
We can use the first three terms of the sequence to form a system of three equations,
T1 = a + b + c = -5
T2 = 4a + 2b + c = 1
T3 = 9a + 3b + c = 11
Solving this system of equations, we get:
a = 3
b = -7
c = -1
Therefore, the expression for the nth term of the sequence is
Tn = 3n^2 - 7n - 1
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geometric relationships
Geometric relationships refer to the ways in which different geometric figures or objects are related to each other. These relationships can include measures of angles,lengths of sides or edges, areas, volumes, and other properties.
Some common geometric relationships include:
Parallel lines: Two lines in a plane are parallel if they never meet, no matter how far they are extended.Perpendicular lines: Two lines in a plane are perpendicular if they intersect at a right angle (90 degrees).Congruence: Two geometric figures are congruent if they have the same size and shape. Congruent figures can be moved or rotated to overlap each other exactly.Similarity: Two geometric figures are similar if they have the same shape but can be different sizes. Similar figures have corresponding angles that are equal and corresponding sides that are in proportion.Pythagorean theorem: In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.Area formulas: The area of a rectangle is equal to the product of its length and width. The area of a triangle is equal to half the product of its base and height. The area of a circle is equal to pi times the square of its radius.Volume formulas: The volume of a rectangular prism is equal to the product of its length, width, and height. The volume of a cylinder is equal to pi times the square of its radius times its height.These are just a few examples of the many geometric relationships that can be studied in mathematics. Understanding these relationships can help us solve problems, make predictions, and create new designs or structures.
The pressure exerted on a table by a brick is 180 N/m² and the area of the base of the brick is 0.03 m². Calculate the force, in N, that the brick exerts on the table. If your answer is a decimal, give it to 1 d.p.
Step-by-step explanation:
We can use the formula for pressure:
pressure = force/area
We know the pressure and the area, so we can rearrange the formula to solve for the force:
force = pressure x area
Plugging in the values, we get:
force = 180 N/m² x 0.03 m² = 5.4 N
Therefore, the force that the brick exerts on the table is 5.4 N
Identify the aspects of this exponential function: y = 350(1 + 0. 75)t
Answer:
Step-by-step explanation:
The initial value, a, is 350, and the growth rate, b, is 1.75 so this is an exponential growth function.
Effect of Price on Supply of Eggs Suppose the wholesale price of a certain brand of medium-sized eggs p (in dollars/carton) is related to the weekly supply x (in thousands of cartons) by the following equation. 625p2 − x2 =100
If 37000 cartons of eggs are available at the beginning of a certain week and the price is falling at the rate of 7¢/carton/week, at what rate is the supply changing? (Round your answer to the nearest whole number. ) (Hint: To find the value of p when x = 37, solve the supply equation for p when x = 37. )
The supply of average eggs is decreasing at a rate of approximately -550 cartons per week as the price decreases 7 cents per carton per week.
The equation given states that the weekly supply x (in thousands of cartons) of a certain brand of medium-sized eggs is related to the wholesale price p (in dollars/carton) by the following equation: 625p2 − x2 =100. In order to determine the rate at which the supply is changing, we must first calculate the value of p when x = 37,000, which is the number of cartons available at the beginning of the week. To do this, we can solve the equation for p when x = 37,000[tex]p = (100 + x2)1/2/625[/tex]. When x=37,000, p=3.65. Therefore, if the price is falling at the rate of 7¢/carton/week, then the new price p = 3.65 - 0.07 = 3.58. When we plug this new value of p into the supply equation, we get[tex]x^2[/tex] = 4,225. Taking the square root of both sides gives us x = 65. Therefore, the supply is decreasing at a rate of approximately -550 cartons per week as the price decreases 7 cents per carton per week.
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Plsss helppppp i need thisss rnnnn
Answer:
a = 21, b = 70, c = 89
Step-by-step explanation:
opposite angles of intersecting lines are the same, 21 = 21
adjacent angles of intersecting lines add up to 180, 180 - 110 = 70
angles of a triangle add up to 180, 180 - 70 - 21 = 89
Answer:
a = 21, b = 70, c = 89
Step-by-step explanation:
A rectangular prism has a height of 1 inch, a width of of an inch, and a 2 1volume of of a cubic inch. What is the length, in inches, of this prism?
The length of the rectangular prism is 1 inch.
The volume of a rectangular prism can be calculated using the formula V = lwh, where l is the length, w is the width, and h is the height. In this case, the volume of the prism is given as 1 cubic inch and the width and height are given as 1 inch and 1 inch, respectively. To solve for the length of the prism, we can rearrange the equation to l = V/wh. Plugging in our given values, we can calculate the length of the prism as l = 1/1 x 1 = 1 inch. Thus, the length of the rectangular prism is 1 inch.Therefore, the length of the rectangular prism is 1 inch.
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is this statement true sometimes always or never ??
The statement: If ∠F is a 60° angle, then cos (F) = sin (F), is never true.
What are sine and cosine functions?Two of the six trigonometric functions used to connect angles to the sides of a right triangle are sine and cosine. The length of the opposite side of an angle divided by the length of the hypotenuse of a right triangle is known as the sine function. The length of the adjacent side of an angle divided by the length of the hypotenuse of a right triangle is known as the cosine function.
We know that,
If angle F is 60 degrees, then the value of cos F = 1/2 and sin F = sqrt(3)/2.
Therefore, cos F is not equal to sin F and the statement is never true.
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What must the values of a and b be for pqrs to be a parallelogram?
Answer:
the a give the x value twords the b and adds up the equation
ITS THE LAST DAY OF THE SEMSTER PLS HELP I AM GONNA FAILLLLLLLL!!!! PLEASE ANYONE HELP!!!!
Answer:
Step-by-step explanation:
I think to get the height is length x width
for distance you need to so SOH CAH TOA
Please help me thanks
Answer:
its 240
Step-by-step explanation: 10 x 8 x 3 = 240 :)
Answer:
a) Volume= length x width x height v=l x w x h
b) 240 cm^3 (cm cubed)
Step-by-step explanation:
Substitute into this formula
v=l x w x h
10 x 8 x 3 = 240
So 240 cm^3 (cm cubed)
Important note: Remember to include units, as they may not be given to you in the exam
460134. 2722596. Qx3zqy7 a shuttle van picks up passengers and drives them to a destination, where they leave the van. Keep a count of the boarding passengers, but don't allow boarding if the van is full. Update the odometer when the van drives
By using a counting method, the driver can keep track of how many passengers have boarded the van, and by measuring the distance driven, they can update the odometer.
The formula for calculating the total number of passengers that can board a shuttle van is:
Total Passengers = Number of Seats in the Van - 1 (Driver)
For example, if the van has 10 seats, the total number of passengers that can board the van is 9 (10 - 1).
To keep track of how many passengers have boarded the van, we can use a simple counting method. Each time a passenger boards the van, the counter should be increased by 1. When the counter reaches the maximum number of passengers the van can hold, the driver should not allow any more passengers to board.
To update the odometer, the driver can measure the distance the van has driven since the last update. This can be done by using a GPS device or by measuring the distance between two points on a map. The total number of miles driven since the last update can then be added to the previous odometer reading to find the new odometer reading.
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The membership fee at a sports club was $700 per year. Each year, the membership fee increased by 6%. Write the equation to find out how many years will it take for the cost of the membership to be $900? Do not solve.
[tex]\qquad \textit{Amount for Exponential Growth} \\\\ A=P(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\dotfill & \$ 900\\ P=\textit{initial amount}\dotfill &700\\ r=rate\to 6\%\to \frac{6}{100}\dotfill &0.06\\ t=years\\\ \end{cases} \\\\\\ 900 = 700(1 + 0.06)^{t} \implies 900=700(1.06)^t[/tex]
If you use a 0.02 level of significance in a (two-tail) hypothesis test, what is your decision rule for rejecting a null hypothesis that the population mean is 950 if you use the Z test? Which of the following decision rules is correct?
A. Reject H0 if -2.05 < Zstat +2.05.
B. Reject H0 if Zstat < -2.05 or Zstat > + 2.05.
C. Reject H0 if Zstat< -2.33 or Zstat > +2.33.
D. Reject H0 if -2.33 < Zstat < +2.33.
The correct decision rule for a test hypothesis with a significance level of 0.02, with two tails, is given as follows:
C. Reject H0 if Zstat< -2.33 or Zstat > +2.33.
What is the relation between the p-value and the conclusion of the test hypothesis?The decision regarding the null hypothesis depends on if the p-value is less or more than the significance level:
If it is more, the null hypothesis is not rejected, meaning that the result obtained on the research study is not statistically significant.If it is less, it is rejected, meaning that the result obtained on the research study is statistically significant.For a two-tailed test with a significance level of 0.02, the critical value is given as follows:
z = 2.33.
Hence the decision rule is given as follows:
|z| < 2.33 -> p-value < 0.02 -> do not reject H0.|z| > 2.33 -> p-value > 0.02 -> reject H0.Hence option C is the correct option for this problem.
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Nick hiked 10 miles up a hill. He is 8 miles east from his starting point. To the nearest degree, at what angle was the incline of the hill?
To the nearest degree, the angle of the incline was approximately 39°.
What is an angle?
An angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.
We can use the inverse tangent function to calculate the angle of the incline. The angle θ is given by:
θ = tan⁻¹(opposite/adjacent)
where opposite is the vertical distance (the height Nick hiked) and adjacent is the horizontal distance (the distance he traveled eastward).
Using the Pythagorean theorem, we can find the height Nick hiked:
height = [tex]\sqrt{(10)^{2} - (8)^2}[/tex]≈ 6
So, the opposite is approximately 6 miles.
Now we can plug in the values to find the angle:
θ = tan⁻¹(6/8)
θ ≈ 38.66°
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One side of a rectangle is three yards longer than twice the length of
If the perimeter is 27 yards, then the length of the other side of the rectangle is 3.5 yards.
Let's start by defining our variables. Let x be the length of one side of the rectangle (in yards), and let y be the length of the other side. We know from the problem statement that one side is three yards longer than twice the length of the other side, so we can express this algebraically as:
x = 2y + 3
The perimeter of a rectangle is given by the formula P = 2(x + y), so we can substitute our expression for x into this formula:
P = 2(2y + 3 + y) = 2(3y + 3) = 6y + 6
We are given that the perimeter of the rectangle is 27 yards, so we can set up an equation:
6y + 6 = 27
Solving for y, we can subtract 6 from both sides and then divide by 6:
6y = 21
y = 3.5
To check our answer, we can plug it back into our equation for x:
x = 2y + 3 = 2(3.5) + 3 = 7 + 3 = 10
So the other side of the rectangle is 10 yards long. We can verify that the perimeter is indeed 27 yards:
P = 2(x + y) = 2(10 + 3.5) = 2(13.5) = 27
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Complete question is:
One side of a rectangle is three yards longer than twice the length of the other side. If the perimeter is 27 yards, what is the length of the other side?
mike has these marks on his report card ; english =82% , french=75% , history =78% , science=80% what mark will he need in math if he wants to earn a mean mark of 80% across all these subjects ?is it possible for him to earn a mean mark of 85%
Mike cannot attain a mean grade of 85% for every one of his studies since a grade higher than 100% is not feasible.
What does the math mean?The sum of all numbers divided by the entire number of values determines the mean (also known as the arithmetic mean, which differs from of the median) of a dataset. This central tendency is usually referred to as the "average".
How can you calculate the mean?Just dividing the total amount of information in a data collection even by sum of all of the values yields it. Both raw data and data that have been compiled into a frequency distribution table can be used for the computation.
mean = (sum of all numbers) / (number of numbers)
80% = (82% + 75% + 78% + 80% + m) / 5
85% = m
The another equation is,
85% = (82% + 75% + 78% + 80% + m) / 5
425% = 315% + m
110% = m
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Ruby sold art supplies but it's on her website she bought the art supplies bundles of drawing paper and utensils for $75 each and included 70% markup to get the selling price what's the retail price including the marker for the art supply bundles
The retail price of each art supply bundle, including the markup, would be $127.50.
What is a markup, and how is it calculated?The gross or whole profit on a certain item or service is what is meant by markup. Moreover, it can be shown as a % of the cost price. For instance, if a product costs Rs. 100 and is sold for Rs. 150, the markup in this case will be 50%.
In studying the fundamentals of business, students come across the words selling price and cost price. A commodity's cost price is the amount paid when buying it from a retailer. The selling price, however, is the cost at which it is offered for sale.
Given that, the products were marked them up by 70%, then the selling price would be:
Selling price = $75 + 70% of $75
Selling price = $75 + $52.50
Selling price = $127.50
Therefore, the retail price of each art supply bundle, including the markup, would be $127.50.
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Two numbers are in the ratio of 7 : 4 and their difference is 33. Find the numbers.
Please do it step by step.
FORM 3A4
MATHS PROJECT
Get into groups of no more than 4 persons. Select a family size of 4/6/10 persons based on the greatest number of persons in the family in your group.
Calculate for the family size selected (allowing 24 inches for each person) the minimum diameter of a circular table to seat all members the minimum length of the side of a square table.
Calculate the area of the top of the circular table and the square table, and explain which is most appropriate for the family.
Assuming the family size selected is 6 persons,
The area of the top of the circular table would be: 1,660.51 In²; and The area of the top of the square table would be: 10,404 In²What is the explanation for the above response?Assuming the family size selected is 6 persons, the minimum diameter of a circular table to seat all members would be:
Minimum diameter = (Number of persons x Minimum distance between two persons) / π
= (6 x 24) / 3.14
= 45.86 inches (rounded up to 46 inches)
The minimum length of the side of a square table to seat all members would be:
Minimum length of side = (Number of persons x Minimum distance between two persons) / √2
= (6 x 24) / 1.41
= 101.41 inches
The area of the top of the circular table would be:
Area = π x (Diameter / 2)²
= π x (46 / 2)²
= 1,660.51 In²
The area of the top of the square table would be:
Area = (Length of side)²
= 102²
= 10,404 In²
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The local fire department has been called to an apartment fire in downtown Atlanta. They need to rescue a family that is on the 3rd floor and want to make sure they brought the correct ladder to perform the rescue. The height they need to reach is 30 feet and they know that they must at least be 20 feet from the bottom of the building to clear the awning. If the fire department brought a 35 foot ladder, Can they save the family
If the fire department brought a 35-foot ladder, Can they save the family: = √(325) feet. And They can't save family because √400 > √325
Given that :- height - 30 feet , ladder - 35 feet
[tex]AC^2 = AB^2+ BC^2[/tex]
BC = √([tex]AC^2-AB^2[/tex])
= √([tex]35^2 -30^2[/tex])
= √( 1225 - 900)
= √(325) feet
but the fire department must be so face from the bream of the building
20 = √[tex](20)^2[/tex]
= √400 > √325
They can't save family because √400 > √325
Pythagorean theorem, the notable mathematical theorem that the amount of squares on the legs of a right triangle is equivalent to the square on the hypotenuse (the side inverse the right point) — or, in recognizable logarithmic documentation, a2 + b2 = c2.
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Find the square root of the difference between the square of thirteen and the square of twelve
The square root of the difference between the square of thirteen and the square of twelve by first calculating the squares of 13 and 12 is 5.
First, we need to find the square root of thirteen and the square of twelve. The square of a number is the number multiplied by itself. So, [tex]$13^2$[/tex] means [tex]$13$[/tex] multiplied by itself and [tex]$12^2$[/tex] means [tex]$12$[/tex] multiplied by itself.
[tex]$13^2[/tex] = 13 \times 13 = 169
[tex]$12^2[/tex] = 12 \times 12 = 144
Next, we need to find the difference between the square of thirteen and the square of twelve by subtracting the smaller number from the larger one.
169 - 144 = 25
Finally, we take the square root of the difference to find the solution. The square root of a number is the value that, when multiplied by itself, gives the original number.
[tex]\sqrt{25} = 5[/tex]
Therefore, the square root of the difference between the square of thirteen and the square of twelve is 5.
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Factorise. 15(k+g)² - 20 (k+g)
Answer:Factorise. 15(k+g)² - 20 (k+g)
Step-by-step explanation:=[15(k+g)-20] (k+g)
=15 k²+15kg-20k+15kg+15g²-20g
=15k²+15g²+30kg-20k-20g
aunt polly asked tom to paint the entire length of a fence,but he only painted 8 yards and then ran away to play. Sid had to finish the job for to. Sid started at noon and by 12:30 p.m.,14 yards were painted including what tom had painted.By 2:00 p.m., 32 yards were painted,and by 2:30 p.m., 38 yards were painted.Let t be the number of minutes that have passed since sid started painting at noon,but not past 4:30 p.m. Write an equation that shows the relation ship between t and p where p is the number of yards of painted fence.
The equation that shows the relation ship between t and p is 0 ≤ t < 270.
What is the Linear equations?A linear equation is an algebraic equation where the highest power of the variable is 1, and the equation represents a straight line when graphed.
Break down the information given in the problem to form an equation that relates the time t and the number of yards of painted fence p:
Tom painted 8 yards before running away, so Sid had to paint the remaining length of the fence, which is L - 8 yards, where L is the total length of the fence.
From noon to 12:30 p.m., Sid painted 14 - 8 = 6 yards.
From 12:30 p.m. to 2:00 p.m., Sid painted 32 - 14 = 18 yards.
From 2:00 p.m. to 2:30 p.m., Sid painted 38 - 32 = 6 yards.
We want to find an equation that relates the time t (in minutes) to the number of yards of painted fence p.
Based on the information above, we can create a piecewise equation that relates t and p:
p = { 8 + 0.2t, 0 ≤ t < 30
14 + 0.6(t - 30), 30 ≤ t < 90
32 + 0.4(t - 90), 90 ≤ t < 150
38 + 0.2(t - 150), 150 ≤ t < 210 }
Explanation of the equation:
The first segment of the piecewise equation, 8 + 0.2t, represents the yards of fence painted by Tom (8) plus the yards painted by Sid in the first 30 minutes (0.2t), where 0 ≤ t < 30.
The second segment, 14 + 0.6 (t - 30), represents the additional yards painted by Sid between 12:30 p.m. and 2:00 p.m., where 30 ≤ t < 90. The expression (t - 30) represents the time elapsed since 12:30 p.m., and the coefficient 0.6 represents the rate at which Sid is painting the fence in yards per minute during this period.
The third segment, 32 + 0.4 (t - 90), represents the additional yards painted by Sid between 2:00 p.m. and 2:30 p.m., where 90 ≤ t < 150. The expression (t - 90) represents the time elapsed since 2:00 p.m., and the coefficient 0.4 represents the rate at which Sid is painting the fence in yards per minute during this period.
The fourth segment, 38 + 0.2 (t - 150), represents the additional yards painted by Sid between 2:30 p.m. and 4:30 p.m., where 150 ≤ t < 210. The expression (t - 150) represents the time elapsed since 2:30 p.m., and the coefficient 0.2 represents the rate at which Sid is painting the fence in yards per minute during this period.
Note that we have limited the time t to be between noon and 4:30 p.m., which corresponds to 0 ≤ t < 270.
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**PLEASE SOMEONE ANSWER IM STRUGGLING** To the nearest hundredth, what is the value x ?
Answer:
A answer that is not yet known.
Step-by-step
For example x is to sub in 21 but you do not know that yet
until you solve the question.
Jacob is planting flowers for his grandmother. This morning, he spent an hour planting annuals and an hour planting perennials, but he planted more annuals than perennials. This afternoon, he has the same number of annuals and perennials left to plant. Which will likely take him more time to plant?
It is likely that it will take Jacob more time to plant the perennials than the annuals.
What is time?Time is an abstract concept that is not tangible and cannot be seen, touched, or heard. It is a measure of the passage of events and is used to set and track schedules, create goals, and even measure the lifespan of an individual. Time is a precious commodity that cannot be replaced or taken back. Once it has passed, it is gone forever. Time is also a powerful force that can bring us joy and sorrow, success and failure. It is a constant reminder of how quickly our lives can change and how important it is to make the most of every moment.
Perennials require more attention when planting because they have a longer lifespan and are meant to come back year after year. Annuals, on the other hand, only last for one season. Therefore, they require less attention when planting. Jacob will need to make sure that he carefully places the perennials in the soil, as well as taking care to not damage the roots. He may also need to take extra time to trim any overgrowth from the plants. This extra care and attention will likely take more time than it would to plant the annuals.
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Determine if 32,48,80and 150 are in proportion
Answer:
No , they are not
Step-by-step explanation:
Checking the ratios:
32/48 = 0.67
48/80 = 0.6
80/150 = 0.53
Since the values are not equal, they are not in proportion.
Answer:
32, 48, 80, and 150 are not in proportion.
Step-by-step explanation:
To determine if 32, 48, 80, and 150 are in proportion, we need to check if the ratio of any two numbers is equal to the ratio of any other two numbers in the set.
Let's check:
Ratio of 32 to 48: 32/48 = 2/3
Ratio of 48 to 80: 48/80 = 3/5
Ratio of 80 to 150: 80/150 = 8/15
Since the ratios are not equal, 32, 48, 80, and 150 are not in proportion.
Please help this is Grade 8 nobody answered it so i am putting it up i will give brainlist you need to solve part 2 the 2 with the circle is all i got
Answer: 12=7
Step-by-step explanation: add one side ten add the other
Yes, this is a solution because the point(s) at which all the lines intersect are the solutions to the system.
one of the volunteers whose sam 77 test result was positive will be chosen at random. to the nearest 0.001 , what is the probability the chosen volunteer does not possess yq 77 ?
The probability the chosen volunteer does not possess yq 77 is 0.655 (nearest to three decimal places).
Given information: One of the volunteers whose sam 77 test result was positive will be chosen at random.
To find: The probability the chosen volunteer does not possess yq 77.
Let P(Y) be the probability that a volunteer possesses yq77 and P(Y') be the probability that a volunteer does not possess yq77. It is given that out of 150 volunteers, 77 tested positive for SAM.
Therefore, the probability of testing positive for SAM is:
P(SAM) = number of volunteers who tested positive for SAM / total number of volunteers= 77/150
Now, using the probability law of total probability, we can find the probability of a volunteer not possessing yq77, as follows:
P(Y') = P(Y'|SAM) P(SAM) + P(Y'|SAM') P(SAM')
It is given that yq77 is found in 60% of volunteers who tested positive for SAM, and in 30% of those who did not test positive for SAM.
Therefore:
P(Y'|SAM) = 0.6, P(Y'|SAM') = 0.3
Now, substituting the given values in the equation,
P(Y') = (0.6 × 77/150) + (0.3 × 73/150)≈ 0.655
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Fully factor the following polynomials. Explain your thought process with your solution.
a. ????(x) = 2x3 − 25x2 + 53x − 30 [3 marks]
b. ????(x) = x3 + 3x − 4 [3 marks]
a. the factors of equation is [tex]f(x)=(x-1)(x-10)(2x-3)[/tex]
b. the factors of equation is [tex]f(x)=(x-1)(x^2+x+4)[/tex]
Define the term polynomial?A polynomial is an expression consisting of variables and coefficients that involves only the operations of addition, subtraction, and multiplication, with non-negative integer exponents.
a. Given function,
[tex]f(x)=2x^3 -25x^2 + 53x - 30[/tex]
By using trial and error method, put values of x =1, x= 10, and x= 3/2 are satisfied values for above function,
[tex]f(x)=(x-1)(x-10)(2x-3)[/tex]
therefore, the factors of equation is (x-1) (x-10) (2x-3)
b. Given function,
[tex]f(x)=x^3 +3x - 4[/tex]
Similarly using trial and error method,
[tex]f(x)=(x-1)(x^2+x+4)[/tex]
therefore, the factors of equation is (x-1) (x² + x + 4)
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Factor of given polynomials are a. (x-1)(x-10)(2x-3), b. (x-1)(x² + x + 4) .
Describe polynomials ?In mathematics, a polynomial is an expression that consists of variables and coefficients, combined using the operations of addition, subtraction, and multiplication, but not division by a variable. The variables in a polynomial can only have non-negative integer exponents.
Polynomials can have different degrees, which is the highest power of the variable in the expression. For example, the polynomial 3x² - 5x + 2 is a quadratic polynomial, since its highest power of x is 2. If the degree of a polynomial is n, then it has at most n roots, which are the values of x that make the polynomial equal to zero.
Polynomials are used in many areas of mathematics and science, including algebra, calculus, physics, and engineering. They are used to model and analyze many different types of phenomena, such as motion, population growth, and electrical circuits.
Polynomials can also be added, subtracted, and multiplied together to form new polynomials. Additionally, polynomials can be factored, which involves breaking them down into simpler polynomials that multiply together to give the original polynomial.
a) f(x)= 2x³ - 25x² + 53x - 30
⇒(x-1)(x-10)(2x-3)
b) f(x)= x³ + 3x - 4
⇒(x-1)(x² + x + 4)
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The complete question is:
Unit 2: Logic & Proof
Homework 2: Compound Statements
on using disjunctive syllogism to derive Q from the second premise and the derived value of A. if P is true, then Q must also be true..
In logic, a compound statement is made up of two or more simple statements, known as premises or assumptions, that are combined using logical operators such as "and," "or," and "not." To prove a compound statement, we use rules of inference to derive a conclusion based on the given premises.
In the first proof question, we are given three premises: P, P or R, and not S or A. We need to prove the statement Q using these premises. To do so, we use the disjunctive syllogism rule, which states that if one of two disjunctive statements is false, then the other must be true. By assuming ~P, we derive ~S from the third premise using the rule of disjunction. Then, using modus ponens, we derive A from the assumption of ~P and the third premise. Finally, we use disjunctive syllogism to derive Q from the second premise and the derived value of A.
In the second proof question, we are given three premises: not P or S, not S, and R or Q. We need to prove the statement P implies Q using a conditional world proof. We assume P is true and derive S using the first premise. Using the second premise and modus tollens, we derive not R. Finally, using disjunctive syllogism, we derive Q from the third premise and the derived value of not R. Therefore, we show that if P is true, then Q must also be true..
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