The double-angle identity for cosine is cos(2x) = cos²(x) - sin²(x).
To derive the reduction identities, we can use this double-angle identity and manipulate it to express cosine or sine of x in terms of cosine or sine of a smaller angle. Here are the steps for deriving the reduction identity for cosine: Start with the double-angle identity: cos(2x) = cos²(x) - sin²(x).
Divide both sides by cos²(x): cos(2x) / cos²(x) = (cos²(x) - sin²(x)) / cos²(x).
Use the identity sin²(x) + cos²(x) = 1 to replace cos²(x) in the denominator: cos(2x) / cos²(x) = (1 - sin²(x)) / cos²(x).
Rearrange the right-hand side: cos(2x) / cos²(x) = 1/cos²(x) - sin²(x) / cos²(x).
Recognize that 1/cos²(x) is equal to sec²(x), and use the identity tan²(x) + 1 = sec²(x) to replace it: cos(2x) / cos²(x) = sec²(x) - tan²(x).
Simplify using the identity cos(x)/sin(x) = cot(x): cos(2x) / cos²(x) = cosec²(x) - cot²(x).
Simplify the left-hand side using the identity cos(2x) = 2cos²(x) - 1: (2cos²(x) - 1) / cos²(x) = cosec²(x) - cot²(x).
Rearrange and simplify: 2 - (1/cos²(x)) = cosec²(x) - cot²(x).
Recognize that 1/cos²(x) is equal to sec²(x) and cosec(x) is equal to 1/sin(x): 2 - sec²(x) = 1/sin²(x) - cot²(x).
Simplify the left-hand side using the identity 1 - sin²(x) = cos²(x): cos²(x) - sec²(x) = 1/sin²(x) - cot²(x).
Recognize that sec(x) is equal to 1/cos(x) and cot(x) is equal to cos(x)/sin(x): cos²(x) - 1/cos²(x) = sin²(x)/cos²²(x) - cos²x)/sin²(x).
Simplify: cos⁴(x) - 1 = sin²(x) - cos⁴(x).
Rearrange: 2cos⁴(x) = sin²(x) + 1.
Finally, recognize that sin²(x) + cos²(x) = 1, and use it to replace sin²(x): 2cos⁴(x) = cos²(x). This is the reduction identity for cosine: cos²(x/2) = (1 + cos(x))/2.
To derive the reduction identity for sine, we can use the same double-angle identity for cosine and manipulate it to express sine of x in terms of sine or cosine of a smaller angle. The steps are similar but may require using different.
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29 Transformations of linear functions C8G
Find g(x), where g(x) is the translation 1 unit right of f(x) = 7x + 4.
Write your answer in the form mx + b, where m and b are integers.
g(x) =
Submit
Step-by-step explanation:
please review the attachment
A company makes short-sleeved and long-sleeved T-shirts. The company checks a random sample of 60 T-shirts of each type. It finds that 4 short-sleeved shirts have problems and 1 long-sleeved shirt has problems. How many T-shirts should the company predict have problems in a shipment of 3,000 shirts of each type?
The company should predict that there will be an estimated 201 short-sleeved T-shirts and 51 long-sleeved T-shirts with problems in a shipment of 3,000 shirts of each type.
To predict the number of T-shirts with problems in a shipment of 3,000 short-sleeved and 3,000 long-sleeved T-shirts, we can use the proportion of T-shirts with problems in the random sample as an estimate of the proportion of T-shirts with problems in the population.
For short-sleeved T-shirts, the proportion of T-shirts with problems in the sample is 4/60 = 0.067. For long-sleeved T-shirts, the proportion of T-shirts with problems in the sample is 1/60 = 0.017.
To estimate the number of short-sleeved T-shirts with problems in a shipment of 3,000, we can multiply the proportion by the total number of short-sleeved T-shirts in the shipment:
Estimated number of short-sleeved T-shirts with problems = 0.067 x 3,000 = 201
Similarly, to estimate the number of long-sleeved T-shirts with problems in a shipment of 3,000, we can multiply the proportion by the total number of long-sleeved T-shirts in the shipment:
Estimated number of long-sleeved T-shirts with problems = 0.017 x 3,000 = 51
Therefore, the company should predict that there will be an estimated 201 short-sleeved T-shirts and 51 long-sleeved T-shirts with problems in a shipment of 3,000 shirts of each type.
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10. Best Buy is having a 15% off sale on brands of computer regularly priced at $2250 How much does Jerome save on the computer the sales tax is 7. 5% what is the amount of on the computer How much is the total cost for the computer What would be the total price for the computer on sale
The total cost of the computer is $2418.75.
The total cost of the computer in sale $2,056.
The computer is regularly priced at $2250.
With the 15% discount, Jerome would save = $2250 × 15%
With the 15% discount, Jerome would save = $2250 × 15/100
With the 15% discount, Jerome would save = $337.50 on the computer
The sales tax is 7.5%.
So the tax on the computer would be = $2250 × 7.5%
The tax on the computer would bee = $2250 × 7.5/100
The tax on the computer would be = $168.75
The total cost for the computer on sale would be = $2250 - $337.50
The total cost for the computer on sale would be = $1,912.5
If the tax is 7.5% .
So he has to pay = $1,912.5 + $1,912.5 × 7.5% = $2,056
Therefore, the total price for the computer on sale would be $2,056,
The price of the computer not on sale = $2250 + $168.75 = $2418.75.
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Estimate ∫20(4x2−2x)dx= using a left Reimann Summ and 4 equal subintervals
The left Riemann Sum approximation for the integral of 20(4x² - 2x)dx over the interval [0, 20] with four equal subintervals is 165250.
In calculus, the integral is a mathematical tool that allows us to find the area under a curve. It is an important concept that is used in many fields of science and engineering. One way to estimate the value of an integral is by using a Riemann Sum, which involves dividing the region under the curve into small rectangles and adding up their areas.
In this case, we will be using a left Riemann Sum with four equal subintervals to estimate the integral of 20(4x² - 2x)dx.
To use a left Riemann Sum, we need to first divide the interval of integration into equal subintervals. In this case, we have four subintervals of width
=> (b - a)/n = (20 - 0)/4 = 5.
The left endpoint of each subinterval will be used as the height of the rectangle that represents the area under the curve in that subinterval.
Next, we need to evaluate the function 20(4x² - 2x) at the left endpoints of each subinterval. These values will be used as the heights of the rectangles. The left endpoints are 0, 5, 10, and 15, so we have:
f(0) = 20(4(0)² - 2(0)) = 0
f(5) = 20(4(5)² - 2(5)) = 1900
f(10) = 20(4(10)² - 2(10)) = 7200
f(15) = 20(4(15)² - 2(15)) = 15150
We can now find the area of each rectangle by multiplying its height by its width (5). The total area is the sum of these areas:
(0)(5) + (1900)(5) + (7200)(5) + (15150)(5) = 165250
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barry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. the sums of the two numbers on each of the three cards are equal. the three numbers on the hidden sides are prime numbers. what is the average of the hidden prime numbers?
The average of the three hidden prime numbers is 5, as the sum of any two of the numbers is equal to 10, and since all three numbers are prime, the only possible values are 3, 5, and 7.
3 + 5 + 7 = 15
15/3 = 5
The average of the three hidden prime numbers is 5. This means that the sum of any two of the numbers on the cards must be equal to 10. Since all three numbers are prime,as the sum of any two of the numbers is equal to 10 the only possible values are 3, 5, and 7. Therefore, the three numbers on the hidden sides of the cards are 3, 5, and 7. To calculate the average, we add the three values together and divide by three. 3 + 5 + 7 = 15, and 15 divided by 3 is 5. Therefore, the average of the hidden prime numbers is 5.
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What is the value of the c when 5c + 23 = c -1
answer is
5(-1) = -5 +23 = 18
patricia's parents buy her too much stuff, so she had 3 times as many as her friend suzy. if patricia has 15 toy trucks, how many does suzy have?
The solution of units problem is Suzy has 5 toy trucks by using the equation 15 = 3 * Suzy's number of toy trucks
According to the given information:
If Patricia has 3 times as many toy trucks as Suzy, we can set up an equation where:
Patricia's number of toy trucks = 3 * Suzy's number of toy trucks
We know that Patricia has 15 toy trucks, so we can substitute that value into the equation:
15 = 3 * Suzy's number of toy trucks
Now we can solve for Suzy's number of toy trucks:
Suzy's number of toy trucks = 15 / 3
Suzy's number of toy trucks = 5
Therefore, Suzy has 5 toy trucks.
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In AJKL, if m ZJ is seven less than mZL and mZK is 21 less than twice mZL, find the measure
of each angle?
The measures regarding the angles of the triangle are given as follows:
x = 26.<P = 116º.<Q = 21º.<R = 43º.How to obtain the angle measures?The angle measures are defined as follows:
<P = 5x - 14.<Q = x - 5.<R = 2x - 9.The sum of the measures of the internal angles of a triangle is of 180º, hence the value of x is obtained as follows:
5x - 14 + x - 5 + 2x - 9 = 180
8x = 208
x = 208/8
x = 26.
Then the measures are given as follows:
<P = 5(26) - 14 = 116º.<Q = 26 - 5 = 21º.<R = 2(26) - 9 = 43º.More can be learned about angle measures at https://brainly.com/question/24607467
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3. Utilizando el Teorema de Pitágoras encuentra el valor que falta. 12cm b= b 13cm 5.4 cm 7.2 cm
The missing value is given as follows:
x = 11.3 cm.
What is the Pythagorean Theorem?The Pythagorean Theorem states that for a right triangle, the length of the hypotenuse squared is equals to the sum of the squared lengths of the sides of the triangle.
The parameters for this problem are given as follows:
The sides are 14 cm and x.The hypotenuse is of 18 cm.Hence the missing value is obtained as follows:
14² + x² = 18²
x = sqrt(18² - 14²)
x = 11.3 cm.
Missing InformationWe have a triangle in which:
The sides are 14 cm and x.The hypotenuse is of 18 cm.More can be learned about the Pythagorean Theorem at brainly.com/question/30203256
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which of these statements about a normally distributed data set is true? a.) the smaller the standard deviation, the less concentrated the data will be. b.) the concentration of the data set is not related to the standard deviation. c.) the smaller the standard deviation, the more concentrated the data will be. d.) the greater the standard deviation, the more concentrated the data will be.
The Statement which is true is (c) the smaller the standard deviation, the more concentrated the data will be.
A standard deviation (or σ) is a measure of how dispersed the data is in relation to the mean. Low standard deviation means data are clustered around the mean, and high standard deviation indicates data are more spread out.
A standard deviation close to zero indicates that data points are close to the mean, whereas a high or low standard deviation indicates data points are respectively above or below the mean.
The smallest possible value for the standard deviation is 0, and that happens only in contrived situations where every single number in the data set is exactly the same
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A brick path between two buildings is a rectangle 30.5m by 4.1m.Flowers grow in 4 square holes in the walkway.Each side of the holes is 1.2m.Calculate the area covered in brick.
Find the area of the polygon in square units.
The area of the figure is solved to be
10 square unitsWhat is the area of ta kite?The area of a kite is solved using the formula
= P * q / 2
Where
p = length of long diagonal
q = length of short diagonal
Using the figure the length can be traced to be
p = 3 - -5 = 3 + 5 = 5
q = 6 - 2 = 4
Area of the figure
= 5 * 4 / 2
= 20 / 2
= 10 square units
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the value of a building is $180,000. the value is expected to increase by 3.5% each year. write a function that represents the value y (in dollars) of the building after x years. the predict the value after 10 years
Determine if the side lengths 9, 9, and 17 form a triangle. If it is a triangle, classify it by its sides.
equilateral triangle
It's not a triangle.
scalene triangle
isosceles triangle
Answer:
It's not a triangle
Step-by-step explanation:
What you do is add the two smallest numbers and hope it's smaller than the longer side
9 + 9 = 18
The longest side is 17, which is LESS than 18. That means it's not a triangle
Answer:
It isnt a triangle
Step-by-step explanation:
I did the quiz and this was the answer.
Jesse spends 12
of his pocket money on Monday. On Tuesday, he spends 23
of what is left. On Wednesday, he spends 14
of what remains. What fraction of the pocket money does he have left? Choose the most reasonable answer. CLEAR CHECK
13
43
66
18
The most reasonable answer for left pocket money to Jesse after spending it on Monday, Tuesday and Wednesday is 1/8.
Let's assume that the amount of pocket money Jesse has is "x".
On Monday, he spends 1/2 of his pocket money, so he has 1/2 of his pocket money left:
Amount left after Monday = x - (1/2)x = (1/2)x
On Tuesday, he spends 2/3 of what is left, which is 2/3 of (1/2)x:
Amount left after Tuesday = (1/2)x - (2/3)(1/2)x = (1/6)x
On Wednesday, he spends 1/4 of what remains, which is 1/4 of (1/6)x:
Amount left after Wednesday = (1/6)x - (1/4)(1/6)x = (1/8)x
Therefore, the fraction of the pocket money he has left is:
Amount left / Original amount = (1/8)x / x = 1/8
So, the most reasonable answer is 1/8.
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____The given question is incorrect, the correct question is given below:
Jesse spends 1/2 of his pocket money on Monday. On Tuesday, he spends 2/3 of what is left. On Wednesday, he spends 1/4 of what remains. What fraction of the pocket money does he have left? Choose the most reasonable answer.
Help meee
Which one is it?
Answer:
the number of shirts sold
Step-by-step explanation:
that's what the question is asking
select the correct answer. what is the solution to this equation? 3^2x = 1/3
Answer:
x= 1/27
Step-by-step explanation:
the correct answer is : X = 1/27
Answer:
- 1/2
Step-by-step explanation:
To solve for x in the equation 3^2x = 1/3, we can start by taking the logarithm of both sides with base 3:
log3 (3^2x) = log3 (1/3)
Using the logarithmic property logb (a^n) = n * logb (a), we can simplify the left side:
2x * log3 (3) = log3 (1/3)
Since log3 (3) = 1, we can simplify further:
2x = log3 (1/3)
To find the value of the logarithm on the right side, we can use the definition of logarithms: logb (a) = c if and only if b^c = a. Since 1/3 = 3^-1, we have:
2x = log3 (3^-1) = -1
Finally, to solve for x, we divide both sides of the equation by 2:
x = -1/2
So the solution to the equation 3^2x = 1/3 is x = -1/2.
Four more than the quotient of a number and 5 is equal to 2 . Use the variable w for the unknown number.
The unknown number's variable w is equal to -10.
How is a variable defined in mathematics?In mathematics, a variable is an alphabet or phrase that represents an unknown quantity, unknown value, or unknown number. The variables are used, particularly when dealing with algebraic expressions or algebra. As an example, the variables x and 9 are constants in the linear equation x+9=4.
The following information can be written as an equation to express it:
4 + w/5 = 2
By deducting 4 from both sides of the equation, we may isolate the variable on one side and solve for w:
w/5 = -2
Then, to find w, we can multiply both sides by 5.
w = -10
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1. Find f(4)
f(1)=8
f(n)=-2f(n-1)-4
The fourth term in the sequence is:
f(4) = -76
How to find the value of f(4)?Here we have a sequence where we know the first term:
f(1) = 8
And the recursive formula is:
f(n) = -2*f(n - 1) - 4
Using the recursive formula we can get the next terms:
f(2) = -2*f(2 - 1) - 4
= -2*f(1) - 4 = -2*8 - 4 = -20
f(3) = -2*f(3 - 1) - 4
= -2*-20 - 4 = 40 - 4 = 36
f(4) = -2*f(4 - 1) - 4
= -2*36 - 4 = -76
That is the value of the fourth term.
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Let A={2,4,6} and B={135}
R={(2,2),(4,4),(6,6)} is a relation on A. Express the relation
The relation:
{x = y : x, y ε A}.
And this relation is a function.
What is a function?A relation between a collection of inputs and outputs is known as a function. A function is a connection between inputs in which each input is connected to precisely one output. Each function has a range, co-domain, and domain.
Given:
A={2,4,6} and B={135}
R={(2,2), (4,4), (6,6)} is a relation on A.
That means, each value of set A connects to the same point.
The relation:
{x = y : x, y ε A}
Hence, {x = y : x, y ε A}.
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Pls help thank you will mark the Brainliest
The exponential function from the decay function given is y = 30.5(0.7)ˣ
What is an exponential functionAn exponential function is a mathematical function that models the growth of a quantity as a power of another quantity, typically as the exponent of a constant called the base. The general form of an exponential function is given by:
f(x) = a^x
where "a" is the base and "x" is the input. The value of "a" must be positive and greater than zero. When "a" is greater than 1, the exponential function grows very quickly, and when "a" is between 0 and 1, the exponential function decreases very quickly.
In this problem, we can find the exponential function by;
when x = 0
f(0) = 30.5
The function can be written as;
f(x) = 30.5(b)ˣ
To find the value of b,
We can take another point and solve;
f(0) = 30.5
f(1) = (21.35)
Looking at the graph, this is a decay function and we can easily find the value as 0.7.
The function is given as y = 30.5(0.7)ˣ
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a two-tailed test is a hypothesis test in which the rejection region is . a. only in the upper tail of the sampling distribution b. in both tails of the sampling distribution c. in one tail of the sampling distribution d. only in the lower tail of the sampling distribution
A two-tailed test is a hypothesis test in which the rejection region is option (B) in both tails of the sampling distribution
The hypothesis test is defined as the statistic approach checking the results of a research study support a particular theory when we apply that result to population
A two-tailed test is a hypothesis test that defined in which the critical area of the distribution is two sided, so the sample will be greater than or less than a certain range of values
Therefore, the rejection region of a two-tailed test is a hypothesis test is in both tails of the sampling distribution.
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#7. Tyler llenó su bañera, se bañó y luego drenó la bañera. La función B da la profundidad del agua,
pulgadas, t minutos después de que Tyler comenzó a llenar la bañera.
Explique el significado de cada declaración en esta situación.
a. B(0) = 0
b. B(1)
C. B(9) = 11
d. B(10) = B(22)
e. B(20) > B(40)
The explanation of each statement in each given situation is given below:
How to solve for thisa. B(0) = 0
depth in inches 0 minutes after filling began is 0; that is at time =0; depth =0
b. B(1) < B(7)
B(1) represents the function one minus after filling began
c. B(9) = 11
depicts that the depth in inches of water 5 minutes after filling is 11
d. B(10) = B(22)
depicts that the depth in inches of water 10 minutes after filling began is 22.
e. B(20) > B(40)
represents the function 20 minutes after filling began
These are the required answers
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The question translated in English is: "#7. Tyler filled his tub, took a shower, and then drained the tub. Function B gives the depth of the water,
inches, t minutes after Tyler started filling the tub.
Explain the meaning of each statement in this situation.
to. B(0) = 0
b. B(1) C.B(9) = 11
d. B(10) = B(22)
and. B(20) > B(40)"
Z
S
L
T
X
V
Figure 1.
Use the diagram to answer the question.
Which of the following are opposite rays?
The rays that are opposite are:
TX and TL
What is a ray?A ray is a line that points in one direction. It has one starting point and no end point.
It is important that we know the difference between a line and a ray.
A line has no endpoints or a starting point.
We have,
From the figure,
The rays are:
TS
TL
TV
TZ
All have a common starting point, T.
We see that,
TX and TS are perpendicular rays.
TS and TL are perpendicular rays.
TX and TL are opposite rays.
Now,
The rays that are opposite are:
TX and TL.
Thus,
TX and TL are opposite rays.
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Subtract 7x²-x-1 from x²+3x+3
Answer: The answer is x²+2x-8
Step-by-step explanation:
A procedure for using sample data to find the estimated regression equation is
a. point estimation. b. interval estimation.
c. the least squares method. d. extrapolation.
Option C. the least squares method is a procedure for using sample data to find the estimated regression equation.
The procedure for using sample data to find the estimated regression equation is called the least squares method. This method involves finding the line that minimizes the sum of the squared vertical distances between the observed data points and the line. The estimated regression equation is used to predict the value of the dependent variable based on the value of the independent variable. This method is commonly used in regression analysis to model the relationship between two variables. It is a useful tool for making predictions and understanding the relationship between variables.
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The picture has the question
The exponential function for the given table will be y=0.01(2)ˣ where x=positive integer.
What exactly are exponential functions?
An exponential function is a mathematical function with the formula f (x) = aˣ, where "x" is a variable and "a" is a constant that is termed the function's base and must be greater than zero. The transcendental number e, which is approximately equal to 2.71828, is the most often used exponential function basis.
Growth that is exponential
The amount expands extremely slowly at first, then rapidly in Exponential Growth. The rate of change accelerates with time. As time passes, the pace of growth accelerates. The quick expansion is designed to represent a "exponential rise".
The exponential growth formula is: y = a (1+ r)ˣ, where r is the growth percentage.
Now,
as given in the table initial value = 0.01
and it is increasing into multiples of 2ˣ where x=positive integer
Then for the value of y for any value of x will be
y=0.01(2)ˣ where x=positive integer
Hence,
The exponential function for the given table will be y=0.01(2)ˣ.
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what is the least number that can be made using each digit exactly once? explain why the value of the 4 is greater than the value of the 6
Answer: The least number that can be made using each digit exactly once is 123456789. The value of 4 is greater than the value of 6 because in the decimal system (base 10), the position of a digit determines its value. The leftmost digit represents the greatest place value (ones, tens, hundreds, etc.) and each subsequent digit represents a smaller place value. In the number 123456789, the 4 is in the thousands place, which has a greater value than the 6, which is in the tens place.
Therefore, the 4 has a greater value than the 6 because it represents a larger place value.
Step-by-step explanation:
4. Find the coordinates of the center and the measure of the diameter for a circle whose equation is
(x+4)² + (y + 5)² = 64
The general equation of a circle is (x-h)² + (y-k)² = r²
Where,
(h,k) = center of circler = radiusHere we are given the equation (x+4)² + (y+5)² = 64 and given this we are asked to find the diameter as well as the coordinates of the center.
Finding the coordinates of the center and diameter.Finding the center
If (h,k) = center, then the coordinates of the center are the given values of h and k but negative. This is because the values are being subtracted from x and y. (x-h)² + (y-k)² = r²
This means that, (x+4)² + (y+5)² = 64 is the same as saying (x-(-4))² + (y-(-5))² = 64 based off of the general equation of a circle.
Notice how the plugged in values are negative rather than positive.
In conclusion, we can say that the center is at (-4,-5) (see attached image below to check answer)
In the equation, 64 takes the place of r²
So r² = 64
Taking the square root of both sides we get r = 8
So the radius is 8.
To convert to diameter we multiply by 2
So diameter = 8 * 2 = 16
In conclusion, a circle with the equation (x+4)² + (y + 5)² = 64, has a center at (-4,-5) and a diameter of 16
I don’t know how to solve 20x³-25x²+4x-5
Answer: The equation 20x³-25x²+4x-5 can be factored by finding its roots, which are the values of x that make the equation equal to zero.
To find the roots, you can use one of the following methods:
Rational Root Theorem
Synthetic Division
Factor Theorem
Binomial Expansion
I would recommend using the Rational Root Theorem, which states that if a polynomial has a rational root, then it can be expressed as a product of linear factors with rational coefficients.
First, list out the possible rational roots of the equation: ±1, ±2, ±5, ±1/2, ±1/5.
Next, test each root by dividing the equation by (x-root), using synthetic division. If the remainder is zero, then the root is a factor of the equation. Repeat this process until you have found all of the roots.
Finally, factor the equation by grouping the roots into pairs and multiplying each pair to obtain a quadratic factor. Factor each quadratic and simplify the expression.
It is important to note that this is a rather advanced topic in algebra and finding the roots of polynomials can be quite challenging. If you need further assistance, I recommend seeking help from a tutor or textbook on algebra.
Step-by-step explanation:
