Use the power-reducing formulas to rewrite the expression in terms of first powers of the cosines of multiple angles.tan4(4x)
Answers
Solution
[tex]\begin{gathered} \tan^2(4x)=\frac{1-\cos(8x)}{1+\cos(8x)} \\ \\ \Rightarrow\tan^4(4x)=\frac{1-2\cos(8x)+\cos^2(8x)}{1+2\cos(8x)+\cos^2(8x)} \\ \\ \text{ since }\cos^2(8x)=\frac{1+\cos(16x)}{2} \\ \\ \Rightarrow\tan^4(4x)=\frac{1-2\cos(8x)+\frac{1+\cos(16x)}{2}}{1+2\cos(8x)+\frac{1+\cos(16x)}{2}} \\ \\ \Rightarrow\tan^4(4x)=\frac{2-4\cos(8x)+1+\cos(16x)}{2+4\cos(8x)+1+\cos(16x)} \\ \\ \Rightarrow\tan^4(4x)=\frac{3-4\cos(8x)+\cos(16x)}{3+4\cos(8x)+\cos(16x)} \end{gathered}[/tex]The answer is:
[tex]\frac{3-4\cos(8x)+\cos(16x)}{3+4\cos(8x)+\cos(16x)}[/tex]Related Questions
Simplify each expression by distributing8(x + 5)
Answers
ok
8(x + 5) Just multiphy 8 by each term
8x + 40 This is the result
Find the measure of a positive angle and a negative angle that are coterminal with 100° sketch of three angles labeling clearly with directional arrows.
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Coterminal angles are different angles that have the same terminal side.
A positive angle has one turn more around so it has a measure of 100°+360° = 460°.
A negative angle will have a measure that is represented in clockwise rotation and be equal to 100° - 360° = -260°.
We can sketch an angle of measure 100°, a positive coterminal angle and a negative coterminal angle as:
The results of a survey show that the percent of adults in a certain town who want to add bike lanes to amajor roadway is in the interval (0.57, 0.65) (9 points)(a) What is the point estimate for the percent who want to add the bike lanes?(b) What is the poll's margin of error?(c) If the town's adult population is 31,526, what is the best estimate for the number of people whowould support the bike lanes?
Answers
If we know the confidence interval for the proportion, the point estimate will be at the center of this interval.
Then, we can calculate the point estimate p as the average between the boundaries of the interval:
[tex]p=\frac{0.57+0.65}{2}=\frac{1.22}{2}=0.61[/tex]The margin of error can be calculated, knowing the interval, as half the difference between the upper boundary and the lower boundary of the interval:
[tex]\text{MOE}=\frac{UB-LB}{2}=\frac{0.65-0.57}{2}=\frac{0.08}{2}=0.04[/tex]The margin of error is 0.04. This margin of error is also the absolute difference between any boundary of the interval and the point estimate.
If the town's population is 31,526, the best estimate for the number of people who
would support the bike lanes is to use the point estimate as the proportion:
[tex]X=p\cdot N=0.61\cdot31526\approx19231[/tex]Answer:
a) The point estimate is p=0.61
b) The margin of error is MOE = 0.04
c) The best estimate is X=19231
*1. If the variable x represents the total number of COVID-19 deaths in the United States since March 1,I 2020, what do the following expressions represent?a. X - 100,000
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The expression represents the number of COVID deaths since March 1 2020 minus 100,000 deaths.
Linda is adding padding to all of the surfaces inside her attic for extra warmth in the winter.She needs to find the approximate surface area of the attic, including the walls, floor, andceiling. The attic is in the shape of a triangular prism. Linda draws the net and writesthe expression below to represent the surface area of the attic. Are Linda's net andexpression correct?15 ft45 ft25 ft40 ft25 ft25 ft25 ft- 15 ft45 ft40 ft15 ftExpression for Surface Area of Attic:45 (40 + 25 + 25) + ] (40 x 15)
Answers
We can formulate an expression for the surface area of the attic like this:
The area of a triangle is given by the following formula:
[tex]A=\frac{b\times h}{2}[/tex]Where b is the base and h is the height of the triangle.
The area of a rectangle is given by the following formula:
[tex]A=w\times l[/tex]Where w is the width and l is the length of the rectangle.
In this case, the attic has three rectangular faces, all of them have a width of 45 ft. two of them have a length of 25 ft and one has a width of 40 ft, then we can calculate the areas of these faces like this:
[tex]\begin{gathered} A1=45\times40 \\ A2=45\times25 \\ A3=45\times25 \end{gathered}[/tex]By summing up these areas, we get the area of the rectangular faces:
[tex]A=45\times40+45\times25+45\times25[/tex]From this expression, we can factor 45 to get:
[tex]A=45\times(40+25+25)[/tex]For the two triangular faces, their height equals 15 ft and the length of the bases equals 40 ft, then their areas are:
[tex]\begin{gathered} A1=\frac{15\times40}{2} \\ A2=\frac{15\times40}{2} \end{gathered}[/tex]By summing them up, we get the area of the triangular faces:
[tex]A=\frac{15\times40}{2}+\frac{15\times40}{2}=15\times40[/tex]By summing the area of the rectangular faces and the area of the triangular faces, we get the expression to calculate the total surface area of the attic, like this:
[tex]A=45(40+25+25)+40\times15=4650[/tex]Then, the net Linda draw is correct. The first term of Linda's expression 45(40+25+25) is correct. The second term of Linda's equation missing a factor of 2. The surface area of Linda's attic is 4650 square feet
Evaluate the function when x= -2,0, and 5 h(x)= -2x+9
Answers
Given:
a function is given as h(x) = -2x + 9
Find:
we have to evaluate the function at x = -2 , 0 and 5.
Explanation:
when x = -2
h(-2) = -2(-2) + 9 = 4 + 9 = 13
when x = 0
h(0) = -2(0) + 9 = 0 + 9 = 9
when x = 5
h(5) = -2(5) + 9 = - 10 + 9 = -1
Therefore, the values of given function h(x) are 13, 9 , -1 at x = -2, 0 , 5 respectively.
Complete the statements about the following numbers: 2/7, 0.1, 0.9, 6/8. Use the + and - buttons to change how many ticks are displayed. The number represents the amount of even segments between 0 and 1. The point closest to the benchmark 1 is at The point closest to the benchmark O is at How would you order these fractions and decimals from least to greatest? Click or tap and drag to move the dot along the number line. 1 B 2 2 +
Answers
To solve the exercise it is easier to convert all the given points to decimals. So,
[tex]\frac{2}{7}=0.23[/tex][tex]\frac{6}{8}=0.75[/tex]Then,
*The point closest to the benchmark 1 is at 0.9.
*The point closest to the benchmark 0 is 0.1.
*Ordering these points from least to greatest you have:
[tex]\begin{gathered} 0.1 \\ 0.23=\frac{2}{7} \\ 0.75=\frac{6}{8} \\ 0.9 \end{gathered}[/tex]please help me!!!!!!
Answers
I attach the table with the results organized correctly.
The correct option is 1.
Suppose that you decide to buy a car for $32,635, including taxes and license fees. You saved $9,000 for a down payment and can get a four year car loan at 6.31%. Find the monthly payment and the total interest for the loan. Equation below.
Answers
SOLUTION:
Step 1:
In this question, we are given the following:
Suppose that you decide to buy a car for $32,635, including taxes and license fees.
You saved $9,000 for a down payment and can get a four-year car loan at 6.31%.
Find the monthly payment and the total interest for the loan.
Step 2:
In this question, we are given:
Cost of the car = $ 32, 635
Down payment = $ 9000
Principal on the Car loan = $ ( 32,635 - 9,000 ) = $ 23, 635
Rate = 6. 31%
Using the PMI Formulae, we have that:
16 The water level of a river was measured each day during a two-week period. The graph models the linear relationship between the water level of the river in feet and the number of days the water level was measured. Water Level of River 28 20 16 Water Level (0) 3 4 6 8 10 12 14 Number of Days Which statement best describes the y-intercept of the graph?
Answers
From the graph, it can be observed that line intersect the y-axis at (0,16) and after that level of water increases with increase in number of days.
So 16 feet represents the initial water level of the river. The correct answer is,
The initial water level was 16 ft.
Johann uses 42 +7 to represent the number of players who are on teams.Explain what 42÷7 means. Enter a number in each box.
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Johann uses 42 +7 to represent the number of players who are on teams.
Explain what 42÷7 means.
_______________________________
42 +7 = the number of players who are on teams
42 = he number of players who are on teams minus 7
42 players
________________________
Dividing by 7 (the number of player per group)
Each team has 7 players
_____________
hi my name is Shila and I'm trying to explain how to solve this problem to my daughter but a but confused can you direct me please?
Answers
As given by the question
There are given that the point
[tex]\frac{8}{6}[/tex]Now,
First, break the point
So,
[tex]\frac{8}{6}=\frac{4}{3}[/tex]According to the above point, there are showing 3 in the denominator
So, between 0 and 1 divide 3 parts in the number line
Then,
A painter has three partially filled paint cans. One contains1 7/8 gallons, the second contains1 1/5 gallons, and the third contains1 3/4 gallons. Which answer is closest to the total amount of paint?
Answers
Explanation
Step 1
convet the mixed numbers into simple fractions
remember
[tex]a\frac{b}{c}=\frac{(a\cdot c)+b}{c}[/tex]then
[tex]\begin{gathered} 1\text{ }\frac{7}{8}=\frac{(1\cdot8+7)}{8}=\frac{15}{8} \\ 1\frac{1}{5}=\frac{(1\cdot5+1)}{5}=\frac{6}{5} \\ 1\frac{3}{4}=\frac{(1\cdot4+3)}{4}=\frac{7}{4} \end{gathered}[/tex]Step 2
now, make the sum to find the total amount
[tex]\begin{gathered} \text{total amount= }\frac{15}{8}+\frac{6}{5}+\frac{7}{4} \\ \text{total amount=}\frac{(15\cdot5\cdot4)+(6\cdot8\cdot4)+(7\cdot5\cdot8)}{160} \\ \text{total amount=}\frac{300+192+280}{160} \\ \text{total amount=}\frac{772}{160} \\ \text{total amount=}\frac{193}{40} \end{gathered}[/tex]I hope this helps you
A multivitamin tablet contains 12.5mg of calcium how much calcium does a bottle of 40 tablets contain write your answer in grams
Answers
One multivitamin tablet contains 12.5mg.
Therefore, 40 tablets will contain
[tex]40\times12.5=500[/tex]40 tablets will contain 500mg.
In grams:
By conversion,
[tex]1g=1000mg[/tex]Hence, 500mg can be converted as
[tex]\frac{500}{1000}=0.5[/tex]Hence, there are 0.5g of calcium in
If 8 more than twice a number is 14, what is 8 times the number?
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Let the unkonwn number be
[tex]=x[/tex]Twice the number means
[tex]\begin{gathered} =2\times x \\ =2x \end{gathered}[/tex]8 more than twice the number means the sum of twice the number and 8
[tex]=2x+8[/tex]8 more than twice a number is 14, will be represented below as
[tex]2x+8=14[/tex]collect similar terms from the equation above to get
[tex]\begin{gathered} 2x+8=14 \\ 2x=14-8 \\ 2x=6 \\ \text{divide both sides by 2} \\ \frac{2x}{2}=\frac{6}{2} \\ x=3 \end{gathered}[/tex]8 times the number will then be,
[tex]\begin{gathered} =8\times x \\ =8\times3 \\ =24 \end{gathered}[/tex]Hence,
The correct answer is OPTION C
Add Solve: n + 7 = 31
Answers
Answer:
n = 24
Explanation:
The initial expression is:
n + 7 = 31
So, to solve the equation, we need to subtract 7 from both sides:
n + 7 - 7 = 31 - 7
n = 24
Therefore, the solution is n = 24
If a person travels 3.5 miles in 30 minutes, what is their speed i miles per hour
Answers
Given:-
If a person travels 3.5 miles in 30 minutes.
To find their speed in miles per hour.
So now we solve using the formula,
[tex]\text{Distance}=\text{speed}\times time[/tex]Substituting the known values. we get,
[tex]3.5=\text{Speed}\times\frac{1}{2}[/tex]Now we solve for speed. so we get,
[tex]\begin{gathered} \text{speed}=3.5\times2 \\ \text{speed}=7 \end{gathered}[/tex]So the required speed is 7miles/hr.
"Radon: The Problem No One Wants to Face" is the title of an article appearing in Consumer Reports. Radon is a gas emitted from the ground that can collect in houses and buildings. At
certain levels it can cause lung cancer. Radon concentrations are measured in picocuries per liter (PCI/L). A radon level of 4 pci/L is considered "acceptable." Radon levels in a house vary
from week to week. In one house, a sample of 8 weeks had the following readings for radon level (in pcI/L).
1.9 2.8 5.7 4.8 1.9 8.6 3.9 7.3
(a) Find the mean, median, and mode. (Round your answers to two decimal places.)
mean
median
mode
(b) Find the sample standard deviation, coefficient of variation, and range. (Round your answers to two decimal places.)
S
CV
range
(c) Based on the data, would you recommend radon mitigation in this house? Explain.
O Yes, since the average value is over "acceptable" ranges, although the median value is not.
O Yes, since the median value is over "acceptable" ranges, although the mean value is not.
O No, since the average and median values are both under "acceptable" ranges.
O Yes, since the average and median values
are both over "acceptable"
ranges.
Answers
a) The mean of the data set is 4.61
The median of the data set is 4.35
The mode of the data set is 1.9
b) Sample standard deviation is 2.58
Coefficient of Variation is 55.96
Range is 6.7
Given,
The data set;
1.9, 2.8, 5.7, 4.8, 1.9, 8.6, 3.9, 7.3
a) We have to find the mean, median and mode
Mean = (1.9 + 2.8 + 5.7 + 4.8 + 1.9 + 8.6 + 3.9 + 7.3) / 8 = 36.9/8 = 4.61Median;Order the data first;
1.9, 1.9, 2.8, 3.9, 4.8, 5.7, 7.3, 8.6
Now,
The data is of even number, so;
Median = [(n/2) + (n/2 + 1)] / 2
Here,
n/2 = 8/2 = 4th term
n/2 + 1 = 5th term
Then,
Median = (3.9 + 4.8) / 2 = 8.7/2 = 4.35
Mode;The mode of the given data is 1.9
b) Sample Standard DeviationHere it is the formula to calculate it:
x = √(∑(xi - x)²/n-1))
sₓ = √(46.85/7) ≈ 2.58
Coefficient of VariationCV is the quotient between sample Standard deviation over Mean and it is used to make comparisons.
CV = sₓ/x × 100 = 2.58/4.61 x 100 = 55.96
RangeThe difference between the highest and the lowest value of this sample
8.6 - 1.9=6.7
Learn more about data set here;
https://brainly.com/question/13708278
#SPJ1
The equation ^2 − 4 − 4^2 + 13 = 0 will produce a hyperbola. How can we tell by simply observing the equation?In what directions do the branches of this hyperbola open? How do you know? Explain. Sketch a graph of this hyperbola, clearly indicating how you have determined thekey characteristics (center, vertices, eccentricity, foci). Give the domain and range of this hyperbola.
Answers
we have the equation
[tex]^2−4−4^2+13=0[/tex]Group similar terms and move the constant to the right side
[tex](^2−4)−4^2=-13[/tex]Complete the square
[tex]\begin{gathered} (y^2-4y+2^2-2^2)-4x^2=-13 \\ (y^2-4y+2^2)-4x^2=-13+2^2 \\ (y^2-4y+2^2)-4x^2=-9 \end{gathered}[/tex]Rewrite as a perfect square
[tex](y-2)^2-4x^2=-9[/tex]Divide both sides by -9
[tex]\begin{gathered} \frac{(y-2)^2}{-9}-\frac{4x^2}{-9}=\frac{-9}{-9} \\ \\ -\frac{(y-2)^2}{9}+\frac{x^2}{\frac{9}{4}}=1 \\ \\ \frac{x^{2}}{\frac{9}{4}}-\frac{(y-2)^{2}}{9}=1 \\ \end{gathered}[/tex]The coordinates of the center are (0,2)
The transverse axis is on the x-axis
a^2=9/4 -----------> a=3/2
b^2=9 -----------> b=3
The vertices are --------> (0+1.5,2) and (0-1.5,2)
so
Vertices at (1.5,2) and (-1.5,2)
Find out the value of c
c^2=a^2+b^2
c^2=(9/4)+9
c^2=45/9
c=√5
Find out the coordinates of the foci
(0+√5,2) and (0-√5,2)
using a graphing tool
The domain is the interval (-infinite, -1.5) U (1.5, infinite)
The range is the interval (-infinite, infinite)
Kathy wants to buy a condominium selling for $96,000. The taxes on the property are $1300 per year, and homeowners' insurance is $336 per year. Kathy's gross monthly income is $4000. She haher van. The bank is requiring 20% down and is charging a 9.5% interest rate with no points. Her bank will approve a loan that has a total monthly mortgage payment of principal, interest, property tthan or equal to 28% of her adjusted monthly income. Complete parts a) through h) below.a) Determine the required down payment.The required down payment is $b) Determine 28% of her adjusted monthly income.28% of her adjusted monthly income is $(Round to the nearest cent as needed.)c) Determine the monthly payments of principal and interest for a 25-year loan.The monthly payment of principal and interest for a 25-year loan is $(Round to the nearest cent as needed.)d) Determine her monthly payment, including homeowners' insurance and taxes.Her total monthly payment, including homeowners' insurance and taxes is $(Round to the nearest cent needed.) Does Kathy qualify for the loan?0 YesO No
Answers
a) the cost of the house is 96000 and the down paidment is the 20% so we can use a rule of 3 to solve it so:
[tex]\begin{gathered} 96000\to100 \\ x\to20 \end{gathered}[/tex]so the equation will be:
[tex]\begin{gathered} x=\frac{96000\cdot20}{100} \\ x=19200 \end{gathered}[/tex]b) her income is $4000 so the 28% will be:
[tex]\begin{gathered} 4000\to100 \\ x\to28 \end{gathered}[/tex]so the equation will be:
[tex]\begin{gathered} x=\frac{4000\cdot28}{100} \\ x=1120 \end{gathered}[/tex]c) the equation that models a loan is:
[tex]C=\frac{P\cdot(0.095\cdot(1+0.095)^n)}{(1+0.095)^{25}-1}[/tex]So we replace the princeiple and find monthly paidment.
[tex]\begin{gathered} C=\frac{76800\cdot(0.095\cdot(1.095)^{25}}{8.67} \\ C=\frac{76800\cdot0.92}{8.67} \\ C=\frac{70540.30}{8.67} \\ C=8136.14 \\ C\approx8136 \end{gathered}[/tex]d)The total monthly paidment will be:
[tex]\begin{gathered} T=8136+\frac{1300}{12}+\frac{336}{12} \\ T=8136+108.33+28 \\ T=8272.33 \\ T\approx8272 \end{gathered}[/tex]SOo the answer is NO she can't afort to buy this house
Judy is buying 6 pint of ice cream for her party at $3.45 each if she has a $20 bill does she have enough to buy the ice cream
Answers
the cost of the ice cream is,
3.45 $
also, she has only 20 $
SO, the number of icecreams bought by
A train travels 165 km in 1.5 hours.
How far will the train travel in 2.2 hours if it maintains the same speed?
Answers
[tex]\huge\red{\mid{\underline{\overline{\textbf{EQUATION AND ANSWER}}}\mid}}[/tex]
_________________
Let's solve this equation using rates,
_________________DefinitionsUnit Rate - A unit rate means a rate for one of something.
Cross Multiplication - In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable.
_________________
Now that we understand the definition we can further solve this equation
[tex]\large\red{\mid{\underline{\overline{\textbf{Values}}}\mid}}[/tex]
[tex]165[/tex] [tex]km[/tex] ⇒ [tex]1.5[/tex] [tex]hr[/tex]
[tex]x\\[/tex] [tex]km[/tex] ⇒ [tex]2.2[/tex] [tex]hr[/tex]
Now we will use cross-multiplication to solve this equation
[tex]\large\red{\mid{\underline{\overline{\textbf{Equtation}}}\mid}}[/tex]
[tex]165 \cdot 2.2\\x\cdot1.5[/tex]
Once solving this equation we get
[tex]1.5x=363[/tex] [tex]km[/tex]
Divide both sides by [tex]1.5[/tex]
[tex]x=242[/tex] [tex]km[/tex]
[tex]\large\red{\mid{\underline{\overline{\textbf{Answer}}}\mid}}[/tex]
A train travels 165 km in 1.5 hours. How far will the train travel in 2.2 hours if it maintains the same speed?The train would've traveled a total of 242 km in 2.2 hours.
Have a good day!How to write a rule for the nth term of the geometric seq
Answers
The nth term of a geometric sequence is expressed as:
[tex]a_n=ar^{n-1}[/tex]were:
• a is the first term
,• r is the common ratio
,• n is the number of terms
If the 2nd term a₂ = 28, then;
[tex]\begin{gathered} 28=ar^{2-1} \\ ar=28 \end{gathered}[/tex]If the 5th term a₅ = 1792, then;
[tex]\begin{gathered} 1792=ar^{5-1} \\ ar^4=1792 \end{gathered}[/tex]Take the ratio of both equations to have:
[tex]\begin{gathered} \frac{ar^4}{ar}=\frac{1792}{28} \\ r^3=64 \\ r=\sqrt[3]{64} \\ r=4 \end{gathered}[/tex]Substitute r = 4 into any of the equations to have:
[tex]\begin{gathered} ar=28 \\ 4a=28 \\ a=\frac{28}{4} \\ a=7 \end{gathered}[/tex]Determine the rule for the nth term of the geometric sequence. Recall that;
[tex]\begin{gathered} a_n=ar^{n-1} \\ a_n=7(4)^{n-1} \end{gathered}[/tex]This gives the nth term of the geometric sequence
The exit is exactly half way between the Ferris wheel and where you parked your car. Give the coordinates of your parking spot.
Answers
GIven data :
The coordinates of ferris wheel is, (2,7).
The coordinates of exit is (4,0).
they have given exactly half way so let us use the midpoint formula,
the mid point formula is,
[tex](\frac{x_1+x_2}{2},\frac{y_1,y_2}{2})\ldots(1)[/tex]take the coordinates as
[tex]\begin{gathered} (x_1,y_1)=(2,7) \\ (x_2,y_2)=(4,0) \end{gathered}[/tex]let us subsitute in eqiuation (1),
[tex]\begin{gathered} (\frac{2+4_{}}{2},\frac{7_{}+0_{}}{2}) \\ (\frac{6}{2},\frac{7}{2}) \\ (3,3.5) \end{gathered}[/tex]thus the coordinates of parking spot is (3,3.5).
If the measures of the angles of a triangle arerepresented by 2x, 3x - 15, and 7x +15, the triangleis1) an isosceles triangle2) a right triangle3) an acute triangle4) an equiangular triangle
Answers
Answer
Option 1 is correct.
The triangle is an isosceles triangle.
Explanation
Noting that the sum of angles in a triangle is 180°.
We can solve for each of the angles in this triangle to obtain the type of triangle it is.
The angles of the triangle are 2x, (3x - 15) and (7x + 15)
2x + 3x - 15 + 7x + 15 = 180°
2x + 3x + 7x - 15 + 15 = 180°
12x = 180°
Divide both sides by 12
(12x/12) = (180°/12)
x = 15°
We can then solve for the measures of the three angles now
2x = 2 (15°) = 30°
3x - 15 = 3 (15°) - 15° = 45° - 15° = 30°
7x + 15 = 7 (15°) + 15° = 105° + 15° = 120°
So, the angles of the triangle are 30°, 30° and 120°
A tringle that has two of its angles equal to each other is called an isosceles triangle.
Hope this Helps!!!
16. Solve this system: y = 3x+1 y= 5x-3
Answers
y=3x+1 (1)
y=5x-3 (2)
To solve this system, we can use the equalize method:
3x+1=5x-3
3+1=5x-3x
4=2x
x=4/2
x=2
Now, substitung x=2 in y=3x+1
y=3(2)+1
y=6+1
y=7
Then, the solution to this system of equation would be: (2, 7)
Convert percent to decimal 51.2% =
Answers
Let's begin by identifying key information given to us:
51.2% = 51.2/100
[tex]\begin{gathered} 51.2\text{ \%}=\frac{51.2}{100} \\ \Rightarrow0.512 \end{gathered}[/tex]How do I find all possible rational zeros in a polynomial function?f(x) = x^4-2x^3-4x^2+2x+3
Answers
The factors are 1 , -1 twice and 3
Here, we want to find the rational roots of;
[tex]f(x)\text{= }x^4-2x^3-4x^2+2x+3[/tex]We can start here by trying out simple numbers if we cannot factorize the polynomial at a go
The given polynomial here can be factorized
We can express it as;
[tex]x^4-2x^3-4x^2+2x+3\text{ = (}x-1)(x+1)^2(x-3)[/tex]So what we have here is to simply equate individual linear factor to zero
Thus, we have the factors as;
1 , -1 twice and 3
3. *Which of the following equations has x intercepts at 4 and -2? (A) y = 3x^2 - 10x - 8 (B) y = x^2 + 2x - 8 (C) y = 3x^2 – 2x – 8 (D) y = x^2 - 2x - 8
Answers
Given data:
The x-intercepts given are 4 and -2.
Substitute 0 for y in the first option.
[tex]\begin{gathered} 0=3x^2-10x-8 \\ 3x^2-12x+2x-8=0 \\ 3x(x-4)+2(x-4)=0 \\ x=4,\text{ -}\frac{2}{3} \end{gathered}[/tex]Substitute 0 for y in the second option.
[tex]\begin{gathered} 0=x^2+2x-8 \\ x^2+2x-8=0 \\ x^2+4x-2x-8=0 \\ x(x+4)-2(x+4)=0 \\ (x-2)(x+4)=0 \\ x=2,\text{ -4} \end{gathered}[/tex]Substitute 0 for y in the third option.
[tex]\begin{gathered} 0=3x^2-2x-8 \\ 3x^2-2x-8=0 \\ 3x^2-6x+4x-8=0 \\ 3x(x-2)+4(x-2)=0 \\ (x-2)(3x+4)=0 \\ x=2,\text{ -}\frac{4}{3} \end{gathered}[/tex]Substitute 0 for y inlast option.
[tex]\begin{gathered} 0=x^2-2x-8 \\ x^2-4x+2x-8=0 \\ x(x-4)+2(x-4)=0 \\ (x-4)(x+2)=0 \\ x=4,\text{ -2} \end{gathered}[/tex]Thus, option (D) is correct.
Find the starting value and the base for the exponential function f(x)=kb^x that passes through the two points:(0,3) and (2,12).The starting value k is: AnswerThe base b is: Answer
Answers
The exponential equation given is,
[tex]f(x)=kb^x[/tex]Given the points
[tex](0,3)\text{ and (2,12)}[/tex]Therefore, the values for k and b will be resolved graphically.
Let us now plot the graph using a graphical calculator
From the graph,
[tex]\begin{gathered} y_1=f(x) \\ a=k=3 \\ b=2 \end{gathered}[/tex]Final answers
[tex]\begin{gathered} k=3 \\ b=2 \end{gathered}[/tex]