Use synthetic division to find all the real zeros of the polynomial.
f(x) = x³ + x² - 8x - 6
Answers
so hmmm we look at hmm x³ + x² - 8x - 6 so how do squeak out a factor out of it, well, using the rational root test and using our p/q checking for factors, we find that likely factors can be ± 6/1 and ±(3, 2, 1) / 1, anyhow so we check some of those and hopefully without boring you to death we find that -3 works, we can always use the remainder theorem to check if that's true, by simply plugin in f(-3) and if it's indeed a factor, that'd give us 0, well, f(-3) = -27 + 9 +24 +6 = 0, holly smokes!! is a factor.
well, all that jazz means that -3 is a factor, namely the factor is really x = -3 or x + 3 = 0 so the factor is (x+3) and for our synthetic division we'll use the -3 version, Check the picture below.
so we end up with the factors of (x+3)(x²-2x-2), now for the 2nd factor what we can do to break it up is use the quadratic formula
[tex]~~~~~~~~~~~~\textit{quadratic formula} \\\\ y=\stackrel{\stackrel{a}{\downarrow }}{1}x^2\stackrel{\stackrel{b}{\downarrow }}{-2}x\stackrel{\stackrel{c}{\downarrow }}{-2} \qquad \qquad x= \cfrac{ - b \pm \sqrt { b^2 -4 a c}}{2 a}[/tex]
[tex]x= \cfrac{ - (-2) \pm \sqrt { (-2)^2 -4(1)(-2)}}{2(1)} \implies x = \cfrac{ 2 \pm \sqrt { 4 +8}}{ 2 } \\\\\\ x= \cfrac{ 2 \pm \sqrt { 12 }}{ 2 }\implies x=\cfrac{2\pm\sqrt{2^2\cdot 3}}{2}\implies x=\cfrac{2\pm 2\sqrt{3}}{2}\implies x=1\pm\sqrt{3} \\\\[-0.35em] ~\dotfill\\\\ ~\hfill {\Large \begin{array}{llll} (x+3)(x-1+\sqrt{3})(x-1-\sqrt{3}) \end{array}}~\hfill[/tex]
now, just to clarify, all those are real roots, and the last two come from simply setting x = 1 ± √3 to 0 to get the factors.
Related Questions
In the decathlon event at large track meets, male athletes compete in a total of 10 events. Their combined performance in each of the events is used to determine the winner. Two of the events are the 200-meter dash and the javelin throw. For 12 athletes at a large international competition, performances in these two events are recorded and placed in a scatterplot. The value of r for this scatterplot is 0.369.
A graph titled decathlon Events has 200-meter dash time (seconds) on the x-axis, and Javelin throw distance (meters) on the y-axis. Points are grouped loosely together in a line with positive slope.
Which of the following best describes the relationship between the variables in the scatterplot?
Those with lower 200-meter times tend to have longer javelin distances. The relationship is strong.
Those with higher 200-meter times tend to have longer javelin distances. The relationship is strong.
Those with lower 200-meter times tend to have longer javelin distances. The relationship is moderate.
Those with higher 200-meter times tend to have longer javelin distances. The relationship is moderate.
answer is D
Answers
Based on the information provided, the best description of the relationship between the variables in the scatterplot is: option D, "Those with higher 200-meter times tend to have longer javelin distances. The relationship is moderate."
What is the relationship about?The scatterplot shows a positive relationship between the two variables, as indicated by the points being grouped loosely together in a line with a positive slope.
This means that as the 200-meter dash time increases, the javelin throw distance also tends to increase. The coefficient of correlation, r, also supports this relationship, as a value of 0.369 indicates a moderate positive correlation between the two variables. Though the correlation is moderate, it's important to note that this does not imply causality.
Therefore, note that, based on the description in the question, the relationship is moderate, not strong. The statement is accurate only when the correlation coefficient is between 0.3-0.7 which is considered as moderate correlation.
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Answer:
d
Step-by-step explanation:
Use a calculator to find cos 149° rounded to the nearest hundredth. Identify the angle of depression
Answers
The value of cos 149 degree as calculated from the calculator is found out to be -0.857167301.
Which on round off is , -0.85.
The formula to calculate the angle of depression is , θ = tan-1 (Opposite Side/Adjacent Side) .
Therefore, the AOD for cos149 degree is found out to be as ,
α = arctan(vertical distance / horizontal distance)
α = arctan(10 / 11.998)
α = arctan(0.83347224537423)
α = 0.69482025147278 ° ≈ 0.69 °
The angle of depression is defined as the point of meeting of a horizontal line with the line between the item and the observer's eye. This angle is determined by two factors , which are their height and distance. If the item is below the observer's level, the angle produced between the horizontal line and the observer's line of sight is known as the angle of depression.
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PLEASE SOMEONE HELPPPPP
Simplify (36)18
O a
Ob
Oc
Od
33
312
324
3108
Answers
Answer:
d. 3^108
Step-by-step explanation:
Just multiply 6 and 18.
Solve the following system of equations graphically on the set of axes below.
y = x + 8
y = -1/2x+5
Plot two lines by clicking the graph. Click a line to delete it.
Answers
By graphing both equations on the same axis and findin where do they intercept, we willsee that the solution is (-2, 6).
How to solve the system of equations?Here we have the following system of equations:
y = x + 8
y = -(1/2)x+5
And we want to solve it graphically.
To solve it in that way, we just need to graph both equations on the same coordinate axis and find the ponit where the graphs intercept, that point will be the solution for the system.
The graph of the system of equations can be seen in the image below:
There, we can see that the lines intercept at the point (-2, 6), so that is the solution of our system.
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Gaya buys 2 oranges and 1 cauliflower she pays £2.00
Esteben buys 4 oranges and 1 cauliflower he pays £2.90
Work out the price of one orange
Answers
Answer:
Step-by-step explanation:6
In a middle-school mentoring program, a number of the sixth graders are paired with a ninth-grade student as a buddy. No ninth grader is assigned more than one sixth-grade buddy. If 13 of all the ninth graders are paired with 25 of all the sixth graders, what fraction of the total number of sixth and ninth graders have a buddy
Answers
Answer:
13/25
I hope this is the correct answer.
Donovan flips a coin a total of 96 times. How many times should he expect the coin to land on heads? Donovan records the results of the next 50 flips. The results are shown in the table. Heads Tails 32 18 Based on Donovan's results, if he flips the coin an additional 75 times, about how many times should he expect the coin to land on tails?
Answers
When Donovan flips a coin a total of 96 times, the number of times he should expect the coin to land on heads will be 48 times.
The times he should expect the coin to land on tail based on the table will be 27 times.
How to illustrate the value?A word problem in mathematics simply refers to a question that is written as a sentence or in some cases more than one sentence which requires an individual to use his or her mathematics knowledge to solve the real life scenario given.
When Donovan flips a coin a total of 96 times, the number of times he should expect the coin to land on heads will be:
= 1/2 × 96
= 48
The number of times he should expect the coin to land on tail based on the table will be:
= 18/50 × 75
= 27 times
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3. Birr 6,500 was divided equally among a certain number of persons. Had there been 15 more persons, each would have got Birr 30 less. Find the original number of persons.
Answers
If the population grows by 15 people. The original number of people was 50, hence the new share of each person was Rs. (which is refused).
What is meant by Mathematical equation?The definition of an equation in algebra is a mathematical statement that proves two mathematical expressions are equal. Consider the equation 3x + 5 = 14, where 3x + 5 and 14 are two expressions that are separated by the symbol "equal."
Let no. of persons = x
Each person get
= 6500/x
If there 15 persons were added more
= 6500/(x + 15)
(6500/x) - 6500/(x + 15) = 30
[tex]\Rightarrow 6500\left[\frac{(\mathrm{x}+15)-(\mathrm{x})}{\mathrm{x}(\mathrm{x}+15)}\right]=30[/tex]
[tex]\Rightarrow 6500\left[\frac{x+15-x}{x^2+15 x}\right]=30[/tex]
[tex]& \Rightarrow 1625\left[\frac{15}{x^2+15 x}\right]=3 \\[/tex]
Then,
[tex]& \Rightarrow \frac{24375}{x^2+15 x}=3 \\[/tex]
Simplifying the above equation,
[tex]& \Rightarrow 3 x^2+45 x=24375 \\[/tex]
[tex]& \Rightarrow 3 x^2+45 x-24375=0 \\[/tex]
[tex]& \Rightarrow x^2+15 n-3250=0 \\[/tex]
[tex]& \Rightarrow x^2+(65 n-50 n)-3250=0 \\[/tex]
Then we get,
[tex]& \Rightarrow x^2+65 n-50 n-3250=0 \\[/tex]
[tex]& \Rightarrow x+65)-50(x+65)=0[/tex]
[tex]\Rightarrow(\mathrm{x}-50)(\mathrm{x}+64)=0[/tex]
x = 50, x = -65
Since the number of people cannot be zero,
the number of people is 50.
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What percentage of the class voted for Maria?
Answers
step-by-step explanation:
At a fancy dinner party, the number of courses that will be served determines how many forks will be set out by the hostess.
c = the number of courses served during dinner
f = the number of forks set out by the hostess
Which of the variables is independent and which is dependent?
Answers
Answer:
c is the independent variable and f is the dependent variable
Step-by-step explanation:
At Florida A&M University, the ratio of students from Florida to students not from Florida is about `3\ :\ 1`.
How many of its `12000` students are from Florida?
Answers
Based on the information given, the number of students from Florida will be; 9000 students.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
A mathematical equation is a statement with two equal sides and an equal sign in between. An equation is, for instance, 4 + 6 = 10. Both 4 + 6 and 10 can be seen on the left and right sides of the equal sign, respectively.
We are given that At Florida A&M University, the ratio of students from Florida to students not from Florida is about 3 : 1
The Total number of students = 12000
Fraction from Florida = 3/4
Fraction of students not from Florida = 1/4
Therefore, the students from Florida will be:
= 3/4 × 12000
= 9000
Hence, the number of students from Florida is 9000.
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What are 4 ways to solve a problem?
Answers
The 4 ways to solve a problem are understanding the problem, devising a plan, carrying out the plan, look back.
A problem is any type of disruption from normality that is impeding progress. A problem can be time-consuming and requires too much energy to solve it.
Polya created his famous four-step process for problem-solving, which is used all over to aid people in problem-solving:
Step 1. Understand the problem.
x = a number
Step 2. Devise a plan (translate).
Twice the difference between a number and 1 is equal to 4 more than that number
Step 3. Carry out the plan (solve).
2(x - 1) = (x + 4)
2x - 2 = x + 4
2x - 2 - x = x + 4 - x
x - 2 + 2 = 4 + 2
x = 6
Step 4. Look back (check and interpret).
If we take twice the difference of 6 and 1, that is the same as 4 more than 6. Thus the final answer is 6.
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The graph below models the value of a $20,000 car fyears after it was purchased.
20000
18000
16000
14000
12000-
10000
8000
6000
4000
2000-
Dollars
Value of Car
52:25
2 4 6 8 10 12 14 16 18 t
Years
Which statement best describes why the value of the car is a function of the number of years since it was purchased?
A. Each car value y, is associated with exactly one time, t.
B. Each time, t, is associated with exactly one car value, y.
C.The rate at which car decreases in value is not constant.
Answers
Your answer should be B. Each time, t, is associated with precisely one car value, y.
There are 14 teams in a basketball league.
Each team plays every other team twice
during the season.
a) Suppose that you want to determine
how many games, in total, are played.
Suggest a simpler problem that you
could solve first.
b) How many games, in total, are played
during the season?
Please I need deep explaining.
Answers
Therefore , the solution of the given problem of combination comes out to be there were 182 games played in all.
Definition of combinationCombinations are operations in mathematics that count the number of possible combinations that can be created from a set of items, where the selection's direction is immaterial. You are free to select any arrangement of the parts. Permutations and combinations can be mixed up.
Here,
There still are 14 teams in all, but each team plays the other twice.
Every team plays almost every team twice, hence the total amount of games played is simply the permutations of 2 out of 14, which is determined by:
₁₄P₂
=14!/((14-2)!)
= (14×13×12!)/ 12!
= 14×13
=182
Consequently, there were 182 games played in all.
Therefore , the solution of the given problem of combination comes out to be there were 182 games played in all.
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Rent-A-Car charges $50 to rent a car plus $0.25 per mile. Save-To-Rent charges $25 to rent a car plus $0.41 per
After how many miles will the cost be the same to rent a car?
miles
Answers
After 156.25 miles will the cost be the same to rent a car. This can be solved by the concept of cross-multiplication.
What is cross-multiplication?Fractions are multiplied by multiplying the numerator of the first fraction by the denominator of the second fraction, and the numerator of the second fraction by the denominator of the first fraction.
In order to get the second fraction's numerator, multiply the first fraction's denominator by it. Cross multiply is another name for the butterfly technique of multiplication. The cross-multiply technique is used to find one or more variables in a fraction.
Let, assume miles be m.
and, Let the total cost be x.
According to the question,
0.25m + 50 = x ........(i)
0.41m + 25 = x ........ (ii)
By solving equation (ii) by (i) we get -
0.16 m - 25 = 0
or, 0.16m = 25
or, m = 25 /0.16
or, m = 156.25
Thus, number of miles = 156.25 miles.
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Is Infinity a interval?
Answers
To represent "infinity" or the notion that an interval lacks an ending, we use the symbol.
Is infinity an open interval?The endpoints of infinity and negative infinity can be seen in two different ways. One the one hand, we can never get to infinity because it is a concept and not a real quantity.
In this sense, the interval is said to be open because it excludes the endpoints infinity and negative infinity. The infinity symbol is placed on the right side since there is no higher endpoint (all values must be larger than or equal to -3). An interval is closed if its boxed end is on -3. Because there is no endpoint on that side, infinity always has a curved end.
To represent "infinity" or the notion that an interval lacks an ending, we use the symbol. The symbol shouldn't be used with a square bracket because it's not a number. Visit this tutorial on set-builder and interval notation for a more thorough discussion of set notation and interval notation.
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5 parents and 3 children cost £ 13.20
2 parents and 1 child cost £ 5.10
Find the cost of 1 child
Find the cost of 1 parent
Answers
Answer:
The cost of 1 child is £ 3.40 and the cost of 1 parent is £ 4.10. To determine this, we can use the following equation:
Cost of 1 child = (13.20 - 5.10) / (5 parents + 3 children)
Cost of 1 parent = (13.20 - 5.10) / (2 parents + 1 child)
Hope this helps!
Step-by-step explanation:
What is the simplified form of this expression?
(-3x2 + 2x − 4) + (4x2 + 5x + 9)
7x2 + 7x + 5
x2 + 7x + 5
x2 + 7x + 13
x2 + 11x + 1
Answers
Answer:
x² + 7x + 5
Step-by-step explanation:
(- 3x² + 2x - 4) + (4x² + 5x + 9)
since both parenthesis are being multiplies by 1 , remove parenthesis
= - 3x² + 2x - 4 + 4x² + 5x + 9 ← collect like terms
= (- 3x² + 4x²) + (2x + 5x) + (- 4 + 9)
= x² + 7x + 5
How do you find the terminal point of a line segment?
Answers
To find the terminal point of a line segment, identify the starting point, measure the length and direction, and use these parameters to calculate the coordinates of the terminal point.
To find the terminal point of a line segment, first identify the starting point of the line. Then, measure the length of the line segment and the direction in which it is going. Finally, use a combination of the starting point coordinates, the length, and the direction to calculate the coordinates for the terminal point of the line.
1. Identify the starting point of the line.
2. Measure the length of the line segment and the direction in which it is going.
3. Use the starting point coordinates, the length, and the direction to calculate the coordinates of the terminal point.
4. The coordinates of the terminal point are the location of the line segment’s endpoint.
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The coordinates of 3 of the vertices of a parallelogram are (-3, 4), (-2, 1), and (2, 6). What is the equation for the
line containing the side opposite the side containing the first two vertices? (Remember, opposite sides of a
parallelogram are parallel.)
Answers
The equation for the side of the line that contains the first two vertices' opposite side is 3x+y=12
Given that,
Three of a parallelogram's vertices have the coordinates (-3, 4), (-2, 1), and (2, 6).
We have to find what is the equation for the side of the line that contains the first two vertices' opposite side.
We know that,
The equation of the line is
y=mx+b
Here, we find m and d
Take the points (-3,4) (-2,1)
m = y₂-y₁ / x₂-x₁
m = 1 - 4 / -2-(-3)
m = -3 / 1
m = -3
Now take point (2,6)
6 = -3(2) + b
6 = -6 + b
6 + 6 = b
12 = b
We get m =-3 and b=12
The equation will be y=-3x+12
3x+y=12
Therefore, the equation for the side of the line that contains the first two vertices' opposite side is 3x+y=12
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A line that includes the points (k,
–
4) and (5,
–
2) has a slope of
–
1. What is the value of k?
Answers
The required value of k is 7 for a line that includes the points (k,–4) and (5,–2).
What is the slope of the line?The slope of a line is defined as the gradient of the line. It is denoted by m
Slope m = (y₂ - y₁)/(x₂ -x₁ )
As per the question, the slope of the line is -1, and the given points are (k, -4) and (5, -2).
So we can substitute these values into the formula:
-1 = (-2 - (-4)) / (5 - k)
-1 = 2 / (5 - k)
-1 × (5 - k) = 2
-5 + k = 2
k = 2 + 5
k = 7
Therefore, the value of k is 7.
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HELP PLEASE 10 POINTS!
pls explain
a. 160 is what percentage of 40?
b. 40 is 160% of what number?
c. What number is 40% of 160?
Answers
Answer:
a) 400% b) 1/4 c) 64
Step-by-step explanation:
Answer:
a. 400%
b. 25
c. 64
Step-by-step explanation:
a. 160 ÷ 40/100
160 × 100 / 40
4 × 100 = 400%
b. 40 ÷ 160%
40 ÷ 160/100
40 × 100/160 = 25
c. 40% × 160
40/100 × 160
4 × 16 = 64
What evidence from Gilgamesh a new English version best supports the conclusion that Gilgamesh is an epic hero?
Answers
In "Gilgamesh: A New English Version," several pieces of evidence support the conclusion that Gilgamesh is an epic hero.
Gilgamesh is the king of Uruk: He is the main character of the story, and his actions and deeds are the focus of the epic. As a king, he holds a position of power and authority and is responsible for the well-being of his people.
Gilgamesh undertakes a quest: He embarks on a journey to find immortality after the death of his dear friend Enkidu. His quest takes him to the ends of the earth and shows his determination and courage.
Gilgamesh faces challenges and monsters: During his journey, Gilgamesh faces several challenges and monsters, such as the giant Humbaba and the Bull of Heaven. His bravery in overcoming these obstacles demonstrates his heroic qualities.
Gilgamesh has supernatural strength and abilities: He is two-thirds god and one-third man, which means he is gifted with supernatural strength, speed and endurance. He is able to perform deeds that would be impossible for a mortal.
Gilgamesh demonstrates wisdom and self-awareness: Throughout his journey, Gilgamesh comes to understand the limitations of human life and the importance of living a meaningful existence. He demonstrates wisdom and self-awareness which are the hallmark of a true epic hero.
These examples of Gilgamesh's character and actions support the conclusion that he is an epic hero and his story is an epic.
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What are the roots of the equation x² - 10x +34 = 0 in simplest
a + bi form?
Answers
x = (10 +/- sqrt(10^2 - 4134)) / (2*1)
This simplifies to:
x = (10 +/- sqrt(100 - 136)) / 2
The square root of a negative number is equal to a complex number in the form a + bi, where a and b are real numbers and i is the square root of -1. Therefore, the roots of the equation x^2 - 10x + 34 = 0 are:
x = (-10 + sqrt(-36)) / 2 = (-5 + 3i) / 2 = -2.5 + 1.5i
x = (-10 - sqrt(-36)) / 2 = (-5 - 3i) / 2 = -2.5 - 1.5i
In simplest a + bi form, the roots of the equation are -2.5 + 1.5i and -2.5 - 1.5i.
combine like terms:
Answers
Like terms for the given data are: 17x² and 3x²; -3xy and -2xy. Combining the like terms we get: 20x² and -5xy
What is like terms?In algebra, the term "equal terms" refers to words with the same variable to the same power.
Similar variables and powers are called similar words in algebra. Matching coefficients is not required. When two or more words do not share the same variable or power, they are called inequal terms. The order of both variables is irrelevant unless there is a power. All of these have the same terminology.
17x² and 3x²
-3xy and -2xy
Combine the same terms obtained.
17x² + 3x² = 20x²
-3xy + (-2xy)
= -3xy -2xy
= -5xy
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Suppose you know the answer to 4/5*(20*1 7/8). Explain how the Commutative and Associative Properties of Multiplication can make the computation easier. Then find the answer
Answers
You can rearrange terms in an expression using the commutative and associative qualities to arrange compatible numbers next to one another and in groups.
What distinguishes an associative property from a commutative property?According to the associative property, numbers can be added to or multiplied while still maintaining the same result. According to the commutative property, even if the numbers are added or multiplied, the solution will remain the same.
Explain:
[tex]\frac{4}{5}[/tex] ×(20×17/8)
20×17/8
340÷8
[tex]\frac{4}{5}[/tex]×[tex]\frac{340}{8}[/tex]
= [tex]\frac{64}{2}[/tex]
= 32
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write days in May to days in a year as a ratio.
Answers
Answer:
Step-by-step explanation:
Days in may = 31
Days in a year = 365
therefore, expected ratio = 31/365
The daily recommended allowance of vitamin c for a sixth grader is 45 mg. 1 orange has about 75% of the recommended daily allowence of vitimin c. How many milligrams are in 1 orange?
Answers
Proportionately, the quantity of milligrams of vitamin c in 1 orange is 33.75 mg.
What is proportion?Proportion refers to the part of a whole contained in another value.
Proportion is a ratio representation, which is fractional in value.'
We depict proportions using decimals, fractions, or percentages.
The daily recommended allowance of vitamin c for a sixth grader = 45 mg
The content of 1 orange to the daily recommended allowance = 75%
75% of 45 mg = 33.75 mg.
Thus, 1 orange will proportionately contain 33.75 mg, which is 75% of 45 mg.
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Donna reads 56 chapters of a book in 7 hours. What is her rate in chapters per hour?
Answers
Answer:
To find what her rate for how many chapters she reads per hour, you want to find out what times 7 gives you 56, or you can simply divide:
56/7 = 8
Donna reads 8 pages per hour.
Step-by-step explanation:
Hope it helps! =D
What’s the area of a square is all sides are 6 feet long?
Answers
Answer:
36 sq ft
Step-by-step explanation:
The area of a square is equal to the length of one side squared. Therefore, if all sides of a square are 6 feet long, the area of the square is 6 feet * 6 feet = 36 square feet.
Answer:
36
Step-by-step explanation:
you multiply just two of the sides hxw and you get your answer