As above, let
$$f(x) = 3\cdot\frac{x^4+x^3+x^2+1}{x^2+x-2}.$$Give a polynomial $g(x)$ so that $f(x) + g(x)$ has a horizontal asymptote of $y=0$ as $x$ approaches positive infinity.
Answers
Answer:
Hello,
Step-by-step explanation:
[tex]\dfrac{f(x)}{3} =\dfrac{x^4+x^3+x^2+1}{(x-1)(x+2)} \\\\=\dfrac{(x^2+3)(x-1)(x+2)-3x+7}{(x-1)(x+2)} \\=x^2+3-\dfrac{3x-7}{(x-1)(x+2)} \\\\=x^2+3-\dfrac{3}{x-1} +\dfrac{1}{(x-1)(x-2)} \\\\\dfrac{f(x)}{3}-\dfrac{3x^2+9}{3} =-\dfrac{3}{x-1} +\dfrac{1}{(x-1)(x-2)} \\\\\\ \lim_{x \to +\infty} (\dfrac{f(x)}{3}-\dfrac{3x^2+9}{3} )\\\\=0+0=0\\\\\\P(x)=-x^2-3[/tex]
Answer:
[tex]g(x)=-3x^2-9[/tex]
Explanation:
[tex]3\frac{x^4+x^3+x^2+1}{x^2+x-2}[/tex]
+[tex]\frac{p(x)(x^2+x-2)}{x^2+x-2}[/tex]
We need p(x) need to be a degree 2 polynomial so the numerator of the second fraction is degree 4. Our goal is to cancel the terms of the first fraction's numerator that are of degree 2 or higher.
So let p(x)=ax^2+bx+c.
[tex]3\frac{x^4+x^3+x^2+1}{x^2+x-2}[/tex]
+[tex]\frac{p(x)(x^2+x-2)}{x^2+x-2}[/tex]
Plug in our p:
[tex]3\frac{x^4+x^3+x^2+1}{x^2+x-2}[/tex]
+[tex]\frac{(ax^2+bx+c)(x^2+x-2)}{x^2+x-2}[/tex]
Take a break to multiply the factors of our second fraction's numerator.
Multiply:
[tex](ax^2+bx+c)(x^2+x-2)[/tex]
=[tex]ax^4+ax^3-2ax^2[/tex]
+[tex]bx^3+bx^2-2bx[/tex]
+[tex]cx^2+cx-2c[/tex]
=[tex]ax^4+(a+b)x^3+(-2a+b+c)x^2+(-2b+c)-2c[/tex]
Let's go back to the problem:
[tex]3\frac{x^4+x^3+x^2+1}{x^2+x-2}[/tex]
+[tex]\frac{ax^4+(a+b)x^3+(-2a+b+c)x^2+(-2b+c)x-2c}{x^2+x-2}[/tex]
Let's distribute that 3:
[tex]\frac{3x^4+3x^3+2x^2+3}{x^2+x-2}[/tex]
+[tex]\frac{ax^4+(a+b)x^3+(-2a+b+c)x^2+(-2b+c)x-2c}{x^2+x-2}[/tex
So this forces [tex]a=-3[/tex].
Next we have [tex]a+b=-3[/tex]. Based on previous statement this forces [tex]b=0[/tex].
Next we have [tex]-2a+b+c=-3[/tex]. With [tex]b=0[/tex] and [tex]a=-3[/tex], this gives [tex]6+0+c=-3[/tex].
So [tex]c=-9[tex].
Next we have the x term which we don't care about zeroing out, but we have [tex]-2b+c[/tex] which equals [tex]-2(0)+-9=-9[/tex].
Lastly, [tex]-2c=-2(-9)=18[/tex].
This makes [tex]p(x)=-3x^2-9[/tex].
This implies [tex]g(x)\frac{(-3x^2-9)(x^2+x-2)}{x^2+x-2}[/tex] or simplified [tex]g(x)=-3x^2-9[/tex]
Related Questions
What is an equation of the line that passes through the points (-2,8) and (1, -1)?
Answers
Answer:
y = -3x + 2
Step-by-step explanation:
y2 - y1 / x2 - x1
-1 - 8 / 1 - (-2)
-9/3
= -3
y = -3x + b
-1 = -3(1) + b
-1 = -3 + b
2 = b
lcm of 12x² and 48xy
Answers
Answer:
576xxxycm
Step-by-step explanation:
1cm×12x×x+48xy=576xxxycm
Use the box plots comparing the number of males and number of females attending the latest superhero movie each day for a month to answer the questions.
Two box plots shown. The top one is labeled Males. Minimum at 0, Q1 at 3, median at 10, Q3 at 15, maximum at 35. The bottom box plot is labeled Females. Minimum at 0, Q1 at 2, median at 6, Q3 at 9, maximum at 14, and a point at 31.
Part A: Estimate the IQR for the males' data. (2 points)
Part B: Estimate the difference between the median values of each data set. (2 points)
Part C: Describe the distribution of the data and if the mean or median would be a better measure of center for each. (4 points)
Part D: Provide a possible reason for the outlier in the data set. (2 points)
PLEASE HELP ME!!! IVE BEEN AVOIDING THIS FOR SO LONG
Answers
(a) The IQR for the males' data is 12, because Q3 is at 15 and Q1 is at 3.
(b) The difference between the median values of each data set is 4 because 10-6 = 4
(c) For the males' data, the median would be a better measure of center, since the data is skewed right. For the females' data, the mean would be a better measure of center, because the data is pretty balanced (except for the outlier).
(d) Maybe a superhero movie had their opening night one day of this month, which would explain the outlier.
Calculate the mode of: 4.6, 3, 8.1, 9, 12, 3, 9, 3.5, 7, 3.
Answers
Answer:
the Mode Is 3
Step-by-step explanation:
You Have To Put the numbers from ascending order to descending order..The Numbers that appears the most is the mode
find the difference 25 - (-3) =
Answers
Answer:
28
Step-by-step explanation:
Subtracting a negative is like adding
25 - -3
25+3
28
Answer:
28
Step-by-step explanation: Not sure about this, but technically the question is written wrong. The "difference" is a form of subtraction, and if you simplify your question you get a question to do with addition.
25-(-3)
= 25+3 (since negative and negative makes positive)
=28
PS Again, not sure about this question.
SOMEONE HELL FAST ILL GIVE U BRAINLIST!!!
Answers
Determine three consecutive odd integers whose sum is 2097.
Answers
Answer:
first odd integer=x
second odd integer=x+2
third odd integer=x+4
x+x+2+x+4=2097
x+x+x+2+4=2097
3x+6=2097
3x=2097-6
3x=2091
3x/3=2091/3
x=697
therefore, x=697
x+2=697+2=699
x+4=697+4=701
Simplify the expression. What classification describes the resulting polynomial?
(8x+3x) - (12x2 - 1)
O A quadratic binomial
O B. quadratic trinomial
ОС.
linear monomial
OD.
linear binomial
Answers
Answer:
The correct choice is option B. quadratic trinomial
Step-by-step explanation:
The answer is in quadratic trinomial because a quadratic trinomial is an expression in the form of ax^2 + bx + c and this is exactly how our final solution looks like.
Help anyone can help me do this question,I will mark brainlest.
Answers
Answer:
60
Step-by-step explanation:
SEE THE IMAGE FOR SOLUTION
HOPE IT HELPS
HAVE A GREAT DAY
The perimeter of a triangle is 24 cm.If it’s area is 24cm2 and it’s one of the three sides is 10 cm,find the sides of remaining
Answers
Answer:
The other sides are 6cm and 8cm
Step-by-step explanation:
Let the other 2 sides be x and y.
Using herons formula, we have an equation
24=sqrt(12*2*(12-x)*(12-y))
Since the perimeter is given, we have x+y=24-10=14. Solving it we have x=6,8 and y=8,6.
2/5 x 1/3 x 3/4 = ( 2/5 x 1/3) x ? And the answer please
Answers
3x2/5=6/5
4.2
6/15x5/15=30/15=2
?= 2.1 or 2 1/10
Wyatt wants to study the relationship between the number of siblings that a student has and the student’s grade-point average using a scatterplot. He believes that if the points form a straight line, then there is a relationship between the two variables. Which scatterplot proves that Wyatt’s statement is not always true?
Answers
Answer:
B.
Step-by-step explanation:
A college chemistry teacher wants to know whether taking a practice
exam will help her students score higher on their final exam. She
randomly selects 50 students from among the four sections of
Introductory Chemistry that she teaches. Then she randomly divides
them into two groups of 25 students each. Group A will take a practice
exam, and group B will not take a practice exam. After all 50 students
take their final exam, she will compare their results.
Which of these are the treatments in this experiment?
A. Taking a morning chemistry class and taking an afternoon
chemistry class
B. Having taken high school chemistry and not having taken high
school chemistry
C. Being one of the 50 students selected and not being one of the 50
students selected
D. Taking a practice exam and not taking a practice exam
Answers
the answer is option b
which of the following is NOT true about a parallelogram?
Answers
Answer:
Any two consecutive angles are complementary.
--------------------
Opposite sides of a parallelogram are parallel, Opposite sides are congruent. Opposite angles are congruent.
OAmalOHopeO
Lilian is building a swimming pool in the shape of a right rectangular prism. The area of the base of the swimming pool is 72 square meters. The depth of the swimming pool is 3 meters. What is the volume of the swimming pool?
Answers
Answer:
216
Step-by-step explanation:
Volume of a rectangular prism = area of base * depth
Area of base: 72
Depth: 3
Volume = 72 * 3 = 216
If you add up Jasford and Jasmines ages you get 35. If Jasford is three years older than Jasmine, how old is Jasmine?
Answers
A typical television newscast has three cameras. The center camera directly faces the news anchor’s desk. The other two cameras are both angled 45 degrees away from the center camera. Suppose each camera has a filed if 60 degrees. What is the total angle covered by three cameras? Explain your reasoning.
Answers
7(2x-7+1 please help
Answers
Answer:
14x-42
Step-by-step explanation:
7(2x-7+1 )
7(2x-6)
14x-42
1-2. Roel earns P 8,000 per month as basic salary. In addition, he earned P 5,725.50 as commission and P 852.50 as overtime pay in January. How much did he earn that month?
3-4. A biscuit factory originally packaged 0.65kg of crackers. To increase profitability, the company reduced the weight by 0.035 kg with a more affordable price. What is the new weight per pack of crackers?
Answers
Answer:
a. he earn 14578.00 in that month.
Simplify the expression using trigonometric identities: sec (–θ) – cos θ.
Answers
Answer:
Step-by-step explanation:
Let's break this down. The secant of a negative angle is the same as the secant of the positive angle. This is because secant is the inverse of cosine, and cosine is an even function. f(-x) = f(x) if the function is even, and f(-x) = -f(x) if the function is odd. Sine is odd and has symmetry about the origin; cosine is even and has y-axis symmetry.
Therefore, sec(-θ) = sec(θ) and we have then
sec(θ) - cos(θ). Since secant is the inverse of cosine, we can write:
[tex]\frac{1}{cos\theta}-cos\theta[/tex] and finding a common denominator:
[tex]\frac{1-cos^2\theta}{cos\theta}[/tex] and using a trig identity:
[tex]\frac{sin^2\theta}{cos\theta}[/tex] and simplify that down a bit by breaking it up:
[tex]\frac{sin\theta}{cos\theta}*sin\theta[/tex] finally boils down to
tanθ · sinθ, the last choice there.
simplify 2- 5/8 and -2 3/5 A. -6 33/40 B. 4- 3/8 C. 4 3/8 d. 6 33/40
Answers
Answer:
Step-by-step explanation:
Convert mixed fraction to improper fraction.
[tex]-2\dfrac{5}{8}*(-2\dfrac{3}{5})=\dfrac{-21}{8}*\dfrac{-13}{5}\\[/tex]
[tex]=\dfrac{(-21)*(-13)}{8*5}\\\\\\=\dfrac{273}{40}\\\\=6\dfrac{33}{40}[/tex]
17. ralph is 3 times as old as sarah. in 4 years ralph will only be twice as old as sara will be then. find ralph’s age now.
Answers
Answer:
Sara is now 4, will be 8 in 4 years. Ralph is now 12, will be 16 in 4 years.
simplify. 5+ 8y^2- 12y^2+ 3y
Answers
Answer:
The answer simplified would be:
-4y^2+3y+5
If you combine the like terms, you will get this answer.
You have to add 8y^2 to -12y^2 and you get -4y^2.
Hope this helps you! :)
Geometry, please answer question ASAP
Answers
Answer:
A
Step-by-step explanation:
The inscribed angle IJK is half the measure of its intercepted arc, so
arc IK = 2 × 75° = 150°
The 3 arcs sum to 360° , then
arc IJ = 360° - (150 + 110)° = 360° - 260° = 100° → A
helphelphelphelphelphelp
Answers
Answer:
E - (8,-9)
F - (9,-9)
G - (9,-4)
Step-by-step explanation:
The first number in the bracket relates to the x-axis
The second number relates to the y-axis
Margo bought the following lengths of ribbon: 3/4 yd, 1/3 yd, 1/2 yrd. How much ribbon did she buy?
Answers
Answer:
she bought 19/12 lengths of ribbon or aka 1 7/12
Step-by-step explanation:
3/4 + 1/3 + 1/2 = 13/12 + 1/2
13/12 +1/2 = 19/12
19/12 simplify is 1 7/12
Margo bought [tex]1\frac{2}{3}[/tex] yards of ribbon.
What is a fraction ?A fraction represents how much part of total quantity is present.Fractions are of two types proper fraction and improper fraction.
According to the question given Margo bought some ribbons of length 3/4 yd, 1/3 yd and 1/2 yd.
To find the total ribbon she bought in yards we have to take the sum of all the fractions.
∴ (3/4) + (1/3) + (1/2) yards.
Now to sum these fractions we have to take LCM of the denominators which is 12.
So, { (3×3) + (1×4) + (1×6) }/12.
= (9 + 4 + 6)/12.
= 19/12 or 1 whole 2/3 yards of ribbon.
learn more about fractions here :
https://brainly.com/question/8482939
#SPJ2
What is the price after taking 25% off $62?
Answers
Answer:
46.5 DOLLARS WILL BE YOUR ANSWER .Answer:
$46.50
Step-by-step explanation:
pls mark brainliest
Write the equation of the line that passes through the point (-1, -2) and is parallel to the
graph of y-3x - 2.
XY
Answers
Answer:
y = -3x - 5
Step-by-step explanation:
In order to form a parallel line, the two lines must have the same slope. Substitute the values into the point-slope formula to make the equation.
[tex]y-y_1=m(x-x_1)\\y-(-2)=-3(x-(-1))\\y+2=-3(x+1)\\y+2=-3x-3\\y=-3x-5[/tex]
Therefore, the equation is y = -3x - 5.
can someone solve this?
-5^4 x 4y^10 x -3y^-1
Answers
Answer:
[tex]7500 {y}^{9} [/tex]
Step-by-step explanation:
[tex] { - 5}^{4} \times {4y}^{10} \times { - 3y}^{ - 1} [/tex]
[tex] = 12 \times {5}^{4} {y}^{9} [/tex]
[tex] = 12 \times 625 {y}^{9} [/tex]
[tex] = 7500 {y}^{9} [/tex]
Does anyone know the answer plz ?
Answers
Answer:
x = 5; y = 5
Step-by-step explanation:
One angle is a right angle, so its measure is 90 degrees.
One angle measures 45 degrees, so the unknown angle must also measure 45 degrees.
x and y are opposite two 45-degree angles, so x = y.
In a 45-45-90 triangle, the ratio of the lengths of the sides is
1 : 1 : sqrt(2)
We have
x : y : 5sqrt(2)
x/1 = 5sqrt(2)/sqrt(2)
x = 5
y = 5
