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
26)
A pile of bricks has 93 bricks in the first row, 89 bricks in the second row, 85 bricks in the third row, and so on.
How many bricks are there in the 12th row?
OA) 49 bricks
OB) 69 bricks
Answers
Answer:
49 bricks
Step-by-step explanation:
because it's getting smaller my 4
Explanation: it’s getting smaller by 4, they already gave us the first three, so now we multiply 4 by 9 and get 36 ( I multiplied 4 by 9 since that’s how many numbers were left to get to 12 from 3) and subtracted 36 by 85 then got 49. Hope this helps ^_^
Instructions: Find the missing side lengths. Leave your answers as radicals in simplest
form.
Answers
Answer:
x = 40
y= 20
Step-by-step explanation:
Since this is a right triangle, we can use trig functions
sin theta = opp / hyp
sin 60 = 20 sqrt(3)/ x
x sin 60 = 20 sqrt(3)
x = 20 sqrt(3)/ sin 60
x = 20 sqrt(3)/ sqrt(3)/2
x = 20 *2
x = 40
tan theta = opp /adj
tan 60 = 20 sqrt(3)/y
y = 20 sqrt(3)/ tan 60
y = 20 sqrt(3) / sqrt(3)
y = 20
Does (x, y) satisfy an equation?
Answers
Answer:
Yeah
Step-by-step explanation:
What is the area of polygon
Answers
You can split it into two sections the rectangle and triangle.
The rectangle is 5 units by 6 units and since the area of a rectangle is b*h; it is 5*6=30 units squared
The triangle isn’t a right triangle but you can still find the base and height, the base is 6 units and the height is 3 units. The area of a triangle is b*h/2; so it is 6*3/2= 9 units squared
Add the two areas together: 9+30= 39 untis squared
Now graph the points C(2, 8) and D(8, 7). (Try entering the coordinates through the input window.) Measure the lengths of and . Do the points C and D lie on the circle? How do you know? Take a screenshot showing the points and their distances from the center, and paste it below.
Answers
Answer:
this was from plato
Step-by-step explanation:
Answer:
Any point on the circle must be at a distance from the center equal to the length of the circle’s radius. In this example, the radius is 3.61 units, AC = 4.24, and AD = 3.61. Point D, which lies on the circle, has the same distance from the center as the length of the radius. Point C, which lies outside the circle, is at a different distance from the center than the length of the radius.
need help with this!!
Answers
Answer:
{ 1,3,4,6,7}
Step-by-step explanation:
Do B∩C first
This is B intersect C which means what they have in common
B∩C = {3,6,7}
Then A∪(B∩C)
A union {3,6,7} which means join together ( combine with no duplicates) the two sets
{ 1,3,4,6,7}
Find sin 0
A. 16/20
B. 12/16
C. 12/20
D. 16/12
Answers
Answer:
16/20
Step-by-step explanation:
Since this is a right triangle
sin theta = opp side / hypotenuse
sin theta = 16/20
Answer:
A.
[tex]{ \tt{ \sin( \theta) = \frac{opposite}{hypotenuse} }} \\ \\ { \tt{ \sin( \theta) = \frac{16}{20} }}[/tex]
“The length of a rectangle is two feet greater than twice its width. If the perimeter is 25 feet, find the width.” Which of the following translations is correct?
Answers
Answer:
b is the answer
Step-by-step explanation:
convert into power notation -1/81
Answers
Answer:
-1/9^2 is the power notation for your questions
Step-by-step explanations
please answer all the questions above.
Answers
Answer:
hope it was helpful!! You are welcome to ask any question
Answer:
Step-by-step explanation:
1) 7 - ( 3+4) = 7 + [- (3+ 4)] = 7 + (-3) + (-4)
(-) is distributed to 3 and 4
A
2) C
I (-3) - 4 I
3) 5 + 3 = 8 miles
B
Identify the equation of the circle that has its center at (9, 12) and passes through the origin.
Answers
Answer:
(x-9)^2 +(y-12)^2 = 225
Step-by-step explanation:
First find the length of the radius
If is the distance from the center to the point
d = sqrt( (x2-x1)^2 + (y2-y1)^2 )
= sqrt( (0-9)^2 + (0-12)^2)
= sqrt ( 81+144)
= sqrt(225)
= 15
The equation for a circle is
( x-h) ^2+ (y-k)^2 = r^2
(x-9)^2 +(y-12)^2 = 15^2
(x-9)^2 +(y-12)^2 = 225
PLS ANSWER YA'LL!
A house owner wishes to cover their roof with solar panels. Each solar panel is of the following size.
->The roof section is rectangular 7.2 m by 4.4 m.
->Each solar panel measures 1400 mm by 850 mm.
There needs to be at least a 30 cm gap left at each edge of the roof section. The house owner thinks he can have at least 15 solar panels fitted to the roof section.
Can u agree? If yes, show the working. (Hint: He can rotate the solar panel)
Answers
Answer:
Yes
Step-by-step explanation:
The roof is Rectangular :
Dimension of roof section : 7.2 m by 4.4 m
Area of roof = length * width
Area of roof = 7.2 * 4.4 = 31.68 m²
Dimension of solar panels = 1400 mm by 850 mm
Converting to m:
1000mm = 1 m
1400/1000 by 850/1000 = 1.4 m by 0.85 m
Area each solar panel = 1.4 * 0.85 = 1.19 m²
Gap left at each edge :
30cm gap about the 4 edges gives a square with side length 30cm:
30cm = 30/100 = 0.3m
Area of gap left = 0.3² = 0.09 m²
Total area that can be covered = (31.68 - 0.09) m² = 31.59 m²
Maximum number of panels that can be placed on roof section :
Total area that can be covered / area of panel
31.59 m² / 1.19 m²
= 26.54 panels
A circle has a radius of 3. An arc in this circle has a central angle of 20°. What is the length of the arc? Either enter an exact answer in terms of or use 3.14 for 1 and enter your answer as a decimal.
Answers
Answer:
[tex]\frac{\pi }{3}[/tex]
[tex]2 * \pi * 3 * \frac{20}{360}[/tex]
120[tex]\pi[/tex]/360 = [tex]\frac{\pi }{3}[/tex]
Step-by-step explanation:
What action represents the "!" symbol on a TI 84 plus CE calculator for a Statistics problem??
Answers
Answer:
The "!" represents a factorial
Step-by-step explanation:
In math, a factorial when there is an ! followed by a number.
For example, 5! = 5 factorial.
To solve factorials, multiply every integer below the number before the factorial, including the number.
5! = 5 x 4 x 3 x 2 x 1 = 120.
This table shows the relationship of the total number of pieces of fruit to the number of bananas.
Why is StartFraction 6 Over 5 EndFraction not equivalent to Three-halves?
Answers
Given:
The table that shows the relationship of the total number of pieces of fruit to the number of bananas.
To find:
Why is [tex]\dfrac{6}{5}[/tex] not equivalent to [tex]\dfrac{3}{2}[/tex].
Solution:
If a, b, c are real numbers, then
[tex]\dfrac{a}{b}=\dfrac{a\times c}{b\times c}[/tex]
The given fraction is [tex]\dfrac{3}{2}[/tex]. It can be written as:
[tex]\dfrac{3\times 2}{2\times 2}=\dfrac{6}{4}[/tex]
The number 3 is multiplied by 2 to get 6. So, the 2 should also be multiplied by 2. The ratio should be [tex]\dfrac{6}{4}[/tex], not [tex]\dfrac{6}{5}[/tex].
Therefore, the correct option is A.
Which expressions are equivalent to the given expression?
Answers
Answer:
The answers are option B and E.
For what value of k are the roots of the quadratic
equation kx²+ 4x+ 1=0 equals and reals."
Answers
Answer:
k ≥ 4
Step-by-step explanation:
A Quadratic equation is given to us and we need to find out the value of k for which the equation has real roots. The given equation is ,
[tex]\rm\implies kx^2 +4x +1=0[/tex]
With respect to Standard form of Quadratic equation :-
[tex]\rm\implies ax^+bx+c=0[/tex]
For real roots ,
[tex]\rm\implies Discriminant = b^2-4ac\geq 0[/tex]
Substitute the respective values ,
[tex]\rm\implies b^2-4ac \geq 0\\[/tex]
[tex]\rm\implies 4^2 - 4(k)(1) \geq 0 \\[/tex]
Simplify the LHS ,
[tex]\rm\implies 16 - 4k \geq 0 \\[/tex]
Add 4k both sides ,
[tex]\rm\implies 4k\geq 16 [/tex]
Divide both sides by 4 ,
[tex]\rm\implies \boxed{\blue{\rm k \geq 4}}[/tex]
Which of the following is a characteristic of science?
Answers
Answer:d
Step-by-step explanation:
Answer:
D
explanation:
i got it right on the test
The median house price in Waterloo Region increased by 3.6% from Jan 1, 2018 to Jan 1, 2019. A home
was purchased in Waterloo Region on April 1, 2019 for $600,000.
(a) Assume this trend continues, write an exponential equation that models the Resale Value of this
home over time.
(b) At this rate, determine the date of the resale price of the home would reach $1 million (Show your
work to accurate to the nearest month)
(c) Use your exponential equation to determine the expected resale value of the home on April 1, 2020.
Answers
Answer:
The right answer is:
(a) [tex]P(t) = P_o \ e^{0.03536t}[/tex]
(b) [tex]t = 14 \ years \ 6 \ months[/tex]
(c) [tex]P(t) = =621,595.6[/tex] ($)
Step-by-step explanation:
Given:
House price increment rate,
= 3.6% annually
(a)
Let the exponential equation will be:
⇒ [tex]P(t) = P_o e^{Kt}[/tex]
here,
t = 0
P = P₀
t = 1 yr
then,
[tex]P(1) = P_o +3.6 \ persent \ P_o[/tex]
[tex]=1.036 \ P_o[/tex]
now,
⇒ [tex]1.036 P_o = P_o \ e^{K.1}[/tex]
[tex]ln(1.036) = K[/tex]
[tex]K = 0.03536[/tex]
Thus, the exponential equation will be "[tex]P(t) = P_o \ e^{0.03536t}[/tex]".
(b)
We know,
[tex]P_o = 600,000[/tex] ($)
[tex]P(t) = 10,00,000[/tex] ($)
∵ [tex]P(t) = P_o \ e^{0.03536t}[/tex]
[tex]1000000=600000 \ e^{0.03536 t}[/tex]
[tex]\frac{5}{3}= e^{0.03536 t}[/tex]
[tex]ln(\frac{5}{3} )=0.03536 t[/tex]
[tex]\frac{\frac{0.5}{0825} }{0.03536} =t[/tex]
[tex]t = 14.45 \ years[/tex]
or,
[tex]t = 14 \ years \ 6 \ months[/tex]
(c)
[tex]P_o=600,000[/tex] ($)
[tex]t = 1 year[/tex]
Now,
⇒ [tex]P(t) = P_o \ e^{0.03536 t}[/tex]
[tex]=600000 \ e^{ 0.03536\times 1}[/tex]
[tex]=621,595.6[/tex] ($)
Find the equation of the line that passes through (1,6) and (3,10).
y=2x+4
y=-2x+4
y=-24-4
Oy-23-4
Answers
Answer:
m = (y2 - y1) / (x2 - x1) = (10 - 6) / (3 - 1) = 2 slope of line
y = m x + b standard form of straight line
y = 2 x + b where b is the intercept at x = 0
b = y - 2 x
b = 6 - 2 = 4 from our first equation
Also, b = 10 - 6 = 4 check from second equation
y = 2 x + 4 is our equation
Check if y = 6 and x = 1 then
y = 2 * 1 + 4 = 6 verifying the first point given
An acute angle of a right triangle measures 30°, and the length of the triangle's hypotenuse is 10 ft. Find the missing angle measure and side lengths.
Answers
Answer:
missing angle <60 and side lengths 5, 5[tex]\sqrt{3}[/tex]
Step-by-step explanation:
we understand from the given information that the triangle in question is a special right triangle
and since this is a special triangle the side lengths follows :
the side length that sees <90 is represented by 2x
the side length that sees <60 is represented by x[tex]\sqrt{3}[/tex]
and the side length that sees <30 is represented by x
2x = 10 so x = 5 and x[tex]\sqrt{3}[/tex] = 5[tex]\sqrt{3}[/tex]
What is the product of the polynomials below?
(5x2 - x-3)(2x+6)
Answers
[tex]\bf \rightarrow \:(5 {x}^{2} - x - 3) \: \: (2x + 6) \\ \\ \bf \small \rightarrow \:10 {x}^{3} - 2 {x}^{2} - 6x + 30 {x}^{2} - 6x - 18 \\ \\ \bf \rightarrow \:10 {x}^{3} + 28 {x}^{2} - 12x - 18[/tex]
༆ Option D is the correct answer༆
A running track has two semi-circular ends with radius 27m and two straights of length 90.6m as shown.
Calculate the total distance around the track rounded to 1 DP.
Answers
Answer:
350.85
Step-by-step explanation:
90.6*2 = 181.2
27*2 = 54 (diameter)
54*pi = 169.646003294
169.646003294 + 181.2 = 350.846003294
350.846003294 = 350.85
please explain this to me.
Answers
Answer:
Equation of line:- y=2x-7
slope(m)=2
slope of parallel line (m)=2
∴ Equation of parallel line:- y=2x+b
it passes through the point (-3,6)
6=2(-3)+b 6+6=b
b=12
∴ y=2x+12
OAmalOHopeO
Which expression is equivalent to ((2x²) (3x) (4x)??
A.24x7
B.48x?
C.9677
D.576x12
Answers
Answer:
24x^4
Step-by-step explanation:
((2x²) (3x) (4x)
Add the exponents when multiplying
2*3*4 x^(2+1+1)
24x^4
Banks and other financial institutions offer incentives for people to keep their money in a savings account.
True or False?
Answers
it's true because it gives interset or compensation amount to the individuals or organizations which motivates people to save their money in a saving account
Bob placed a 17- foot ramp against the side of a house so the ramp rested on a ledge that is 8 feet above the ground. How far was the base of the ramp from the house?
Answers
Hope this helps! Please make me the brainliest, it’s not necessary but appreciated, I put a lot of effort and research into my answers. Have a good day, stay safe and stay healthy.
What could be the coefficient of x once the variable term is isolated on one side of the equation? Check all that apply.
3x - 6(5x + 3) = 9x + 6
1. Distribute: 3x - 30x - 18 = 9x + 6
2. Combine like terms: -27x - 18 = 9x + 6
–36
–27
–24
24
27
36
Answers
Answer:
-36,36
Step-by-step explanation:
3x - 6(5x + 3) = 9x + 6
Distribute:
3x - 30x - 18 = 9x + 6
Combine like terms:
-27x - 18 = 9x + 6
There are two possible ways to isolate x
On the left
Subtract 9x from each side
-27x - 18 -9x = 9x-9x + 6
-36x -18 = 6
On the right
Add 27x from each side
-27x - 18 +27x = 9x+27x + 6
-18 =36x+ 6
You are riding your bike. At 8:00am you have ridden your bike 23 miles. By 9:00pm you have ridden 179 miles. Find the rate of change in miles per hour. If needed, round your answer to the nearest whole number.
Answers
Answer:
12 miles/hour
Step-by-step explanation:
8am to 9pm = 13 hours
in that time we were driving 179-23 = 156 miles.
so, our speed was 156 miles / 13 hours.
now simplify it to our standard miles/hour format :
156/13 = 12
therefore, or standardized speed was
12 miles/hour
Graph the inequality.
1
Answers
Answer:
hope it helps
Step-by-step explanation:
