Identify the domain, vertical asymptotes and horizontal asymptotes of the following rational function: f(x)= \frac{3x-4}{x^3-16x} Domain is all real numbers except x\neq Answer , Answer and AnswerVertical asymptote at x= Answer , Answer and AnswerHorizontal asymptote at y= Answer
Answers
Answer
Domain is all real numbers except x ≠ 0, -4, and 4
Vertical asymptote at x = 0, -4, and 4
Explanation
Given function:
[tex]f(x)=\frac{3x-4}{x^3-16x}[/tex]Note: The domain of a function is a set of input or argument values for which the function is real and defined.
For the function to be real; the denominator must not be equal zero, i.e.
[tex]\begin{gathered} x^3-16x\ne0 \\ x(x^2-16)\ne0 \\ x(x-4)(x+4)\ne0 \\ x\ne0,x-4\ne0,\text{ and }x+4\ne0 \\ \therefore x\ne0,x\ne4,\text{ and }x\ne-4 \end{gathered}[/tex]Hence, the domain is all real numbers except x ≠ 0, -4, and 4.
Note: A vertical asymptote with a rational function occurs when there is division by zero.
Hence, the vertical asymptote at x = 0, -4, and 4
Related Questions
“ Judy has a bag with 12 DVD’s, 12 marbles, 11 books, and 1 orange. What is the ratio of books to marbles? What is the ratio of DVD’s to the total number of items in the bag? What percentage of the items in the bag are DVD’s? “
Answers
First, let's calculate the total number of items:
[tex]12+12+11+1=36[/tex]The ratio of books to marbles is calculated by dividing the number of books by the number of marbles:
[tex]ratio=\frac{books}{\text{marbles}}=\frac{11}{12}[/tex]The ratio of DVD's to the total number of items is:
[tex]\text{ratio}=\frac{\text{dvds}}{\text{total}}=\frac{12}{36}=\frac{1}{3}[/tex]The percentage of dvd's from the total is:
[tex]\frac{1}{3}=0.3333=33.33\text{\%}[/tex]The table shows the total cost c for the number of aquarium tickets purchased t. Write an equationthat can be used to find the cost c oft aquarium tickets. Use the equation and complete the table tofind the cost of 7 tickets.7Number of Tickets, tCost, cWrite an equation3$29.2510 12$97.50 $117.00(Use the operation symbols in the math palette as needed. Use integers or decimals for any numbers in the equatioDo not include the $ symbol in your answer.)
Answers
We can model the cost and number of tickets by a linear equation of the form
[tex]c=mt+b[/tex]Where c is the cost, t is the number of tickets.
m is the slope of the equation and b is the y-intercept.
First, let us find the slope which is given by
[tex]m=\frac{c_2-c_1}{t_2-t_1}[/tex]You can take any two pairs of values from the table.
[tex]m=\frac{117-97.50}{12-10}=\frac{19.5}{2}=9.75[/tex]The slope is 9.75 and the equation becomes
[tex]c=9.75t+b[/tex]Now we need to find the y-intercept (b)
Choose any one pair of values from the table and substitute them into the above equation and solve for b.
Let's choose (12, 117)
[tex]\begin{gathered} c=9.75t+b \\ 117=9.75(12)+b \\ 117=117+b \\ b=117-117 \\ b=0 \end{gathered}[/tex]The y-intercept is 0 so the equation is
[tex]c=9.75t[/tex]Now to find the cost of 7 tickets, simply substitute t = 7 into the above equation
[tex]\begin{gathered} c=9.75t \\ c=9.75(7) \\ c=68.25 \end{gathered}[/tex]Therefore, the cost of 7 tickets is $68.25
4. Solve the polynomial.
7x³ + 21x² - 63x = 0
Answers
After solving the given polynomial (7x³ + 21x² - 63x = 0), the value of x are (x = 0) and {x = [(-3 ± 3√5)/2]}
What is a polynomial?An expression that consists of variables, constants, and exponents and is combined using mathematical operations like addition, subtraction, multiplication, and division is referred to as a polynomial (No division operation by a variable). A polynomial is a mathematical expression made up of coefficients and indeterminates that uses only the operations addition, subtraction, multiplication, and powers of positive integers of the variables. x² 4x + 7 is an illustration of a polynomial with a single indeterminate x.So, 7x³ + 21x² - 63x = 0:
Now, solve for x as follows:
7x³ + 21x² - 63x = 07x(x² + 3x - 9) = 0Zero factor principal, if ab = 0, then a = 0 and b = 0.
x = 0 and x² + 3x - 9 = 0Now, x² + 3x - 9 = 0:
x = [(-3 ± 3√5)/2]x = 0Therefore, after solving the given polynomial (7x³ + 21x² - 63x = 0), the value of x are (x = 0) and {x = [(-3 ± 3√5)/2]}
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Question 1. Write the equation of the line that goes through the points (-2,1) and (4,2).
Answers
Slope-intercept equation:
y=mx+b
Where:
m= slope
b=y- intercept
Point 1 = (x1,y1) = (-2,1)
Point 2 = (x2,y2)= (4,2)
First, find the slope by applying the formula:
[tex]m=\text{ }\frac{y2-y1}{x2-x1}=\frac{2-1}{4-(-2)}=\frac{1}{6}[/tex]Now we have:
y=1/6x+b
Replace x,y by a point ( for example point 1 (-2,1)) and solve for b:
1 = 1/6 (-2) +b
1= -1/3 +b
1+1/3 = b
b= 4/3
Final equation:
y= 1/6x+4/3
Let set E be defined as follows:
E = {english, math, french, art}
Which of the following are subsets of set
E
Answers
The subsets of E is all the above .
What are subsets of set ?If every component present in Set A is also present in Set B, then Set A is said to be a subset of Set B. To put it another way, Set B contains Set A. As an illustration, if set A has the elements X, Y, and set B contains the elements X, Y, and Z, then set A is the subset of set B.
If every element in a set A is also an element in a set B, then the set A is a subset of the set B. The set A is therefore contained within the set B. AB is used to represent the subset connection. For instance, if the sets A and B are equal, AB but BB, respectively.
Let the event E = {english, math, french, art}
The subsets of E is all the above .
null set is also subset of E
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Let p = x^2 + 6.Which equation is equivalent to (22 + 6)^2 – 21 = 4x^2 + 24 in terms of p?Choose 1 answer:А) p^2 + 4p - 21 = 0B) p^2 - 4p - 45 = 0C) p^2 - 4p - 21 = 0D) p^2 + 4p - 45 = 0
Answers
Given:
[tex](22+6)^2-21=4x^2+24[/tex][tex]\text{Let p = x}^2+6[/tex]Let's solve the equation in terms of p:
[tex]undefined[/tex]¿Por qué NO puede encontrar el punto medio de una línea?
Answers
Las líneas en un plano cartesiano son infinitas, no tienen un punto de inicio o final, por lo que no es posible determinar un punto medio para ellas.
Parveen wanted to make a temporary shelter for her car, by making a box-like structure with tarpaulin that covers all the four sides and the top of the car (with the front face as a flap which can be rolled up). Assuming that the stitching margins are very small, and therefore negligible, how much tarpaulin would be required to make the shelter of height 2.5 m, with base dimensions 4m×3m ?
Answers
Given:
Length = 4m
Width= 3m
Height = 2.5 m
Therefore, the surface area of rectangle prism is 2lh+2bh+lb
[tex]\begin{gathered} 4\times2.5\times2+3\times2.5\times2+4\times3=10\times2+5\times3+12 \\ =20+15+12 \\ =47 \end{gathered}[/tex]Hence, the required answer is 47m^2.
1. The price for a new iPhone today is $829. In 2010 it was $299 for the new iPhone. What is the percent of change in the price of an iPhone from 2010 to today?
Answers
Given:
a.) The price for a new iPhone today is $829.
b.) In 2010 it was $299 for the new iPhone.
To be able to determine the percent change of price, we will be using the following formula:
[tex]\text{ Percent of Change = }\frac{Price\text{ today - Price in 2010}}{\text{Price in 2010}}\text{ x 100}[/tex]We get,
[tex]\text{ = }\frac{\text{ \$829 - \$299}}{\text{ \$299}}\text{ x 100}[/tex][tex]\text{ = }\frac{\text{ \$530}}{\text{ \$299}}\text{ x 100}[/tex][tex]\text{ = 1.77257525084 x 100 = 1.77 x 100}[/tex][tex]\text{ Percent of Change = 177\% ; an increase}[/tex]Therefore, the percent of change in the price from 2010 and today is an increase of 177%.
Nathalie is finishing a workout on the treadmill. She speeds up before slowing down to a stop. Nathalie draws a graph to represent her workout.
Answers
As the x-axis increases uniformly the y-axis increase and decreases so the horizontal axis must be labeled with time and the vertical axis with speed so option (A) is correct.
What is a graph?A graph is a diametrical representation of any function between the dependent and independent variables.
The graph is easy to understand the behavior of the graph.
The graph of a treadmill workout has been plotted.
We all know that the speed of the treadmill keep fast initially but after some time the speed reduces and it goes to zero lineary.
Therefore,the horizontal axis wich is uniform changes cause to vertical axis with first increase and then decrease shown.
Hence "As the x-axis increases uniformly the y-axis increase and decreases so the horizontal axis must be labeled with time and the vertical axis with speed".
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Determine the prime factorization of 350
Answers
SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Define Prime factorization.
Prime factorization is a way of expressing a number as a product of its prime factors. A prime number is a number that has exactly two factors, 1 and the number itself. Examples of prime numbers are 2,3,5,7...etc.
STEP 2: Find the prime factors of the given number
Prime factorization of any number means to represent that number as a product of prime numbers.
We start by dividing the number by the lowest possible prime numbers.
STEP 3: Express 350 as a product of its prime factors
[tex]\begin{gathered} \text{Prime factors}=2,5,5,7 \\ \text{Product of prime factors=}2\times5\times5\times7 \\ =2\times5^2\times7 \end{gathered}[/tex]Hence, the prime factorization of 350 is given as:
[tex]2\times5^2\times7[/tex]
A window washer drops a tool from their platform 155 ft high. The polynomial -16r2 + 155 tells us the height, in feet, of
the tool / seconds after it was dropped. Find the height, in feet, after t = 1.5 seconds.
Answers
At t = 1.5 sec the tool is at the height of 119 feet.
Given, A window washer drops a tool from their platform 155 ft high.
The polynomial -16r² + 155 tells us the height, in feet, of the tool / seconds after it was dropped.
we are asked to determine the height, in feet, after t = 1.5 seconds.
we know that h(t) = -16r² + 155
hence at t=1.5 sec, height is = ?
⇒ h(1.5) = -16t² + 155
⇒ h(1.5) = -16(1.5)² + 155
⇒ h(1.5) = -16(2.25) + 155
⇒ h(1.5) = -36 + 155
⇒ h(1.5) = 119
at t=1.5 sec the tool is at the height of 119 feet.
Hence we get the height as 119 feet.
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Which interval notation represents a function with a domain of all real numbers greater than or equal to 4?A.) -35 D.) y>0 E.) Y<4
Answers
If the domain is all real numbers greater than or equal to 4, the interval will be
[tex]x\ge4[/tex]Quincy has a jewelry business in which he designs and sells bracelets. His daily profit, Q(x), can be modeled by the function Q(x) = 7.25x − 36.25, where x is the number of bracelets he sells. What is the value of Q(5), and what is its interpretation?
Q(5) = 0; If Quincy sells 0 bracelets, he will earn $5.
Q(5) = 0; If Quincy sells 5 bracelets, he will earn $0.
Q(5) = 5.69; If Quincy sells 5.69 bracelets, he will earn $5.
Q(5) = 5.69; If Quincy sells 5 bracelets, he will earn $5.69.
Answers
The value of Q(5) is zero and represents the zero profit on selling 5 bracelets thus option (B) is correct.
What is a function?A certain kind of relationship called a function binds inputs to essentially one output.
The machine will only accept specified inputs, described as the function's domain, and will potentially produce one output for each input.
As per the given,
Daily profit function Q(x) = 7.25x − 36.25 where x is the number of bracelets he sells.
At x = 5 ( selling 5 bracelets)
Q(5) = 7.25(5) - 36.25
Q(5) = 36.25 - 36.25 = 0
It means selling 5 bracelets doesn't give any profit.
Hence "The value of Q(5) is zero and represents the zero profit on selling 5 bracelets".
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The safe load, L, of a wooden beam of width w, height h, and length l, supported at both ends, varies directly as the product of the width and the square of the height, and inversely as the length. A wooden beam 5 inches wide, 8 inches high, and 216 inches long can hold a load of 7670 pounds. What load would a beam 3 inches wide, 5 inches high, and 240 inches long of the same material, support? Round your answer to the nearest integer if necessary.
Answers
we know that
L=KW(h^2)/l
we have that
W=5 in
h=8 in
l=216 in
L=7670 pounds
step 1
Find the value of K (constant of proportionality)
substitute the given values in the equation
7670=K(5)(8^2)/216
7670=k(1.4815)
k=5,177.25
step 2
we have the equation
L=(5,177.25)W(h^2)/l
for
W=3 in
h=5 in
l=240 in
substitute in the equation and solve for L
L=(5,177.25)(3)(5^2)/240
L=1,617.89 pounds
Round your answer to the nearest integer
so
L=1,618 pounds
Fill in the missing numbers to complete the linear equation that gives the rule for this table.x: 1, 2, 3, 4y: 8, 28, 48, 68Y = ?x + ?
Answers
we have a table that describe the line and we need to finde the slope and the intercept with the y axis, so the slope can be found with this equation:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]So I use the numbers in the table to fill the equation so:
[tex]\begin{gathered} m=\frac{28-8}{2-1} \\ m=\frac{20}{1} \\ m=20 \end{gathered}[/tex]now for the intercept we replace x=0 and use the coordinate (1,8) so:
[tex]20=\frac{y-8}{0-1}[/tex]and we solve for y so:
[tex]\begin{gathered} -20=y-8 \\ -20+8=y \\ -12=y \end{gathered}[/tex]So the equation is:
[tex]y=20x+(-12)[/tex]ave read 14 pages in 28 minutes how much pages can she read for 50 minutes
Answers
Answer:
Step-by-step explanation:
14x2=28
50 divided by 2 = 25 pages
14pages=28mins
page=2mins
so
pages =50/2
=25
Identify each of the following statements as true or false in relation to confidence intervals (CIs).
Answers
Let's analyze each sentence to check if it is true or false:
First:
This sentence is true, the confidence interval is an interval where the true mean is likely to be.
Second:
This sentence is true, with a sample size smaller than 30, it is better to use the t-distribution instead of the normal distribution.
Third:
This sentence is true, the confidence interval is not a 100% guarantee that the true mean will be inside it.
Fourth:
Ti s sentence is true, this theorem states that when getting a large enough sample of a distribution with mean and standard deviation, the sample will be approximately normally distributed.
Fifth:
This sentence is false, because the number of degrees of freedom is 1 less than the sample size, so it would be 10.
Therefore the answer is:
True, True, True, True, False.
Find the rate of change of each linear function 1. y = x - 7
Answers
Rate of change = 1
Explanations:The given linear function is:
y = x - 7
The rate of change of the function is gotten by finding the derivative (dy/dx) of the function
dy/dx = 1
The rate of change = 1
An airplane takes off from an airport that is 144 ft above sea level. The airplane flies at 30,000 ft. To avoid a storm , the airplane goes up to 35,000 ft. Immediately after passing the storm, the airplane returns to its original altitude. Finally , the airplane lands at an airport that is 1,998 ft above sea level . What integer represents the airplanes changes in altitude to avoid the storm ? Immediately after passing the storm ? the integer □ represents the change in altitude to feet to avoid the storm.the integer □ represents the change in altitude in feet immediately after passing the storm.
Answers
What integer represents the airplanes changes in altitude to avoid the storm ?
changes = 35000 - 30000
= 5000 ft
the integer 5000 represents the change in altitude to feet to avoid the storm.
the integer -5000 represents the change in altitude in feet immediately after passing the storm.
Simplify.8(10 m)ANSWER CHOICES:80 m18 m810 m80 + m
Answers
To simplify this, we need to apply distributive property.
Given: 8(10 m)
Expand the parenthesis:
[tex]\begin{gathered} 8\text{ }\ast\text{ 10m} \\ =\text{ 80m} \end{gathered}[/tex]ANSWER:
[tex]80m[/tex]For the function f(x)=3x2−4x−4,a. Calculate the discriminant.b. Determine whether there are 0, 1, or 2 real solutions to f(x)=0.
Answers
Answer:
a) Using the formula for the discriminant we get:
[tex]\begin{gathered} \Delta=(-4)^2-4(3)(-4), \\ \Delta=16+48, \\ \Delta=64. \end{gathered}[/tex]The discriminant is 64.
b) Based on the above result we know that the f(x)=0 has 2 real solutions,
I need help with this questions I don’t. Get it
Answers
You will need 275 ml of the 90% solution
Explanation:Let the amount of the 90% alcohol be x
Amount of the 30% alcohol solution = 385 ml
The amount of the mixture = 385 + x
(30% of 385) + (90% of x) = 55% of (385+x)
[tex]\begin{gathered} (\frac{30}{100}\times385)+(\frac{90}{100}\times x)=\frac{55}{100}\times(385+x) \\ \\ (0.3\times385)+(0.9\times x)=0.55(385+x) \\ \\ 115.5+0.9x=211.75+0.55x \\ \\ 0.9x-0.55x=211.75-115.5 \\ \\ 0.35x=96.25 \\ \\ x=\frac{96.25}{0.35} \\ \\ x=275 \\ \\ \end{gathered}[/tex]You will need 275 ml of the 90% solution
Let f(t) = 3 + 2, g(x) = -x^2?, andhe) = (x - 2)/5. Find the indicated value:24. h (g(5))
Answers
The Solution to Question 24:
Given the function below:
[tex]\begin{gathered} g(x)=-x^2 \\ h(x)=\frac{x-2}{5} \end{gathered}[/tex]We are asked to find the value of h(g(5)).
Step 1:
We shall find g(5) by substituting 5 for x in g(x).
[tex]g(5)=-5^2=-25[/tex]So that:
[tex]h(g(5))=h(-25)[/tex]Similarly, we shall find h(-25) by substituting -25 for x in h(x).
[tex]h(-25)=\frac{-25-2}{5}=\frac{-27}{5}[/tex]Therefore, the correct answer is
[tex]\frac{-27}{5}[/tex]Drag each tile to the correct box. Not all tiles will be used. Arrange the steps to solve the equation x + 3 − 2 x − 1 = - 2 . Simplify to obtain the final radical term on one side of the equation. Raise both sides of the equation to the power of 2. Apply the Zero Product Rule. Use the quadratic formula to find the values of x. Simplify to get a quadratic equation. Raise both sides of the equation to the power of 2 again.
Answers
The value of x = 16 + 4[tex]\sqrt{15}[/tex]
Given,
To solve the equations :
[tex]\sqrt{x+3} - \sqrt{x -1}[/tex] = -2
Solve by the given steps :
Now, According to the question:
Step 1: Simplify to obtain the radical form on one side of the equation:
[tex]\sqrt{x+3} - \sqrt{x -1}[/tex] = -2
Step 2: Raise both sides of the equation to the power of 2
[tex](\sqrt{x+3} - \sqrt{x -1})^2 = (-2)^2[/tex]
x + 3 + 2x - 1 -2 [tex]\sqrt{(x+3)(2x -1)}[/tex] = 4
3x - 2 = 2 [tex]\sqrt{(x+3)(2x -1)}[/tex]
[tex](3x - 2)^2 = [2\sqrt{(x+3)(2x -1)}]^2[/tex]
9[tex]x^{2}[/tex] - 12x + 4 = 4 (2[tex]x^{2}[/tex] + 5x -3)
Step 3: Apply the zero product rule, Simplify to get a quadratic equation :
[tex]x^{2}[/tex] - 32x +16 = 0
Step 4: Use the quadratic formula to find the values of x :
[tex]x^{2}[/tex] - 32x + 16 =0
x = 16 + 4[tex]\sqrt{15}[/tex] and x = 16 - 4[tex]\sqrt{15}[/tex]
x = 16 - 4[tex]\sqrt{15}[/tex] (It is rejected)
So, the value of x = 16 + 4[tex]\sqrt{15}[/tex]
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Answer: Raise both sides of the equation to the power of 2
simplify to obtain the final radical term on one side of the equation
raise both sides of the equation to the power of 2 again
simplify to get a quadratic equation
use the quadratic formula to find the xvalues
Step-by-step explanation:
A small college is forming a planning committee from 10 administrators, 16 faculty members, and 9 staff members. In how many ways can a planning committee be formed if there are 3 members from each group?
Answers
The number of ways that the planning committee can be formed if there are 3 members from each group is 6545 ways.
What are combinations?Combinations are also referred to as selections. Combinations imply the selection of things from a given set of things. In this case, we intend to select the objects. This can be illustrated by ⁿCr
The combination formula is illustrated thus:
ⁿCr = n! / ((n – r)! r!)
n = the number of items.
r = how many items are taken at a time
The number of people will be:
= 10 + 16 + 9
= 35
The number of ways will be:
= ³⁵C₃
= 35! / (35 - 3)! 3!
= 35! / 32! 3!
= 35 × 34 × 33 / 3 × 2
= 6545 ways
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6545 ways can a planning committee be formed if there are 3 members from each group.
What is combination?Combination is a way of selecting items from a collection where the order of selection does not matter.
The formula for combination is ⁿCr = n! / ((n – r)! r!)
Where n is total number of objects and r is number of objects we have to choose.
The committee has 10 administrators, 16 faculty members, and 9 staff members.
The total number of persons
10+16+9
35
Now we need to select 3 persons from 35 persons
n=35 and r=3
³⁵C3 = 35! / ((35 – 3)! 3!)
=35! / ((32)! 3!)
=35×34×33×32! / (32! 3!)
=35×34×33 /6
=6545 ways
Hence in 6545 ways can a planning committee be formed if there are 3 members from each group.
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the Center is (2,0) the circle passes through the point (4.5,0) What is the Radius?
Answers
The radius of the circumference would be
x2 = 4.5
x1 = 2
r = x2 - x1
r = 4.5 - 2.0
r = 2.5
The radius would be 2.5
A car wheel has a radius of 35 cm.(a) What is the circumference of the wheel? Give your answer correct to 2 decimal places.(b) If the wheel rotates 100 000 times, how far does the car travel?
Answers
Explanation
(a) The formula for the circumference of a circle is as follows:
[tex]C=2\pi r[/tex]Where r is the radius of the circle. So, we have:
[tex]X=2\pi r=2\cdot3.1415\ldots\cdot35=219.9114\ldots\approx219.91[/tex]So, the circumference is approximately 219.91 cm.
(b) Assuming the wheel is always in contact and every rotation make sthe exact same length of travel, every rotation will make the car travel approximately 219.91 cm.
If the wheel rotates 100,000 times, the car will travel 100,000 times as many, so it will travel:
[tex]100,000\cdot219.91=21,991,000[/tex]So, the car will travel approximately 21,991,000 cm which is equivalente to 219.91 km.
Answer
(a) the circumference is approximately 219.91 cm
(b) the car will travel approximately 21,991,000 cm or 219,91 km.
we know,
circumference of the wheel= 2 pi r
=2*22/7*35
= 220cm (it covers 220 cm in one rotation)
Now if the wheel rotates 100000 times,
it will travel = 220* 1000000
= 220000000 cm
= 220000000/100m
=2200000m
Answer the questions below about the quadratic function.g(x) = 3x² + 12x+8Does the function have a minimum or maximum value?MinimumMaximumWhere does the minimum or maximum value occur?x =0What is the function's minimum or maximum value?
Answers
Plot the function on the graph.
From the graph it can be observed that graph of function opening upwards and it has minimum value at x = -2.
Thus function has minimum value.
The minimum value of the function occurs at x = -2. So mimimum value of function occurs at x = -2.
The value of the function at x = -2 is -4. So function's minimum value is -4.
Please help this is due tomorrow!!
Answers
The expression 2x⁷· y⁴ would be equivalent to the given polynomial expression.
What is a polynomial?A polynomial is defined as a mathematical expression that has a minimum of two terms containing variables or numbers. A polynomial can have more than one term.
The given polynomial expression below is:
⇒ 10x⁵y⁷/5x⁵y · 3x⁴y⁸/3x⁻³y¹⁰
Apply the division operation in the constant terms
⇒ 2x⁵y⁷/x⁵y · x⁴y⁸/x⁻³y¹⁰
Apply the arithmetic operation in the Exponents of the same base variables
⇒ 2y⁶ · x⁷y⁻²
⇒ 2y⁶⁻² · x⁷
⇒ 2y⁴ · x⁷
⇒ 2x⁷· y⁴
Therefore, the expression 2x⁷· y⁴ would be equivalent to the given polynomial expression.
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