The coefficient of the [tex]\(x^4\) term in \((x + 3)^{12}\)[/tex] is 495, calculated using binomial coefficients.
To find the coefficient of the [tex]\(x^4\)[/tex] term in the expansion of [tex]\((x + 3)^{12}\)[/tex], you can use the binomial theorem. According to the binomial theorem, the expansion of [tex]v[/tex] is given by:
[tex]\[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\][/tex]
Where [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient, equal to [tex]\(n\) choose \(k\)[/tex], which is defined as:
[tex]\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\][/tex]
In this case, [tex]\(n = 12\) and \(y = 3\).[/tex] We're interested in the term where the exponent of [tex]\(x\) is 4, so \(n - k = 4\) or \(k = 12 - 4 = 8\).[/tex]Thus, we need to find the coefficient when [tex]\(k = 8\)[/tex]. So, the coefficient of the [tex]\(x^4\)[/tex] term is:
[tex]\[\binom{12}{8} = \frac{12!}{8!(12-8)!}\][/tex]
Calculating this:
[tex]\[\binom{12}{8} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = \frac{11880}{24} = 495\][/tex]
So, the coefficient of the [tex]\(x^4\)[/tex] term in the expansion of [tex]\((x + 3)^{12}\)[/tex] is 495.
Simplify the following expression. (3 + 8 x ) + 7 x
3 + 15
x 18
x 18 + x
11 x + 7
The simplified expression of 4 +9 "/" 3 "/" 2 x 3 + (3 + 1) x 4 is 17, following the order of operations or PEMDAS.
To simplify the given expression, follow the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
The original expression is 4 +9 / 3 / 2 x 3 + (3 + 1) x 4. First, address operations inside parentheses: (3+1) equals 4. The expression becomes 4 +9 / 3 / 2 x 3 + 4 x 4.
Next, handle the multiplication and division from left to right: 9 / 3 equals 3, and 2 x 3 equals 6. Then, 4 x 4 equals 16. This turns our expression into 4 + 3 - 6 + 16.
Finally, perform addition and subtraction from left to right: 4 + 3 equals 7, 7 - 6 equals 1, and 1 + 16 equals 17. Therefore, the simplified expression is 17.
What is an equation of a line, in point-slope form, that passes through (1, −7) and has a slope of −2/3 ?
Calc find the maximum of f(x, y) = y2 + xy â x2 on the square 0 ⤠x ⤠6, 0 ⤠y ⤠6.
To find the maximum of the function f(x, y) = y² + xy – x² on the square 0 ≤ x ≤ 6, 0 ≤ y ≤ 6, we need to find the critical points by taking the partial derivatives with respect to x and y.
To find the maximum of the function f(x, y), we need to find the critical points by taking the partial derivatives with respect to x and y:
[tex]f_x[/tex] = y - 2x
[tex]f_y[/tex] = 2y + x
Setting these derivatives equal to zero and solving for x and y, we get the critical point (2,2). To determine if this point is a maximum, minimum, or point of inflection, we can use the second derivative test. Taking the second partial derivatives:
[tex]f_{xx[/tex] = -2, [tex]f_{yy[/tex] = 2, [tex]f_{xy[/tex] = 1
Using the discriminant D = [tex]f_{xx} \times f_{yy} - {(f_{xy})}^2[/tex], we have D = [tex](-2) \times 2 - {1}^2[/tex] = -4. Since D is negative, the critical point (2,2) is a saddle point, which means there is no maximum or minimum on the given square.
The yard behind the Cindy’s house is rectangular in shape
and has a perimeter of 72 feet. If the length of the yard
is 18 feet longer than the width w of the yard, what is the
area of the yard, in square feet?
You decide to treat yourself to some of your favorite cookies. The box that you bought has 24 cookies. You eat 2 cookies each day as a snack. Which equation shows the number of cookies that you have left at the end of each day?
Answer:
-2X+24
Step-by-step explanation:
Answer:
[tex]y=-2x+24[/tex]
Step-by-step explanation:
Let
x------> the number of days
y----->the number of cookies which are left at the end of each day
we know that
[tex]y=-2x+24[/tex]
verify
For [tex]x=0\ days[/tex]
[tex]y=-2(0)+24=24\ cookies[/tex] -----> original amount of cookies
For [tex]x=1\ days[/tex]
[tex]y=-2(1)+24=22\ cookies[/tex]
For [tex]x=12\ days[/tex]
[tex]y=-2(12)+24=0\ cookies[/tex]
What reason explains why (-6,8) must also be a point on the graph?
What is 0.306 in word form?
Answer:
Three hundred six thousandths.
Step-by-step explanation:
We are asked to write 0.306 in word form.
We can see that our given number has 3 tenths, 0 hundredths and 6 thousandths.
To convert our given number in word form, we will write the number of thousandths in 0.306.
The number of thousandths for our given number are three hundred six thousandths, therefore, our number is three hundred six thousandths.
what is 139.47 rounded to the nearest tenth?
Ned took a test with 25 questions. He lost 4 points for each of the 6 questions he got wrong and earned an additional 10 points for answering a bonus question correctly. How many points did Ned recieve or lose overall?
What Is two thirds of a straight angle
Two thirds of a straight angle is calculated as 120 degrees.
Two thirds of a straight angle can be calculated by first understanding that a straight angle is equal to 180 degrees. When we want to find a fraction of a given angle, we multiply the fraction by the total degree measure of that angle.
In this case, to find two thirds of a straight angle, we multiply 180 degrees by 2/3.
Here is a step-by-step explanation of the calculation:
Understand that a straight angle equals 180 degrees.
Multiply 180 degrees by the fraction 2/3:
(180 degrees) * (2/3) = 120 degrees.
The owner of a candy store wants to mix some peanuts worth $3 per pound, some cashews worth $10 per pound, and some Brazil nuts worth $9 per pound to get 50 pounds of a mixture that will sell for $6.80 per pound. She uses 10 fewer pounds of cashews than peanuts. How many pounds of each did she use?
what is the graph of the equation 4x-5y=15
The correct answer I think is
translate to an algebraic expression 49 more than t
the graph of y=x^2 is shifted 4 units to the right, vertically stretched by a factor of 6, reflected over the x axis and shifted 7 units downward
Answer:y=-6Vx-7
Step-by-step explanation:
A certain vibrating system satisfies the equation u′′ + γu′ + u = 0. find the value of the damping coefficient γ for which the quasi period of the damped motion is 50% greater than the period of the corresponding undamped motion.
A family pays $45 each month for cable television. The cost increases 7%.
a. How many dollars is the increase?
b. What is the new monthly cost?SHO
W YOUR WORK
write division problem that will have a 2-digit quotient and another division problem that will have a 3-digit quotient. Explain how you chose the divisor and dividends.
A campground has cabins that can each hold 28 campers. There are 148 campers that are visiting the campground. How many cabins are full if there are 28 campers in each cabin? Please help i'm desperate and i'm marking most brainliest.
To find the number of full cabins, divide the total number of campers by the number of campers in each cabin. Round the answer up to the nearest whole number.
Explanation:To find the number of full cabins, you can divide the total number of campers by the number of campers in each cabin.
148 campers / 28 campers per cabin = 5.2857 cabins
Since we can't have fractional cabins, we round up to the nearest whole number, which is 6.
Therefore, there are that are full.
Final answer:
There are 6 cabins that are full.
Explanation:
To find the number of cabins that are full, we need to divide the total number of campers by the number of campers each cabin can hold. In this case, there are 148 campers and each cabin can hold 28 campers.
We can solve this problem by dividing 148 by 28: 148 ÷ 28 = 5.2857.
Since we cannot have a fraction of a cabin, we round up to the nearest whole number. Therefore, there are 6 cabins that are full.
Simplify (25x2 − 14xy + 4) − (7x2 − 9y) + (2xy + 7)
Answer:
18x2 − 12xy + 9y + 11
Step-by-step explanation:
complete the solution of the equation. find the value of y when x equals 15.
-x+2y=-1
Will give brainliest.
The table below represents the displacement of a bird from its nest as a function of time: Time (hours) x Displacement from nest (miles) y 0 12 1 20 2 28 3 36 4 44 Part A: What is the y-intercept of the function, and what does this tell you about the bird? (4 points) Part B: Calculate the average rate of change of the function represented by the table between x = 1 to x = 3 hours, and tell what the average rate represents. (4 points) Part C: What would be the domain of the function if the bird continued to fly at this rate until it traveled 172 miles from the nest? (2 points)
What is the sign of-5×-1/-2
A basket of laundry is being separated. Divide 48 pieces into 2 clothing groups so the ratio is 1 to 3
• capital one is advertising a 60-month, 5.99% apr motorcycle loan. if you need to borrow $8000 to purchase your dream harley davidson, what will your monthly payment be?
Answer:
The monthly payment is $ 154.29
Step-by-step explanation:
Given : time period = 60 months
Annual interest rate = 5.99%
Amount to be borrowed = $8000
We have to calculate the monthly payment.
We know ,
[tex]P=\dfrac{A\cdot \frac{i}{12}}{1-(1+\frac{i}{12})^{-n}}[/tex]
We have,
interest rate = 5.99% = 0.0599
Amount = $8000
time period = 60 months
P is monthly payment.
[tex]P=\dfrac{8000\left(\frac{0.059}{12}\right)}{1-\left(1+\frac{0.059}{12}\right)^{\left(-60\right)}}[/tex]
Simplify, we get,
P = 154.29
Thus, the monthly payment is $ 154.29
You order an entree for $12. You pay 0.78 in taxes. What is the tax rate
Which are factors of x2 – 4x – 5? Check all that apply. (x – 5) (x – 4) (x – 2) (x + 1) (x + 5)
Describe how the graph of y= x2 can be transformed to the graph of the given equation. y = x2 - 20
in triangle ABC, L is the midpoint of BC, M is the midpoint of AB and N is the midpoint of AC. If MN=8, ML 5 and NL=6, the perimeter of BMNC is?
I really need help with this question!
how do u find the initial value on a graph