Expand the following using either the Binomial Theorem or Pascal’s Triangle. You must show your work for credit. (x - 5)5

Answers

Answer 1

well, using the binomial theorem, is far simpler, since you get the coefficient as you go and only the ones you need.

[tex]\bf (x-5)^5\implies \begin{array}{llll} term&coefficient&value\\ -----&-----&-----\\ 1&+1&(x)^5(-5)^0\\ 2&+5&(x)^4(-5)^1\\ 3&+10&(x)^3(-5)^2\\ 4&+10&(x)^2(-5)^3\\ 5&+5&(x)^1(-5)^4\\ 6&+1&(x)^0(-5)^5 \end{array}[/tex]

notice, with the binomial theorem, you start the first element with the highest exponent, in this case 5, and the second element with 0, and then you gradually decrease it by 1 to the first one and increase it by 1 to the second one.

That part is rather simple, now, to get the coefficient, the 1st coefficient is 1 for one, to get the next one is "the product of the current coefficient and the exponent of the first element, divided by the exponent of the second element on the next term"...which is a mouthful...but anyway ... for example

how did we get 10 for the 3rd term? 5 * 4 / 2

how did we get 10 again for the 4th term though? 10 * 3 / 3

and the 5th coefficient? 10 * 2 / 4

and so on.

so, just combine those, and that's the expansion for that binomial.

Answer 2

Answer:

The expansion is [tex](x-5)^5= x^5-25x^4+250x^3-1250x^2+3125 x-3125[/tex]

Step-by-step explanation:

Given : Expression [tex](x-5)^5[/tex]

To find : Expand the following using either the Binomial Theorem or Pascal’s Triangle?

Solution :

Applying binomial theorem,

Defined as [tex](p+q)^n=\sum_{r=0}^{n} ^nC_rp^{n-r} q^r[/tex]

Where, [tex]^nC_r=\frac{n!}{(n-r)!r!}[/tex]

Now, expanding [tex](x-5)^5[/tex] by binomial formula,

We have,

[tex](x-5)^5=\sum_{r=0}^{5} ^5C_rx^{5-r} (-5)^r[/tex]

Open the summation,

[tex]\Rightarrow ^5C_0 x^{5-0} (-5)^0 + ^5C_1 x^{5-1} (-5)^1 + ^5C_2 x^{5-2} (-5)^2 + ^5C_3 x^{5-3} (-5)^3 \\+ ^5C_4x^{5-4} (-5)^4 + ^5C_5x^{5-5} (-5)^5[/tex]

[tex]\Rightarrow (1)(x^5)(1)+(5)(x^4)(-5)+(10)(x^3)(25)+(10)(x^2)(-125)+(5)(x^1)(625)+(1)(1)(-3125)[/tex]

[tex]\Rightarrow x^5-25x^4+250x^3-1250x^2+3125 x-3125[/tex]

Therefore, The expansion is [tex](x-5)^5= x^5-25x^4+250x^3-1250x^2+3125 x-3125[/tex]

Related Questions

Janice deposited $750 in a savings account that earns 3.5% simple interest. How much interest has Janice earned by the end of the first year?

A.$776.25
B.$214.29
C.$2,625.00
D.$26.25

Answers

Janice deposited $750 and a savings account earns 3.5% simple interest.
r = 3.5% = 0.035
I = $750 * 0.035
I = $26.25
Answer:
Janice has earned
D ) $26.25

If 1 cm = 0.394 inches, how many inches are in 1 kilometer?

Answers

Since 1 km is 100,000 cm there are 39,400 inches in a kilometer

Answer: There are 39400 inches in 1 kilometers.

Step-by-step explanation:

Since we have given that

1 cm = 0.394 inches

As we know that

1 m = 100 cm

1 km = 1000 m

1000 m = 100×1000=100000 cm

1 km = 100000 cm

So, it becomes,

[tex]1\ km=0.394\times 100000\\\\1\ km=39400\ inches[/tex]

Hence, there are 39400 inches in 1 kilometers.

I need help with equation number 6

Answers

Combine like terms
-2x-4=-2x-4

Since the equations are the same, the answer is infinitely many solutions.

round 38562.7 to 1 significant figure

Answers

= 488000
(real number)

= 4.88 × 105
(scientific notation)

= 4.88e5

maybe this will help ?

to round it to 1 significant figure there needs to be only 1 figure that is above 1 the rest must be zero so the answer is 40000

Brandon is shopping at Old Navy during a sale. All shorts are $16 each and all t-shirts are $10 each.He has $100 to spend and would like to purchase at least 2 pairs of shorts.
a) Write a system of linear inequalities to represent
this situation, then graph.
b)Using your graph, give two possible combinations
of shorts and Brandon can buy.

Answers

Let
x = the number of shorts bought
y = the number of t-shirts bought

A pair of shorts costs $16 and a t-shirt costs $10. Brandom has $100 to spend.
Therefore
16x + 10y ≤ 100
This may be written as
y ≤ - 1.6x + 10 (1)

Brandon wants at least 2 pairs of shorts. Therefore
x ≥ 2 (2)

Graph the equations y = -1.6x + 10 and x = 2.
The shaded region satisfies both inequalities.

Answer:
Two possible solutions are
(a) 3 pairs of shorts and 4 t-shirts,
(b) 4 pairs of shorts and 2 t-shirts.

Two possible solutions are (a) [tex]3[/tex] pairs of shorts and [tex]5[/tex] t-shirts

(b) [tex]4[/tex] pairs of shorts and [tex]3[/tex] t-shirts.

[tex]x=the\; number\; of shorts\\y=the\; number\; of\ t-shirt\; bought[/tex]

So equation formed is

[tex]16x+10y\leq 100[/tex]

[tex]10y\leq 100-16x\\y\leq 10-1.6x[/tex]---Eq [tex]1[/tex]

Bradon wants at least [tex]2[/tex] pair of shorts.Therefore

[tex]x\geq 2[/tex]--Eq [tex]2[/tex]

Now put,

[tex]x=3 \\y=10-1.6(3)\\y=10-4.8\\y=5\\[/tex]

Hence [tex]3[/tex] shorts and [tex]5[/tex] t-shirts

Now put,

[tex]x=4 \\y=10-1.6(4)\\y=10-6.4\\y=3\\[/tex]

Now graph the equation,

Learn more about inequalities here:

https://brainly.com/question/15137133?referrer=searchResults

How do I solve 28.8=18x

Answers

28.8 = 18x....divide both sides by 18, eliminating the 18 on the right side
28.8 / 18 = x
1.6 = x <==

Solve each equation. Then check your solution. 3-a/7=-2

Answers

3-a/7=-2 (problem)
3-a=-14 ( multiply by 7 on both sides)
-a=-14 (subtract 3 from both sides)
a=14 (divide both sides by negative 1)
Answer: a=14

Janet Fog's uncle, Pete Moore, promised her a gift of $30,000 upon her graduation from law school or $1500 every quarter for the next four years. if the money could be invested at 12% compounded quarterly, which offer should Janet choose?

Answers

First calculate the future value of the annuity
The formula to find the future value of an annuity ordinary is
Fv=pmt [((1+r/k)^(kn)-1)÷(r/k)]
Fv future value?
PMT quarterly payment 1500
R interest rate 0.12
K compounded quarterly 4
N time 4 years
Fv=1,500×(((1+0.12÷4)^(4×4)
−1)÷(0.12÷4))
=30,235.32

Now compare the amount of the annuity with amount of the gift
30,235.32−30,000=235.32
So as you can see the amount of the annuity is better than the amount of the gift by 235.32

Second offer is better

Hope it helps!

What is the value of x?

Answers

(n - 2 ) 180 = sum of interior angles

n = 6, because it is the number of angles the polygon has

(6-2) 180 = 720

720 - 120 - 100 - 128 - 133 - 112 = x

x = 127

hope this helps

the sum of the interior angles equal 720

112 + 133 +128 +100 + 120 = 593

720 - 593 = 127

x = 127 degrees

If f is a function defined by the equation yequals=f(x) then x is called the

Answers

X is the independent variable (think 'input variable'); all of the values of x which are valid (ones that don't create division by zero or square root of a negative value) is the domain.

A jet plane, flying 120 mph faster than a propeller plane, travels 3520 miles in 3 hours less time than the propeller plane takes to fly the same distance. how fast does each plane fly?

Answers

Final answer:

The propeller plane flies at a speed of 880 mph, and the jet plane flies at a speed of 1000 mph.

Explanation:

Let the speed of the propeller plane be represented by x mph. Since the jet plane is flying 120 mph faster, its speed can be represented as x + 120 mph.

We know that the distance traveled by both planes is the same, so we can set up the equation:

x * t = (x + 120) * (t - 3)

where t is the time taken by the propeller plane.

Simplifying the equation, we get:

x * t = xt + 120t - 3x - 360

Rearranging the terms, we have:

120t - 3x = 360

Since we have two variables, we need another equation. We can use the formula for distance, which is given by:

Distance = Speed * Time

For the propeller plane, we have:

D = x * t

For the jet plane, we have:

D = (x + 120) * (t - 3)

Since the distance is the same, we can equate these two equations:

x * t = (x + 120) * (t - 3)

x * t = xt + 120t - 3x - 360

Rearranging the terms, we get:

120t - 3x = 360

Solving the system of equations, we find that x = 880 mph and x + 120 = 1000 mph.

Find a cubic function with the given zeros. Square root of five., - Square root of five., -7

Answers

[tex]\bf x= \begin{cases} \pm \sqrt{5}\\ -7 \end{cases}\implies \begin{cases} x=\sqrt{5}\implies &(x-\sqrt{5})=0\\ x=-\sqrt{5}\implies &(x+\sqrt{5})=0\\ x=-7\implies &(x+7)=0 \end{cases} \\\\\\ \stackrel{\textit{difference of squares}}{(x-\sqrt{5})(x+\sqrt{5})}(x+7)=\textit{original polynomial} \\\\\\ (x^2-5)(x+7)=y\implies x^3+7x^2-5x-35=y[/tex]

One half of a number is 8 less than the number itself

Answers

The question is a little vague but I believe the answer would be 16. 16 divided in half is 8. And that is 8 less than 16. < Not the answer for this question.

I didn't understand the question until Maria answered, she is completely right! Sorry for the misunderstanding.

1/2 of x is 2/3 of x minus 8

(1/2)x = (2/3)x -8

clear fraction by multiplying by 6

3x = 4x -48

x = 48

While in europe, if you drive 113 km per day, how much money would you spend on gas in one week if gas costs 1.10 euros per liter and your car's gas mileage is 41.0 mi/gal ? assume that 1euro=1.26dollars?

Answers

113 x 7 = 791 km per week
791 km ~ 491.5 miles

491.5/41 = 12 gallons used

1 galloon ~ 3.8 liters

45.6 liters used

45.6 x 1.10 euros = 50.16 Euros used or $63.20 dollars used.

Answer:

[tex]\$63.04[/tex]

Step-by-step explanation:

Distance traveled in [tex]1[/tex] day = [tex]113 km[/tex]

Distance traveled in [tex]1[/tex] week = [tex]113*7 = 791 km[/tex]

Mileage of car = [tex]41.0 mi/gal = \frac{41*1.60}{3.78} = 17.35 km/liter[/tex]

Total gas consumed = [tex]\frac{791}{17.35} = 45.49liter[/tex]

Cost of [tex]1liter[/tex] gas = [tex]1.10 euros = 1.10*1.26 = \$1.386[/tex]

Money spent on gas in a week = [tex]45.49*1.386 = \$63.04[/tex]

What is the value of x ?

Answers

2x-14=x+2
x=16
Your welcome

what is the value of x

a machine manufactures wheels with a diameter of 70cm.It is acceptable for the diameter of a wheel to be withen 0.02 cm of this value .write and sopve an absolute value equation to find the minimum acceptable diameters

Answers

The ranges of the diameter of the wheels that the company manufactures can be calculated by adding or subtracting the given tolerance from the desired diameter. The lower limit is calculated by subtracting the allowance while the upper limit is obtained by adding the allowance.
lower limit = 70 cm - 0.02 cm = 6.98 cm
upper limit = 70 cm + 0.02 cm = 70.02 cm
The absolute value equation for this scenario is,
0.02 = I70 - xI

What is the measure of angle S?

Answers

132 + 56 + 79 = 267

360 - 267 = 93

answer is 93 degrees

Jason's age is 1/2 of his brother's age. His brother's age can be represented by the expression 12a - 9. Write an expression that can be used to represent the sum of their ages.
A) 12a - 5.5
B) 18a - 13.5
C) 18a + 21
D) (12a-9)/2

Answers

Answer:

the answer is B)18a-13.5

Step-by-step explanation:

How much is 8% of 1249? round the answer to the nearest hundredth

Answers

First of all you make the multiplier:
8/100 = 0.08

Then you multiply the number by the multiplier:
1249*0.08 = 99.92

Hope this helps! :)

It is 99.92. I'm sure that thats the answer.

help please! Note: Use ELIMINATION!!!

Answers

the answers are 24 and -2

Multiplying the second equation by 4/3 and adding it to the first, we get 36=9x and by dividing both sides by 9, we get x=4. Next, we can plug that into the second equation (it doesn't matter which equation) to get 3*4-3y=18=12-3y. Subtracting 12 from both sides, we get 6=-3y and by dividing both sides by -3 we get y=-2

2,226 divided by 53 how do you solve this??

Answers

The answer is 42 becuase you are making 53 groups of 2226 you can also simply solve it by useing long division

2226 divided by 53 is equal to42

A certain roller coaster can accommodate 1600 riders per hour a digital sign states that there are 345 people in line if you get in line about how many minutes will it take you to get to the front of the line

Answers

Given that the roller coaster can accommodate 1600 riders per hour, thus the roller coaster can accomodate 1600 / 60 = 26.67 riders per minute.

Given that there are 345 people in the line, the number of minutes it will take for the 345 persons to get into the roller coaster is given by 345 / 26.67 = 12.94 minutes.

Therefore, it will take you 12.94 minutes to get to the front of the line.

(01.02) kim earns $30 for babysitting on friday nights. she makes an average of $1.25 in tips per hour. write the function of kim's earnings, and solve for how much she would make after 3 hours. (1 point) y = 1.25x + 30; $33.75 y = 30x + 1.25; $43.75 y = 30x + 1.25; $91.25 y = 1.25x + 30; $31.25

Answers

y = 1.25x + 30 <== ur function

for 3 hrs.....x = 3
y = 1.25(3) + 30
y = 3.75 + 30
y = 33.75 <=== for 3 hrs

Solve for x
7x-0.15=2x+0.6
Explain

Answers

if you subtract 2x from each side you get 5x-0.15=0.6 because the subtraction of 2x on the right of the equal sign cancels out. Then, if you add 0.15 to both sides you will get 5x=0.75. Finally, if you divide both sides by 5 you get x = 0.15.

The difference of the squares of two positive consecutive even integers is 20 . find the integers. use the fact that, if x represents an even integer, then xplus 2 represents the next consecutive even integer.

Answers

If x is the first number, x+2 is the next consecutive even integer. The difference (subtraction) of their squares equals 20. Therefore:
(x+2)²-x²=20...expand this:
x²+4x+4-x²=20
x²+4x+4-x²=20
4x+4=20
4x=16
x=4. Therefore the next consecutive even integer is x+2=6.
To check: 6²-4² must equal 20
36-16=20 check!

The positive even numbers are 4 and 6.

Equation

State of quality between two expressions consisting of variable and/or number.

Given

The difference of the squares of two positive consecutive even integers is 20.

To find

The number will be

How do find the numbers?

The difference of the squares of two positive consecutive even integers is 20.

Let x be the even number and positive consecutive even is (x+2)

Then according to the condition

[tex]\begin{aligned} (x+2)^{2} -x^{2} &=20\\x^{2} +4x+4-x^{2} &= 20\\4x &= 20-4\\4x &= 16\\x &= \dfrac{16}{4}\\x &= 4\end{aligned}[/tex]

Then the positive even numbers are 4 and 6.

More about the equation link is given below.

https://brainly.com/question/2263981

help me please geometry

Answers

If we translate the top figure (ABC) 2 units to the left and reflect over the x axis, then we end up with the bottom figure (A'B'C'). This proves the figures to be congruent.

This is something to do with trial and error really. See attached.

How would I solve 1/5=125^x

Answers

Hi there,

This question is a bit weird lol, but anyway here is how to do it and the answer:

1/5 = 123² x
1/5 = 15625x
15625x = 1/5

Hence, X = 1/5

Hope this helps :)

This is a log equation. To solve

125^x = 1/5
x log 125 = log 1/5
x = log (1/5) / log 125
[tex]x = -\frac{1}{3} [/tex]

To show that we are correct. Input -1/3 for x [tex]125^{-1/3}[/tex] with a calculator and it will = 1/5, meaning it is correct.

[tex]125^{-1/3} = \frac{1}{5}[/tex]

This is what I've done so far but does anyone know what I do next or did I make a mistake

Answers

remmeber, you can do anything to an equation as long as you do it to both sides

for inequalities, if you multiply or divide both sides by a negive, flip the direction of the inequality sign

pemdas always applies
but also the commutative property and assiociative property

so
2(5y+13)-6<20
add 6 both sides
2(5y+13)<26
divide both sides by 2 (easier that distributing)
5y+13<13
minus 13 both sides
5y<0
y<0 is the solution

A regular hexagon rotates counterclockwise around its center. It turns through angles greater than 0° and less or equal to 360°. At how many different angles of a hexagon map onto itself?

Answers

there are 6 sides to a hexagon

360/6 = 60 degrees

so it will map onto itself at 60, 120, 180, 240, 300 & 360

there are 6 different angles

What is the common term for the set of all points with x- coordinate zero?

A. X axis

B. Y axis

C. Coordinate plane

D. Origin

Answers

I think the correct answer would be option B. The common term for the set of all points with the x-coordinate that is zero is the y-axis. All of the points that is found in the y-axis contains zero as the x-coordinate of the point. So, it should be the y-axis the correct answer.

Answer:

Option B is the correct answer.

Step-by-step explanation:

We have X axis, Y axis and origin in coordinate plane.

Origin is the point for which both x and y coordinate is zero.

X axis is the set of all points with y- coordinate zero.

Y axis is the set of all points with x- coordinate zero.

Here we are asked to find the common term for the set of all points with x- coordinate zero.

That is Y axis

Option B is the correct answer.

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