1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Dmitry_Shevchenko [17]
3 years ago
6

Expand the following using the Binomial Theorem and Pascal’s triangle. (x + 2)6 (x − 4)4 (2x + 3)5 (2x − 3y)4 In the expansion o

f (3a + 4b)8, which of the following are possible variable terms? Explain your reasoning. a2b3; a5b3; ab8; b8; a4b4; a8; ab7; a6b5

Mathematics
2 answers:
ivolga24 [154]3 years ago
7 0
\bf (2x+3)^5\implies 
\begin{array}{llll}
term&coefficient&value\\
-----&-----&-----\\
1&&(2x)^5(+3)^0\\
2&+5&(2x)^4(+3)^1\\
3&+10&(2x)^3(+3)^2\\
4&+10&(2x)^2(+3)^3\\
5&+5&(2x)^1(+3)^4\\
6&+1&(2x)^0(+3)^5
\end{array}

as you can see, the terms exponents, for the first term, starts at highest, 5 in this case, then every element it goes down by 1, till it gets to 0

for the second term, starts at 0, and every element it goes up by 1, till it gets to the highest

now, to get the coefficient, they way I get it, is "the product of the current coefficient and the exponent of the first term, divided by the exponent of the second term plus 1"

notice the first coefficient is always 1

so...how did we get 10 for the 3rd element?  well, 5*4/2
how did we get 10 for the fourth element?  well, 10*2/4


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


\bf (3a+4b)^8\implies 
\begin{array}{llll}
term&coefficient&value\\
-----&-----&-----\\
1&&(3a)^8(+4b)^0\\
2&+8&(3a)^7(+4b)^1\\
3&+28&(3a)^6(+4b)^2\\
4&+56&(3a)^5(+4b)^3\\
5&+70&(3a)^4(+4b)^4\\
6&+56&(3a)^3(+4b)^5\\
7&+28&(3a)^2(+4b)^6\\
8&+8&(3a)^1(+4b)^7\\
9&+1&(3a)^0(+4b)^8
\end{array}

and from there, you can simplify the elements of the expansion by combining the coefficients

like for example, the 7th element of (3a+4b)⁸ will then be 1032192a²b⁶
erik [133]3 years ago
6 0

Answer:

The possible terms are a^5b^3,b^8,a^4b^4,a^8,ab^7.

Step-by-step explanation:

Binomial Theorem:

(a+b)^n=^nC_0a^n+^nC_1a^{n-1}b+...+^nC_{n-1}a^1b^{n-1}+^nC_nb^n

Using Binomial Theorem, we get

(x + 2)^6=^6C_0a^n+^6C_1a^{5}b+...+^6C_{5}a^1b^{5}+^5C_5b^5

(x + 2)^6=x^6 + 12 x^5 + 60 x^4 + 160 x^3 + 240 x^2 + 192 x + 64

Similarly,

(x-4)^4=x^4 - 16 x^3 + 96 x^2 - 256 x + 256

(2x+3)^5=32 x^5 + 240 x^4 + 720 x^3 + 1080 x^2 + 810 x + 243

(2x-3y)^4=16 x^4 - 96 x^3 y + 216 x^2 y^2 - 216 x y^3 + 81 y^4

The Pascal’s triangle is given below. In Pascal’s triangle nth row represents the coefficients of the expression of (a+n)^{n-1}.

In the expression (x + 2)^6, a=x, b=2 and n=6. Using 7th row of Pascal’s triangle we get

(x + 2)^6=1(x^6)(2^0)+6(x^5)(2^1)+15(x^4)(2^2)+20(x^3)(2^3)+15(x^2)(2^4)+6(x^1)(2^5)+1(x^0)(2^6)

(x + 2)^6=x^6 + 12 x^5 + 60 x^4 + 160 x^3 + 240 x^2 + 192 x + 64

Similarly,

In (x-4)^4 a=x, b=-4 and n=4. Using 5th row of Pascal’s triangle we get

(x-4)^4=x^4 - 16 x^3 + 96 x^2 - 256 x + 256

In (2x+3)^5 a=2x, b=3 and n=5. Using 6th row of Pascal’s triangle we get

(2x+3)^5=32 x^5 + 240 x^4 + 720 x^3 + 1080 x^2 + 810 x + 243

In (2x-3y)^4 a=2x, b=-3y and n=4. Using 5th row of Pascal’s triangle we get

(2x-3y)^4=16 x^4 - 96 x^3 y + 216 x^2 y^2 - 216 x y^3 + 81 y^4

In binomial expansion of (3a + 4b)^8 the degree of each term must be 8. It means the sum of powers of a and b is equal to 8.

For a^2b^3 the dgree of term is 2+3=5. So, it is not possible variable term.

Therefore the possible terms are a^5b^3,b^8,a^4b^4,a^8,ab^7.

You might be interested in
Digital music distribution provides an opportunity for everyone to get their music heard. In order to get on a service like iTun
Fiesta28 [93]

Based on the charge to the artist and the amount the streaming services pay, the following are true:

  1. -50.
  2. 5.
  3. y = 5x - 50
  4. .
  5. 10,000 streams.
  6. 190,010,000 streams

<h3>What model can be derived for the streaming data?</h3>

The y-intercept would be the amount that the artist has before they start making any money. In this case that amount is -$50 because they will owe the distributor.

The slope is the change per unit which is $5.00 here.

The formula is therefore:

y = mx + b

y = Slope × x + b

y = 5 × x + (-50)

y = 5x - 50

The number of streams to pay off the distributor is when the profit is $0:

0 = 5x - 50

50 = 5x

x = 10 units

Number of streams:

= 10 units x 1,000

= 10,000 streams

To make a profit of $950,000, the units are:

950,000 = 5x - 50

5x = 950,000 + 50

x = (950,000 + 50) / 5

= 190,010 units

Number of streams is:

= 190,010 x 1,000

= 190,010,000 streams

Find out more on linear equations at brainly.com/question/1884491.

#SPJ1

3 0
2 years ago
Plz plz help help help plz plz help plz
uranmaximum [27]

Answer:

since your adding to hours each time youll have 11, then the answer would be d the last one :)

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
( a-8) x b^3<br><br><br><br> Answer this
VARVARA [1.3K]

Answer:

ab^3x - 8b^3x

Step-by-step explanation:

(a - 8)b^3x

5 0
3 years ago
2.2 geometry i need the answers
Strike441 [17]

Answer:

Where’s the picture

Step-by-step explanation:

8 0
3 years ago
2 water taps A B fill a tank in 12 hrs. Tap A takes 10 hrs less than tap B to till tank separately. Find time taken by tap B to
Hatshy [7]

Answer:

11 hours

Step-by-step explanation:

Setting up a system of equations and using the substitution method will give you the number of hours used by each tap.  Let a = time for tap A and b = time for tap B:

a + b = 12  (A and B fill a tank in 12 hours)

a = b - 10 (tap A takes 10 hours less than tap B)

Substitute 'b - 10' for 'a' in the first equation:

b - 10 + b = 12

Combine like terms: 2b - 10 = 12

Add 10 to both sides: 2b - 10 + 10 = 12 + 10 or 2b = 22

Divide by 2: 2b/2 = 22/2

Solve for b: b = 11 hours

5 0
3 years ago
Other questions:
  • Calculate the distance between (0,3) and (3,10) to 1 decimal place
    13·1 answer
  • Can someone check my answers please, I am brushing up on some skills. So far for number one I got 589 and for two I got 116.
    15·1 answer
  • Julio is paid 1.4 times is normal hourly rate for each hour he works over 32 hours a week last week he worked 43 hours and earne
    9·1 answer
  • If triangle ABC = triangle PQR and triangle RPQ = triangle XYZ, which of the following angles is congruent angle CAB?
    13·2 answers
  • Find x. Round your answer to the nearest tenth of a degree.<br> Lo<br> 40
    13·1 answer
  • Find the 10th term in a sequence whose general term is an = n2 – 8. Question 7 options: A) 92 B) 108 C) 12 D) 113
    12·2 answers
  • Find the shaded area. Round your answer to the nearest tenth.
    8·1 answer
  • 8. Find the area and the perimeter of
    7·1 answer
  • Calculate the percentage increase or decrease in each of
    15·1 answer
  • Adam is training a group of cyclists. Each cyclist completes a 10-kilometer time trial during training. Adam records the times,
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!