Expand this Bionomial.(x-3y)^7?

Question

Steps on how to solve this problem.

Answer 1

pascal's triangle will give you coeffs of (a+b)*7 as
1, 7, 21,35,35,21,7,1
instead of b you have (-3y)
so coeffs will be multiplied by 1,-3,9,-21 ...

x^7 -21 x^6y + .....

Answer 2

(a + b)^7 = a^7 + 7 a^6 b + 21 a^5 b^2 + 35 a^4 b^3 + 35 a^3 b^4 + 21 a^2 b^5 + 7 a b^6 + b^7

The needed powers of -3 are: 1, -3, 9, -27, 81, -243, 729, -2187

So (x - 3y)^7 =

x^7 - 21 x^6 y + 189 x^5 y^2 - 945 x^4 y^3 + 2835 x^3 y^4 - 5103 x^2 y^5 + 5103 x y^6 - 2187 y^7

Answer 3

use the binomial theorem
x=a
-3y=b
(a+b)^7=
(1a^7)+
(7*a^6*b)+
(21*a^5*b^2)+
(35*a^4*b^3)+
(35*a^3*b^4)+
(21*a^2*b^5)+
(7*a*b^6)+
(1*b^7)
write this out on paper and substitute
for clarification search wikipedia for both
the binomial theorem and pascals triangle

Answer 4

Expansion of (x+y)^n

1. The first term is x^n
2. The last term is y^n
3. The sum of the exponents in each term is n.
4. The exponent of x decreases by 1
5. The exponent of y increases by 1
6. The number of terms is n+1
7. The coefficient of terms equidistant are equal

Binomial formula:

D = AB/(C+1)

where:
A is the coefficient of the previous term
B is the exponent of x of previous term
C is the exponent of y of previous term
D is the coefficient of next term

y^r term = [n(n-1)(n-2)...(n-r+1)x^(n-r)y^r]/r! ....or

y^r term = nCr x(n-r) y^r

In your problem, there must be 8 terms
1st term: x^7
2nd term: 7C1x^(7-1)[(-3y)^1]
= -21x^6y
3rd term: 7C2x^(7-2)[(-3y)^2]
= 189x^5y^2
4th term: 7C3x^(7-3)[(-3y)^3]
= -945x^4y^3
5th term: 7C4x^(7-4)[(-3y)^4]
= 2835x^3y^4
6th term: 7C5x^(7-5)[(-3y)^5]
= -5103x^2y^5
7th term: 7C6x^(7-6)[(-3y)^6]
= 5103x^5y^2
8th term or last term: (-3y)^7
= -2187y^7

therefore:
(x-3y)^7=(x^7)-(21x^6y)+(189x^5y^2)-(945x^4y^3)+(2835x^3y^4)-(5103x^2y^5)+(5103x^5y^2)-(2187y^7)