How do I expand (x + 1)^8?

Question

How do I expand (x + 1)^8?

Answer 1

Binomial theorem or Pascal's Triangle

To expand (x + 1)^n, use Pascal's triangle to get the coefficients

n = 0 | 1

n = 1 | 1 1

n = 2 | 1 2 1

n = 3 | 1 3 3 1

n = 4 | 1 4 6 4 1

n = 5 | 1 5 10 10 5 1

n = 6 | 1 6 15 20 15 6 1

n = 7 | 1 7 21 35 35 21 7 1

n = 8 | 1 8 28 56 70 56 28 8 1

(x + 1)^8 = x^8 + 8x^7 + 28x^6 + 56x^5 + 70x^4 + 56x^3 + 28x^2 + 8x +1

Answer 2

one of the efficient ways to expand this is by the pascal's triangle

(a+b)^8=(a^8)+8(a^7)(b)+28(a^6)(b^2)+

56(a^5)(b^3)+70(a^4)(b^4)+56(a^3)(b^5)

+28(a^2)(b^6)+8(a)(b^7)+(b^8)

therfore

(x + 1)^8=(x^8)+8(x^7)+28(x^6)+

56(x^5)+70(x^4)+56(x^3)+28(x^2)

+8(x)+1

Answer 3

(1 + x)^8

Terms are :-

1

8x

[ (8) (7) / 2 ! ]  x² = 28 x²

[ (8)(7)(6) / 3 ! ] x³ = 56 x³

[ 8 7 6 5 / 4! ] x^4 = 70 x^4

[ 8 7 6 5 4 / 5! ] x^5 = 56x^5

[ 8 7 6 5 4 3  / 6! ] x^6 = 28 x^6

[ 8 7 6 5 4 3 2 / 7!] x^7 = 8 x^7

x^8

1 + 8x + 28x² + 56x³ + 70x^4 + 56x^5 + 28x^6 + 8x^7 + x^8

CHECK

_______________1

__________1_________1

_________1_____2______1

_______1____3_____3_____1

_____1____4_____6____4____1

___1____5____10____10___5___1

__1___6____15____20___15__6___1

1___7___21____35___35___21__7__1

1_8___28___56____70___56___28_8__1

Answer 4

x^8 +8x^7 + 28x^6 + 56x^5 + 70x^4 + 56x^3 + 28x^2 + 8x + 1

I use Pascal's Triangle http://www.lincoln.k12.nv.us/alamo/high/...

Answer 5

(x+1)^2(x+1)^2(x+1)^2(x+1)^2