Pascal's Triangle Question

Using Pascal's Triangle only, and not binomial expansion, how would you work this question out:

[ 1 + (x - 1)^2]^3

This is what I have been doing so far, and I know I'm definitely making a mistake because the answer I get is different from the answer in the back of my text book.

My working out is as follows:

[1 + (x - 1)^2]^3

1 + (x - 1)^6

1 + [-1(1 - x)]^6

1 + (1 - x)^6 This part is wrong I think, but I need help here.

1 + (1 + 6(-x) + 15(-x)^2 + 20(-x)^3 + 15(-x)^4 + 6(-x)^5 + 1(-x)^6]

1 + 1 - 6x + 15x^2 - 20x^3 + 15x^4 - 6x^5 + x^6

2 - 6x + 15x^2 - 20x^3 + 15x^4 - 6x^5 + x^6

The actual answer is: 8 - 24x + 36x^2 - 32x^3 + 18x^4 - 6x^5 + x^6

Sorry about the length of this post but I'm getting really :mad:

about this!!!

Reply 1

Slumpy

16

I don't agree with your first step.

Reply 2

Ben77mc

14

Narik

Using Pascal's Triangle only, and not binomial expansion, how would you work this question out:

[ 1 + (x - 1)^2]^3

This is what I have been doing so far, and I know I'm definitely making a mistake because the answer I get is different from the answer in the back of my text book.

My working out is as follows:

[1 + (x - 1)^2]^3

1 + (x - 1)^6

1 + [-1(1 - x)]^6

1 + (1 - x)^6 This part is wrong I think, but I need help here.

1 + (1 + 6(-x) + 15(-x)^2 + 20(-x)^3 + 15(-x)^4 + 6(-x)^5 + 1(-x)^6]

1 + 1 - 6x + 15x^2 - 20x^3 + 15x^4 - 6x^5 + x^6

2 - 6x + 15x^2 - 20x^3 + 15x^4 - 6x^5 + x^6

The actual answer is: 8 - 24x + 36x^2 - 32x^3 + 18x^4 - 6x^5 + x^6

Sorry about the length of this post but I'm getting really :mad:

about this!!!

Put it in latex please?

Reply 3

rnd

6

Narik

My working out is as follows:

[1 + (x - 1)^2]^3

1 + (x - 1)^6

(a+b)^3=a^3 + 3 a^2 b + 3 a b^2 + b^3

Do you see the 1 3 3 1 ?

rather than a+b^3 which is what you seem to have done. (or even a^3 + b^3)

Reply 4

Small123

15

Narik

Using Pascal's Triangle only, and not binomial expansion, how would you work this question out:

[ 1 + (x - 1)^2]^3

This is what I have been doing so far, and I know I'm definitely making a mistake because the answer I get is different from the answer in the back of my text book.

Pascals:

...1
..1 1
.1 2 1
1 3 3 1 Thus use this line.
My working out is as follows (AMENDED):