Use pascal's triangle to find an expression for (a+b)^6 without doing any multiplication.?

Question

Use this to figure out with polynomial equals to (x-1)^6. Careful about the -1

Answer 1

Just write down the seventh (6+1) row of Pascals triangle.
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
So (a+b)^6 = a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + b^6.
Substitute b = -1 (and change a to x):
x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1.

Answer 2

Paschal's triangle ... 1st 7 lines

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

So (a+b)^6
= a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + b^6

Thus (x -1)^6 = (x + (-1))^6
= x^6 + 6x^5*(-1) + 15x^4*(-1)^2 + 20x^3*(-1)^3 + 15x^2*(-1)^4 + 6x*(-1)^5 + (-1)^6
= x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1