Use Pascal's triangle to expand the binomial. (d−5y)6
To expand the binomial (d - 5y)^6 using Pascal's triangle, we use the sixth row which contains the coefficients that will be applied to each term in the binomial expansion. The 6th row of Pascal's triangle (starting with row 0) is:
1 6 15 20 15 6 1
Let's write down the general formula for the binomial expansion of (a - b)^n using the binomial theorem:
(a - b)^n = a^n - (nC1)a^(n-1)b + (nC2)a^(n-2)b^2 - ... + (-1)^n * b^n
Here, nCk represents the binomial coefficients, which are exactly the numbers found in Pascal's triangle.
Now, let's apply it to our specific case:
(d - 5y)^6 = d^6 - (6C1)d^5(5y) + (6C2)d^4(5y)^2 - (6C3)d^3(5y)^3 + (6C4)d^2(5y)^4 - (6C5)d(5y)^5 + (6C6)(5y)^6
Plugging in the coefficients from Pascal's triangle:
= d^6 - 6d^5(5y) + 15d^4(5y)^2 - 20d^3(5y)^3 + 15d^2(5y)^4 - 6d(5y)^5 + (5y)^6
Now let's simplify it:
= d^6 - 30d^5y + 375d^4y^2 - 2500d^3y^3 + 9375d^2y^4 - 18750dy^5 + 15625y^6
This is the expanded form of the binomial (d - 5y)^6 using Pascal's triangle.