Use Pascal's triangle to expand the binomial.%0D%0A%0D%0A(d−5y)6 show all your work

Answer 1

To expand the binomial (d - 5y)^6 using Pascal's triangle, we can follow these steps:

Step 1: Begin by writing the binomial raised to the 6th power:

(d - 5y)^6

Step 2: Look at the 6th row of Pascal's triangle to determine the coefficients for each term:

1 6 15 20 15 6 1

Step 3: Start with the first term and increase the exponent of d by 1, while decreasing the exponent of 5y by 1 for each subsequent term.

Term 1:
1 * d^6 * (-5y)^0 = d^6 * 1 = d^6

Term 2:
6 * d^5 * (-5y)^1 = 6 * d^5 * (-5y) = -30d^5y

Term 3:
15 * d^4 * (-5y)^2 = 15 * d^4 * 25y^2 = 375d^4y^2

Term 4:
20 * d^3 * (-5y)^3 = 20 * d^3 * -125y^3 = -2500d^3y^3

Term 5:
15 * d^2 * (-5y)^4 = 15 * d^2 * 625y^4 = 9375d^2y^4

Term 6:
6 * d^1 * (-5y)^5 = 6 * d * -3125y^5 = -18750dy^5

Term 7:
1 * d^0 * (-5y)^6 = 1 * (-3125y)^6 = 15625y^6

Step 4: Combine all the terms:

(d - 5y)^6 = d^6 - 30d^5y + 375d^4y^2 - 2500d^3y^3 + 9375d^2y^4 - 18750dy^5 + 15625y^6

This is the expanded form of (d - 5y)^6 using Pascal's triangle.