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Use Pascal’s triangle to expand the binomial.
(d–5y)6
To expand the binomial (d–5y)6 using Pascal's triangle, we need to find the coefficients of the terms in the expansion. Pascal's triangle is a triangular arrangement of numbers where each number is the sum of the two numbers directly above it.
The 6th row of Pascal's triangle is 1, 6, 15, 20, 15, 6, 1
To expand (d–5y)6, we can write it as:
1(d)6 + 6(d)5(-5y) + 15(d)4(-5y)2 + 20(d)3(-5y)3 + 15(d)2(-5y)4 + 6(d)(-5y)5 + 1(-5y)6
Simplifying each term, we get:
(d)6 + 6(d)5(-5y) + 15(d)4(25y2) + 20(d)3(-125y3) + 15(d)2(625y4) + 6(d)(-3125y5) + (-625y)6
This can be further simplified as:
d^6 - 30d^5y + 750d^4y^2 - 10,000d^3y^3 + 93,750d^2y^4 - 187,500dy^5 + 156,250y^6
Therefore, the expansion of (d–5y)6 using Pascal's triangle is:
d^6 - 30d^5y + 750d^4y^2 - 10,000d^3y^3 + 93,750d^2y^4 - 187,500dy^5 + 156,250y^6