To expand the binomial (d – 3)^6 using Pascal's Triangle, we can use the following pattern:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
The numbers in each row of Pascal's Triangle represent the coefficients of the expanded binomial. The powers of "d" decrease from left to right, while the powers of "-3" increase. To expand (d – 3)^6, we use the coefficients from the 6th row of Pascal's Triangle, which are:
1, 6, 15, 20, 15, 6, 1
We substitute these coefficients into the equation:
(d – 3)^6 = 1(d^6) + 6(d^5)(-3) + 15(d^4)(-3)^2 + 20(d^3)(-3)^3 + 15(d^2)(-3)^4 + 6(d)(-3)^5 + 1(-3)^6
Expanding the terms and simplifying, we get:
= d^6 - 18d^5 + 135d^4 - 540d^3 + 1215d^2 - 1458d + 729
Therefore, (d – 3)^6 expands to d^6 - 18d^5 + 135d^4 - 540d^3 + 1215d^2 - 1458d + 729.
Use Pascal's Triangle to expand the binomial.
(d – 3)6
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