Use Pascal's Triangle to expand the binomial. (d – 3)6 (1 point)
To expand the binomial (d - 3)^6 using Pascal's Triangle, we can follow these steps:
1. Write down the sixth row of Pascal's Triangle.
The sixth row of Pascal's Triangle is: 1 5 10 10 5 1
2. Split the binomial into two terms: d^6 and -3^6.
3. Starting with the first term, use the row of Pascal's Triangle to determine the coefficients for each term.
Term 1: d^6 * 1 = d^6
Term 2: d^5 * (-3) * 5 = -15d^5
Term 3: d^4 * (-3)^2 * 10 = 90d^4
Term 4: d^3 * (-3)^3 * 10 = -270d^3
Term 5: d^2 * (-3)^4 * 5 = 405d^2
Term 6: d * (-3)^5 * 1 = -243d
Term 7: (-3)^6 * 1 = 729
4. Combine all the terms:
(d - 3)^6 = d^6 - 15d^5 + 90d^4 - 270d^3 + 405d^2 - 243d + 729
To expand the binomial expression (d - 3)^6 using Pascal's Triangle, follow these steps:
Step 1: Determine the power of the binomial, which in this case is 6.
Step 2: Write down the rows of Pascal's Triangle that correspond to the power of the binomial. For a binomial with power 6, we need to consider rows 0 to 6 of Pascal's Triangle.
Pascal's Triangle:
Row 0: 1
Row 1: 1, 1
Row 2: 1, 2, 1
Row 3: 1, 3, 3, 1
Row 4: 1, 4, 6, 4, 1
Row 5: 1, 5, 10, 10, 5, 1
Row 6: 1, 6, 15, 20, 15, 6, 1
Step 3: Write down the coefficients for each term in the expansion.
The first term has a coefficient of 1 * (d^6) * (-3)^0, which simplifies to d^6.
The second term has a coefficient of 6 * (d^5) * (-3)^1, which simplifies to 6d^5 * (-3).
The third term has a coefficient of 15 * (d^4) * (-3)^2, which simplifies to 15d^4 * 9.
The fourth term has a coefficient of 20 * (d^3) * (-3)^3, which simplifies to 20d^3 * (-27).
The fifth term has a coefficient of 15 * (d^2) * (-3)^4, which simplifies to 15d^2 * 81.
The sixth term has a coefficient of 6 * d * (-3)^5, which simplifies to 6d * (-243).
The seventh term has a coefficient of 1 * (-3)^6, which simplifies to 729.
Step 4: Write down the final expanded form of the binomial.
Putting it all together, the expanded form of (d - 3)^6 is:
d^6 - 18d^5 + 135d^4 - 540d^3 + 1215d^2 - 1458d + 729.
To expand the binomial (d - 3)^6 using Pascal's Triangle, we can apply the binomial theorem. The binomial theorem states that for any positive integer n, the expansion of (a + b)^n can be expressed as:
(a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n−1) * b^1 + C(n, 2) * a^(n−2) * b^2 + ... + C(n, n) * a^0 * b^n
In this case, (d - 3)^6, we can see that the first term is d^6 and the second term is -3^6. The powers of d decrease while the powers of -3 increase. To find the coefficients, we can use the values from Pascal's Triangle.
Pascal's Triangle is a triangular array of numbers in which each number is the sum of the two numbers directly above it. The triangle starts with 1 at the top and each row is generated by adding adjacent numbers from the previous row. Here's an example of Pascal's Triangle:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
...
To find the coefficients for expanding (d - 3)^6, we need to use the 6th row of Pascal's Triangle. So, the coefficients for each term will be: 1, 6, 15, 20, 15, 6, and 1.
Expanding (d - 3)^6:
(d - 3)^6 = C(6, 0) * d^6 * (-3)^0 + C(6, 1) * d^5 * (-3)^1 + C(6, 2) * d^4 * (-3)^2 + C(6, 3) * d^3 * (-3)^3 + C(6, 4) * d^2 * (-3)^4 + C(6, 5) * d^1 * (-3)^5 + C(6, 6) * d^0 * (-3)^6
Putting it all together, we get:
(d - 3)^6 = (1 * d^6) + (6 * d^5 * -3) + (15 * d^4 * 9) + (20 * d^3 * -27) + (15 * d^2 * 81) + (6 * d * -243) + (1 * 729)
Simplifying, the expanded form of (d - 3)^6 is:
d^6 - 18d^5 + 135d^4 - 540d^3 + 1215d^2 - 1458d + 729