Using Pascal’s Triangle, write out the expansion of (a+1)^3 .(1 point)
The expansion of (a+1)^3 using Pascal's Triangle is as follows:
(a+1)^3 = 1(a^3) + 3(a^2)(1^1) + 3(a^1)(1^2) + 1(1^3)
Simplifying this expression, we get:
(a+1)^3 = a^3 + 3a^2 + 3a + 1
To expand (a+1)^3 using Pascal's Triangle, follow these steps:
Step 1: Write out the first few rows of Pascal's Triangle:
1
1 1
1 2 1
1 3 3 1
Step 2: Determine the coefficients for each term in the expansion. The coefficients can be found by using the numbers in each row of Pascal's Triangle.
The first term in the expansion is (a)^3, which has a coefficient of 1.
The second term in the expansion is 3*(a^2)*(1), which has a coefficient of 3.
The third term in the expansion is 3*(a)*(1^2), which also has a coefficient of 3.
The fourth term in the expansion is (1^3), which has a coefficient of 1.
Step 3: Write out the terms of the expansion using the coefficients and variables.
(a+1)^3 = 1*(a)^3 + 3*(a^2)*(1) + 3*(a)*(1^2) + 1*(1^3)
Simplifying the terms, we get:
(a+1)^3 = a^3 + 3a^2 + 3a + 1
To expand (a+1)³ using Pascal's Triangle, you need to locate the fourth row of the triangle, as the exponent of (a+1) is 3. The rows in Pascal's Triangle are numbered starting from row 0.
The fourth row of Pascal's Triangle is 1 3 3 1. Each number represents the coefficients of the terms in the expansion of (a+1)³.
To write out the expansion, you multiply the terms in (a+1) by their corresponding coefficients from the fourth row:
(a+1)³ = 1a³ + 3a² + 3a + 1
Therefore, the expansion of (a+1)³ is 1a³ + 3a² + 3a + 1.