Using Pascal’s Triangle, what is the fourth term in the expansion of (a+3)^4
when the expanded polynomial is written in standard form?
To find the fourth term in the expansion of (a + 3)^4 when written in standard form, we can use Pascal's triangle to determine the coefficients of the terms in the expansion.
The coefficients in the expansion of (a + b)^n are given by the entries in the nth row of Pascal's triangle.
The fourth row of Pascal's triangle is 1 4 6 4 1.
In the expansion of (a + 3)^4, the coefficients of the terms will be in the order: 1, 4, 6, 4, 1.
Therefore, the fourth term in the expansion of (a + 3)^4 is 6a^2 * 3^2 = 6a^2 * 9 = 54a^2.
To find the fourth term in the expansion of (a+3)^4, we can use 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 first term in each row is 1, and each row represents the coefficients of the binomial expansion.
The fourth term in the expansion of (a+3)^4 corresponds to the combination in the fourth row of Pascal's Triangle. The fourth row of Pascal's Triangle is 1, 4, 6, 4, 1.
The powers of (a+3) decrease from 4 to 0 as the powers of a increase from 0 to 4. So, the fourth term is the coefficient of (a^1)*(3^3) which is 4.
Therefore, the fourth term in the expansion of (a+3)^4, when written in standard form, is 4*(a^1)*(3^3), which simplifies to 4a*(27), or 108a.
To find the fourth term in the expansion of (a+3)^4, we can use Pascal's Triangle and the Binomial Theorem.
The Binomial Theorem states that for any two numbers, a and b, and a positive integer n, the expansion of (a + b)^n can be written as the sum of the coefficients multiplied by the powers of a and b.
In this case, we have the expression (a + 3)^4. To expand it, we need to find the coefficients of each term. We can use Pascal's Triangle to find these coefficients.
Pascal's Triangle is a triangular arrangement of numbers in which each number is the sum of the two numbers directly above it.
The fourth row of Pascal's Triangle is 1 4 6 4 1. These numbers represent the coefficients of the terms in the expansion of (a + 3)^4.
In the expansion of (a + 3)^4, the powers of a and 3 follow a pattern. Starting from the highest power of a (a^4) and decreasing by 1 for each subsequent term until we reach a^0, the powers of a are 4, 3, 2, 1, 0.
Similarly, the powers of 3 increase from 0 to 4. So, the powers of 3 are 0, 1, 2, 3, 4.
Using the coefficients from Pascal's Triangle and the powers of a and 3, we can write out the expanded polynomial:
(a + 3)^4 = 1*a^4 + 4*a^3*(3) + 6*a^2*(3^2) + 4*a^1*(3^3) + 1*(3^4)
The fourth term in this expansion, when written in standard form, is 4*a^1*(3^3), which simplifies to: 108a.
Therefore, the fourth term in the expansion of (a + 3)^4 is 108a.