%5E5-binomial-theorem.jpeg "(x+3)^5 Binomial Theorem")
The Binomial Theorem: Expanding (x+3)^5
The binomial theorem is a fundamental concept in algebra that provides a formula for expanding powers of a binomial expression. In this article, we will explore the binomial theorem and its application in expanding the expression (x+3)^5.
What is the Binomial Theorem?
The binomial theorem is a mathematical formula that describes the algebraic expansion of powers of a binomial expression. It states that for any positive integer n, the following equation holds:
(a + b)^n = ∑(n choose k) * a^(n-k) * b^k
where n is a positive integer, a and b are real numbers, and the summation is taken over all values of k from 0 to n.
Expanding (x+3)^5 using the Binomial Theorem
Now, let's apply the binomial theorem to expand the expression (x+3)^5. Using the formula above, we can write:
(x+3)^5 = ∑(5 choose k) * x^(5-k) * 3^k
where k takes on values from 0 to 5.
Calculating the Terms
To expand the expression, we need to calculate the values of the terms for each value of k.
k = 0
(5 choose 0) * x^5 * 3^0 = 1 * x^5 * 1 = x^5
k = 1
(5 choose 1) * x^4 * 3^1 = 5 * x^4 * 3 = 15x^4
k = 2
(5 choose 2) * x^3 * 3^2 = 10 * x^3 * 9 = 90x^3
k = 3
(5 choose 3) * x^2 * 3^3 = 10 * x^2 * 27 = 270x^2
k = 4
(5 choose 4) * x^1 * 3^4 = 5 * x^1 * 81 = 405x
k = 5
(5 choose 5) * x^0 * 3^5 = 1 * 1 * 243 = 243
The Expanded Expression
Now that we have calculated the terms, we can write the expanded expression as:
(x+3)^5 = x^5 + 15x^4 + 90x^3 + 270x^2 + 405x + 243
This is the expanded form of the expression (x+3)^5 using the binomial theorem.
Conclusion
In this article, we have explored the binomial theorem and its application in expanding the expression (x+3)^5. By applying the formula and calculating the terms, we were able to obtain the expanded expression. The binomial theorem is a powerful tool in algebra that can be used to expand powers of binomial expressions, and its applications are numerous in mathematics and other fields.
