%5E3-expand.jpeg "(x+1)^3 Expand")
Expanding (x+1)^3
In algebra, expanding a binomial expression like (x+1)^3 means multiplying it out to get a polynomial expression. In this article, we will learn how to expand (x+1)^3 using the binomial theorem.
What is the Binomial Theorem?
The binomial theorem is a mathematical formula that describes the expansion of powers of a binomial expression. It states that:
$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k}b^k$
where $a$ and $b$ are terms, $n$ is a positive integer, and $\binom{n}{k}$ is the binomial coefficient.
Expanding (x+1)^3
To expand (x+1)^3, we can use the binomial theorem with $a=x$, $b=1$, and $n=3$. Then, we get:
$(x+1)^3 = \sum_{k=0}^3 \binom{3}{k} x^{3-k}1^k$
Now, let's calculate each term:
Term 1: k=0
$\binom{3}{0} x^{3-0}1^0 = 1x^3 = x^3$
Term 2: k=1
$\binom{3}{1} x^{3-1}1^1 = 3x^2 \cdot 1 = 3x^2$
Term 3: k=2
$\binom{3}{2} x^{3-2}1^2 = 3x^1 \cdot 1^2 = 3x$
Term 4: k=3
$\binom{3}{3} x^{3-3}1^3 = 1x^0 \cdot 1^3 = 1$
The Final Expansion
Now, we can add up all the terms to get the final expansion of (x+1)^3:
$(x+1)^3 = x^3 + 3x^2 + 3x + 1$
And that's it! We have successfully expanded (x+1)^3 using the binomial theorem.
