%5E3-expand-formula.jpeg "(x+1)^3 Expand Formula")
Expanding the Formula: (x+1)^3
When working with algebraic expressions, expanding formulas is an essential skill to master. In this article, we will explore the expansion of the formula (x+1)^3.
The Binomial Theorem
To expand the formula (x+1)^3, we will use the Binomial Theorem, which states that:
$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
where $\binom{n}{k}$ is the binomial coefficient, which can be calculated as:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
Expanding (x+1)^3
To expand (x+1)^3, we can plug in the values a = x, b = 1, and n = 3 into the Binomial Theorem formula:
$(x+1)^3 = \sum_{k=0}^3 \binom{3}{k} x^{3-k} 1^k$
Calculating the Binomial Coefficients
Using the formula for binomial coefficients, we can calculate the values of $\binom{3}{k}$:
$\binom{3}{0} = \frac{3!}{0!(3-0)!} = 1$
$\binom{3}{1} = \frac{3!}{1!(3-1)!} = 3$
$\binom{3}{2} = \frac{3!}{2!(3-2)!} = 3$
$\binom{3}{3} = \frac{3!}{3!(3-3)!} = 1$
Expanding the Formula
Now, we can plug in the calculated values of the binomial coefficients into the formula:
$(x+1)^3 = \binom{3}{0} x^{3-0} 1^0 + \binom{3}{1} x^{3-1} 1^1 + \binom{3}{2} x^{3-2} 1^2 + \binom{3}{3} x^{3-3} 1^3$
Simplifying the Expression
Simplifying the expression, we get:
$(x+1)^3 = x^3 + 3x^2 + 3x + 1$
Conclusion
And that's it! We have successfully expanded the formula (x+1)^3 using the Binomial Theorem. The resulting expression is:
$(x+1)^3 = x^3 + 3x^2 + 3x + 1$
This formula can be used in various algebraic manipulations and applications.
