%5E3-expanded.jpeg "(2x+4)^3 Expanded")
Expanded Form of (2x+4)^3
In this article, we will discuss the expanded form of the cubic expression (2x+4)^3. To find the expanded form, we will use the binomial theorem and the formula for the cube of a binomial.
Binomial Theorem
The binomial theorem states that for any positive integer n, the expression (a+b)^n can be expanded as:
$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
where n is a positive integer, a and b are real numbers, and \binom{n}{k} is the binomial coefficient.
Expanding (2x+4)^3
To expand (2x+4)^3, we will use the binomial theorem with a = 2x, b = 4, and n = 3. Substituting these values into the formula, we get:
$(2x+4)^3 = \sum_{k=0}^3 \binom{3}{k} (2x)^{3-k} 4^k$
Calculating the Binomial Coefficients
To calculate the binomial coefficients, we can use the formula:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
where n! is the factorial of n. For n = 3 and k = 0, 1, 2, 3, we get:
$\binom{3}{0} = \frac{3!}{0!(3-0)!} = \frac{6}{6} = 1$ $\binom{3}{1} = \frac{3!}{1!(3-1)!} = \frac{6}{2} = 3$ $\binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{6}{4} = 3$ $\binom{3}{3} = \frac{3!}{3!(3-3)!} = \frac{6}{6} = 1$
Expanding the Expression
Now, we can substitute the binomial coefficients into the formula:
$(2x+4)^3 = 1(2x)^3(4)^0 + 3(2x)^2(4)^1 + 3(2x)^1(4)^2 + 1(2x)^0(4)^3$
Simplifying the expression, we get:
$(2x+4)^3 = 8x^3 + 24x^2(4) + 12x(16) + 64$
Final Answer
The expanded form of (2x+4)^3 is:
$(2x+4)^3 = \boxed{8x^3 + 96x^2 + 192x + 64}$
This is the final answer.
