(2x+1)^3
what is your question in this problem ?
recall the multipliers of Pascal's Triangle: 1 3 3 1
Now use those in the expansion.
To expand the expression (2x+1)^3, we can use the binomial theorem or the distributive property. Let's use the distributive property to expand it step by step.
First, recall the formula for expanding a binomial raised to a power:
(a + b)^n = C(n,0) * a^n * b^0 + C(n,1) * a^(n-1) * b^1 + C(n,2) * a^(n-2) * b^2 + ... + C(n,n) * a^0 * b^n
Here, a = 2x and b = 1, and n = 3.
Now, let's substitute these values into the formula:
(2x + 1)^3 = C(3,0) * (2x)^3 * 1^0 + C(3,1) * (2x)^2 * 1^1 + C(3,2) * (2x)^1 * 1^2 + C(3,3) * (2x)^0 * 1^3
Next, we'll determine the values of the binomial coefficients using the combination formula:
C(n,r) = n! / (r! * (n-r)!)
In our case, n = 3 and r takes the values of 0, 1, 2, and 3. Calculating these coefficients, we have:
C(3,0) = 3! / (0! * (3-0)!) = 1
C(3,1) = 3! / (1! * (3-1)!) = 3
C(3,2) = 3! / (2! * (3-2)!) = 3
C(3,3) = 3! / (3! * (3-3)!) = 1
Let's substitute these values into the expanded expression:
(2x + 1)^3 = 1 * (2x)^3 * 1^0 + 3 * (2x)^2 * 1^1 + 3 * (2x)^1 * 1^2 + 1 * (2x)^0 * 1^3
Simplifying further, we have:
(2x + 1)^3 = 8x^3 + 12x^2 + 6x + 1
Therefore, the expanded form of (2x+1)^3 is 8x^3 + 12x^2 + 6x + 1.