Question 1206241
your looking at (2x + 2) ^ 4


this is the same as (2x + 2) ^ 2 * (2x + 2) ^ 2


(2x + 2) ^ 2 equals:
(2x + 2) * (2x + 2) which equals:
2x * (2x + 2) + 2 * (2x + 2) which equals:
4x^2 + 4x + 4x + 4 which equals:
4x^2 + 8x + 4.


(4x^2 + 8x + 4) ^ 2 equals:
(4x^2 + 8x + 4) * (4x^2 + 8x + 4) which equals:
4x^2 * (4x^2 + 8x + 4) + 8x * (4x^2 + 8x + 4) + 4 * (4x^2 + 8x + 4).


4x^2 * (4x^2 + 8x + 4) = 16x^4 + 32x^3 + 16x^2.
8x * (4x^2 + 8x + 4) = 32x^3 + 64x^2 + 32x.
4 * (4x^2 + 8x + 4) = 16x^2 + 32x + 16.


total is 16x^4 + 32x^3 + 16x^2 + 32x^3 + 64x^2 + 32x + 16x^2 + 32x + 16.


combine like terms to get:


16x^4 + 64x^3 + 96x^2 + 64x + 16


i used an online binomial expansion calculator to confirm.
it gave me the same answer.
see below.


<img src = "http://theo.x10hosting.com/2024/022501.jpg">


the correct answer is 96, but that's the coefficient of the x^2 term, not the x term.


here's my worksheet from the formula.


<img src = "http://theo.x10hosting.com/2024/022502.jpg">


here's the calculator.


<a href = "https://www.symbolab.com/solver/binomial-expansion-calculator/expand%20%5Cleft(2x%2B2%5Cright)%5E%7B4%7D?or=input" target = "_blank">https://www.symbolab.com/solver/binomial-expansion-calculator/expand%20%5Cleft(2x%2B2%5Cright)%5E%7B4%7D?or=input</a>



here's some references.


<a href = "https://www.le.ac.uk/users/dsgp1/COURSES/MATHSTAT/2binome.pdf" target = "_blank">https://www.le.ac.uk/users/dsgp1/COURSES/MATHSTAT/2binome.pdf</a>


<a href = "https://www.math10.com/en/algebra/probabilities/binomial-theorem/binomial-theorem.html#:~:text=The%20Binomial%20Theorem%20Using%20Pascal's%20Triangle&text=(a%20%2B%20b)n%20%3D,st%20row%20of%20Pascal's%20triangle." target = "_blank">https://www.math10.com/en/algebra/probabilities/binomial-theorem/binomial-theorem.html#:~:text=The%20Binomial%20Theorem%20Using%20Pascal's%20Triangle&text=(a%20%2B%20b)n%20%3D,st%20row%20of%20Pascal's%20triangle.</a>