%5E4.jpeg "(2x-5y)^4")
Expanding (2x - 5y)^4 using the Binomial Theorem
The expression (2x - 5y)^4 can be expanded using the Binomial Theorem. This theorem provides a formula to expand any binomial raised to a positive integer power.
Here's how it works:
The Binomial Theorem:
For any real numbers a and b, and any non-negative integer n, we have:
(a + b)^n = <sup>n</sup>C<sub>0</sub> a<sup>n</sup> b<sup>0</sup> + <sup>n</sup>C<sub>1</sub> a<sup>n-1</sup> b<sup>1</sup> + <sup>n</sup>C<sub>2</sub> a<sup>n-2</sup> b<sup>2</sup> + ... + <sup>n</sup>C<sub>n-1</sub> a<sup>1</sup> b<sup>n-1</sup> + <sup>n</sup>C<sub>n</sub> a<sup>0</sup> b<sup>n</sup>
where <sup>n</sup>C<sub>r</sub> represents the binomial coefficient, calculated as:
<sup>n</sup>C<sub>r</sub> = n! / (r! * (n-r)!)
Applying the Binomial Theorem to (2x - 5y)^4:
- Identify a and b: In our case, a = 2x and b = -5y.
- Determine n: The power is 4, so n = 4.
- Calculate binomial coefficients: We need to calculate <sup>4</sup>C<sub>0</sub>, <sup>4</sup>C<sub>1</sub>, <sup>4</sup>C<sub>2</sub>, <sup>4</sup>C<sub>3</sub>, and <sup>4</sup>C<sub>4</sub>.
- <sup>4</sup>C<sub>0</sub> = 4! / (0! * 4!) = 1
- <sup>4</sup>C<sub>1</sub> = 4! / (1! * 3!) = 4
- <sup>4</sup>C<sub>2</sub> = 4! / (2! * 2!) = 6
- <sup>4</sup>C<sub>3</sub> = 4! / (3! * 1!) = 4
- <sup>4</sup>C<sub>4</sub> = 4! / (4! * 0!) = 1
- Substitute values and simplify:
(2x - 5y)^4 = 1 * (2x)<sup>4</sup> (-5y)<sup>0</sup> + 4 * (2x)<sup>3</sup> (-5y)<sup>1</sup> + 6 * (2x)<sup>2</sup> (-5y)<sup>2</sup> + 4 * (2x)<sup>1</sup> (-5y)<sup>3</sup> + 1 * (2x)<sup>0</sup> (-5y)<sup>4</sup>
Simplifying the expression:
(2x - 5y)^4 = 16x<sup>4</sup> + 4 * 8x<sup>3</sup> * (-5y) + 6 * 4x<sup>2</sup> * 25y<sup>2</sup> + 4 * 2x * (-125y<sup>3</sup>) + 625y<sup>4</sup>
(2x - 5y)^4 = 16x<sup>4</sup> - 160x<sup>3</sup>y + 600x<sup>2</sup>y<sup>2</sup> - 1000xy<sup>3</sup> + 625y<sup>4</sup>
Therefore, the expanded form of (2x - 5y)^4 is 16x<sup>4</sup> - 160x<sup>3</sup>y + 600x<sup>2</sup>y<sup>2</sup> - 1000xy<sup>3</sup> + 625y<sup>4</sup>.