%5E5.jpeg "(2x-5y)^5")
Expanding (2x-5y)^5 using the Binomial Theorem
The binomial theorem is a powerful tool for expanding expressions of the form (a + b)<sup>n</sup>. Let's use it to expand (2x - 5y)<sup>5</sup>.
The Binomial Theorem
The binomial theorem states:
(a + b)<sup>n</sup> = <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> is the binomial coefficient, calculated as:
<sup>n</sup>C<sub>r</sub> = n! / (r! * (n-r)!)
Applying the Theorem to (2x-5y)<sup>5</sup>
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Identify a and b: In our case, a = 2x and b = -5y.
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Determine n: n = 5.
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Calculate the binomial coefficients:
- <sup>5</sup>C<sub>0</sub> = 1
- <sup>5</sup>C<sub>1</sub> = 5
- <sup>5</sup>C<sub>2</sub> = 10
- <sup>5</sup>C<sub>3</sub> = 10
- <sup>5</sup>C<sub>4</sub> = 5
- <sup>5</sup>C<sub>5</sub> = 1
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Substitute values into the binomial theorem:
(2x - 5y)<sup>5</sup> = 1 (2x)<sup>5</sup> (-5y)<sup>0</sup> + 5 (2x)<sup>4</sup> (-5y)<sup>1</sup> + 10 (2x)<sup>3</sup> (-5y)<sup>2</sup> + 10 (2x)<sup>2</sup> (-5y)<sup>3</sup> + 5 (2x)<sup>1</sup> (-5y)<sup>4</sup> + 1 (2x)<sup>0</sup> (-5y)<sup>5</sup>
- Simplify:
(2x - 5y)<sup>5</sup> = 32x<sup>5</sup> - 800x<sup>4</sup>y + 8000x<sup>3</sup>y<sup>2</sup> - 50000x<sup>2</sup>y<sup>3</sup> + 156250xy<sup>4</sup> - 3125y<sup>5</sup>
Final Result
Therefore, the expanded form of (2x - 5y)<sup>5</sup> is 32x<sup>5</sup> - 800x<sup>4</sup>y + 8000x<sup>3</sup>y<sup>2</sup> - 50000x<sup>2</sup>y<sup>3</sup> + 156250xy<sup>4</sup> - 3125y<sup>5</sup>.