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Use Pascal's triangle to expand the binomial \((x - 5y)^5\).

1. \(x^5 - 5x^4(5y)\)
2. \(\binom{5}{2}x^3(5y)^2 - \binom{5}{3}x^2(5y)^3\)
3. \(\binom{5}{4}x(5y)^4 - (5y)^5\)

Now, simplify each term:

1. \(x^5 - 25x^4y\)
2. \(+ 100x^3y^2 - 250x^2y^3\)
3. \(+ 500xy^4 - 3125y^5\)

Thus, the expanded form is:

\(x^5 - 25x^4y + 100x^3y^2 - 250x^2y^3 + 500xy^4 - 3125y^5\)

Answer :

Final answer:

To expand the binomial (x - 5y)⁵ using Pascal's triangle, we use the fifth row of Pascal's triangle and apply the binomial theorem, giving us the expanded form x⁵ - 5x⁴y + 10x³y² - 10x²y³ + 5xy⁴ - y⁵.

Explanation:

To use Pascal's triangle to expand the binomial (x - 5y)⁵, we look at the row of Pascal's triangle corresponding to the exponent, which is 5.

The row for n=5 is 1, 5, 10, 10, 5, 1. We use the binomial theorem to expand the expression, with each term taking coefficients from Pascal's triangle, the variable x decreasing in power, and the variable y increasing in power, all multiplied appropriately.

  • The first term is x⁵.
  • The second term is – 5 times x⁴ times y (since the sign alternates).
  • The third term is 10x³y² (note the square of -5y is 25y²).
  • The fourth term is – 10x²y³.
  • The fifth term is 5xy⁴.
  • The final term is – y⁵.

Combining all the terms gives us the expanded form:

(x - 5y)⁵ = x⁵ - 5x⁴y + 10x³y² - 10x²y³ + 5xy⁴ - y⁵

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