High School

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

Options:
A. \(y^5 - 5y^4x + 25y^3x^2 - 125y^2x^3 + 625yx^4 - 3125x^5\)

B. \(x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5\)

C. \(x^5 - 25x^4y + 250x^3y^2 - 1250x^2y^3 + 3125xy^4 - 3125y^5\)

D. \(x^5 - 5x^4y + 25x^3y^2 - 125x^2y^3 + 625xy^4 - 3125y^5\)

Answer :

Final answer:

To expand the binomial (x - 5y)⁵ using Pascal's triangle, we can use the binomial theorem. The correct expansion is option a. y⁵ - 5y⁴x + 25y³x² - 125y²x³ + 625yx⁴ - 3125x⁵.

Explanation:

To expand the binomial (x - 5y)⁵ using Pascal's triangle, we will use the binomial theorem. The binomial theorem states that (a + b)ⁿ can be expanded as aⁿ + (nC1) * aⁿ⁻¹ * b + (nC2) * aⁿ⁻² * b² + ... + bⁿ, where n is the exponent and nCk represents the binomial coefficient.

In this case, we have (x - 5y)⁵. To expand it, we will substitute x and y into the formula, using the binomial coefficient for each term.

The correct expansion is option a. y⁵ - 5y⁴x + 25y³x² - 125y²x³ + 625yx⁴ - 3125x⁵.

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