Question

Use Pascal's triangle to expand (x² - 3y)⁴(x - 3y)⁴

a) x⁸ - 12x⁶y + 54x⁴y² - 81x²y³ + 27y⁴
b) x⁸ - 24x⁶y + 108x⁴y² - 162x²y³ + 81y⁴
c) x⁸ - 18x⁶y + 81x⁴y² - 121x²y³ + 64y⁴
d) x⁸ - 16x⁶y + 72x⁴y² - 108x²y³ + 36y⁴

Answer 1

To expand (x² - 3y)⁴(x - 3y)⁴, we use Pascal's triangle to determine the coefficients and apply them to each term of the binomial expansions. The correct expansion is b) x⁸ - 12x⁶y + 54x⁴y² - 108x²y³ + 81y⁴.

Step-by-step explanation:

The student is asking for the expanded form of the expression (x² - 3y)⁴(x - 3y)⁴ using Pascal's triangle. To find the coefficient of each term, we can look at the 4th row of Pascal's triangle which is 1, 4, 6, 4, 1. We will then raise each part of the binomials to the appropriate power and multiply by the corresponding coefficient.

For the first binomial (x² - 3y)⁴, the expanded form will be:

• x⁸
• - 4(x⁶)(3y)
• + 6(x⁴)(3y)²
• - 4(x²)(3y)³
• + (3y)⁴

For the second binomial (x - 3y)⁴, we can treat it as already expanded since it's raised to the power of 1.

Therefore, the expanded form combining both binomials, and simplifying will result in:

• x⁸
• - 12x⁶y
• + 54x⁴y²
• - 108x²y³
• + 81y⁴

So the correct expanded form is x⁸ - 12x⁶y + 54x⁴y² - 108x²y³ + 81y⁴, which is option b).