Question

Expand the following:

(x – 2y + 3)³

Answer 1

Given: Expand (x – 2y + 3)³

Step-wise calculation:

Let z = x – 2y.

Then the expression becomes (z + 3)³.

Using the binomial expansion formula for (a + b)³ = a³ + 3a²b + 3ab² + b³, we have:

(z + 3)³ = z³ + 3z² × 3 + 3z × 3² + 3³

(z + 3)³ = z³ + 9z² + 27z + 27

Substitute back (z = x – 2y):

= (x – 2y)³ + 9(x – 2y)² + 27(x – 2y) + 27

Now expand each term:

1. Expand (x – 2y)³: 

= x³ – 3x²(2y) + 3x(2y)² – (2y)³

= x³ – 6x²y + 12xy² – 8y³

2. Expand (x – 2y)²: 

= x² – 2 × x × 2y + (2y)² 

= x² – 4xy + 4y²

Multiply by 9:

9(x² – 4xy + 4y²) = 9x² – 36xy + 36y²

3. Expand 27(x – 2y): 

= 27x – 54y

4. Constant term: (27)

Adding all parts together gives:

x³ – 6x²y + 12xy² – 8y³ + 9x² – 36xy + 36y² + 27x – 54y + 27