Question

Find the product:

(x – 2y + 5z) (x² + 4y² + 25z² + 2xy + 10yz – 5xz)

Answer 1

Given: (x – 2y + 5z)(x² + 4y² + 25z² + 2xy + 10yz – 5xz)

Step-wise calculation:

1. Let the first expression be A = x – 2y + 5z.

2. The second expression inside parentheses is:

B = x² + 4y² + 25z² + 2xy + 10yz – 5xz

3. Multiply A by each term in B: 

(x – 2y + 5z)(x²) = x³ – 2yx² + 5zx², 

(x – 2y + 5z)(4y²) = 4xy² – 8y³ + 20y²z, 

(x – 2y + 5z)(25z²) = 25xz² – 50yz² + 125z³, 

(x – 2y + 5z)(2xy) = 2x²y – 4y²x + 10yzx, 

(x – 2y + 5z)(10yz) = 10xyz – 20y²z + 50yz², 

(x – 2y + 5z)(–5xz) = –5x²z + 10yxz – 25z²x.

4. Now add all these terms together:

x³ – 2yx² + 5zx² + 4xy² – 8y³ + 20y²z + 25xz² – 50yz² + 125z³ + 2x²y – 4y²x + 10yzx + 10xyz – 20y²z + 50yz² – 5x²z + 10yxz – 25z²x.

5. Group like terms be cautious with variables and multiplication order, but remember multiplication is commutative over real numbers:

• (x³) term: (x³).
• (x²y) terms: (–2yx² + 2x²y = 0) they cancel out.
• (x²z ) terms: (5zx² – 5x²z = 0) they cancel out.
• (xy²) terms: (4xy² – 4y²x = 0) cancel out.
• (y³) term: (–8y³).
• (y²z) terms: (20y²z – 20y²z = 0) cancel out.
• (xyz) terms: (10yzx + 10xyz + 10yxz = 30xyz).
• (xz²) terms: (25xz² – 25z²x = 0) cancel out.
• (yz²) terms: (–50yz² + 50yz² = 0) cancel out.
• (z³) term: (125z³).

6. After cancellations, the expression reduces to x³ – 8y³ + 30xyz + 125z³.