Question

The product $(3x+2y+1)(x+4y+5)(7x-y+2)$ is expanded. What is the sum of the coefficients of the terms which contain a nonzero power of $y$?

Answer 1

264

Explanation:

To find the sum of the coefficients of the terms that contain a non-zero power of y in the expanded expression (3x + 2y + 1)(x + 4y + 5)(7x - y + 2), we first need to expand the expression and then identify the terms with non-zero powers of y.

First, expand the first two factors:

`\begin{aligned}&\phantom{w.}(3x+2y+1)(x+4y+5) \\\\&= 3x(x+4y+5) + 2y(x+4y+5) + 1(x+4y+5)\\\\&=3x^2+12xy+15x+2xy+8y^2+10y+x+4y+5\\\\&=3x^2+14xy+16x+8y^2+14y+5\end{aligned}`

Now, multiply this expression with the third factor (7x - y + 2):

`\begin{aligned}&\phantom{w..}(3x^2+14xy+16x+8y^2+14y+5)(7x - y + 2)\\\\&=21x^3-3x^2y+6x^2+98x^2y-14xy^2+28xy+112x^2-16xy+32x\\&\;\;\;\;+56xy^2-8y^3+16y^2+98xy-14y^2+28y+35x-5y+10\\\\&=21x^3+118x^2+67x+95x^2y+42xy^2+110xy-8y^3+2y^2+23y+10\end{aligned}`

Identify the terms with non-zero powers of y:

• 
  `95x^2y`
• 
  `42xy^2`
• 
  `110xy`
• 
  `-8y^3`
• 
  `2y^2`
• 
  `23y`

Sum the coefficients of these terms:

`95+42+110-8+2+23=264`

Therefore, the sum of the coefficients of the terms with non-zero powers of y is 264.