Mathematico | Mathematico

A window is made from a wooden frame of thickness $1 - x$ and $9$ square glass panes of width $10 + x$ , where $0 < x < 1$ .

Window diagram

Find a fully factorised expression for the total area of the wooden frame only.

The expression can be given in the form

- k (x + a) (x - b), a, b, k \in N

Give the value of

(a + 1) (b + 1) (k + 1)

Hint

The total height of the square window is

4 (1 - x) + 3 (10 + x)

and so the total area of the window is this value squared.

If we subtract the $9$ glass panes, we are left with the area of the wooden frame.

Hint

After expanding and simplifying everything, you should get

- 8 x^{2} - 248 x + 256

Remove a common factor of $- 8$ before continuing with the factorisation.

Solution

The total height of the square window is

4 (1 - x) + 3 (10 + x) = 34 - x

and so the total area of the window is

(34 - x)^{2} = x^{2} - 68 x + 1156

To get the area of the wooden frame only, we subtract the $9$ glass panels:

\begin{aligned} x^{2} - 68 x + 1156 - 9 (10 + x)^{2} \\ = & x^{2} - 68 x + 1156 - 9 (x^{2} + 20 x + 100) \\ = & - 8 x^{2} - 248 x + 256 \\ = & - 8 (x^{2} + 31 x - 32) \\ = & - 8 (x + 32) (x - 1) \end{aligned}