The perimeter of a rectangle is 16x³ - 6x² + 12x + 4. If one of its sides is 8x² + 3x, find the other side.

Given:

Perimeter of a rectangle = 16x³ - 6x² + 12x + 4

Length (Known side) = 8x² + 3x

Breadth = ?

We know the formula,

Perimeter of a rectangle = 2(Length + Breadth)

$\Rightarrow \text {Breadth} = \dfrac{\text{Perimeter - 2(Length)}}{2}$

Multiply Length by 2:

2(Length) = 2(8x² + 3x) = 16x² + 6x

Now, subtract 2(Length) from Perimeter:

$\begin{array}{rcccccc} 16x^3 & - & 6x^2 & + & 12x & + & 4 \\ +0 & + & 16x^2 & + & 6x & + & 0 \\ -\phantom{0} & - & & - & & - \\ \hline 16x^3 & - & 22x^2 & + & 6x & + & 4 \\ \hline \end{array}$

Now we have:

Breadth = $\dfrac{16x^3 - 22x^2 + 6x + 4}{2}$ = 8x³ - 11x² + 3x + 2

∴ The other side of the rectangle is 8x³ - 11x² + 3x + 2.