Applications of Polynomials

The box-cutting problem is the most common applied polynomial in HSC Extension 1. Here is the full worked solution for a 30 cm sheet.

Problem: A square sheet of side 30 cm has squares of side $x$ cut from each corner. The sides are folded up to make an open box. Find $x$ for a volume of 1000 cm³.

Setting up the model:

• Let $x$ = side length of cut-out square (cm).
• Physical constraint: $0 < x < 15$ (since $30 - 2x > 0 \Rightarrow x < 15$).
• Base side after cutting: $30 - 2x$; height: $x$.
• Volume: $V = x(30-2x)^2 = 1000$.

Solving:

$x(900 - 120x + 4x^2) = 1000$

$4x^3 - 120x^2 + 900x - 1000 = 0$

$x^3 - 30x^2 + 225x - 250 = 0$   (divide by 4)

Test $x = 10$: $1000 - 3000 + 2250 - 250 = 0$ ✓

Factorise: $(x - 10)(x^2 - 20x + 25) = 0$

$x = 10$ or $x = \dfrac{20 \pm \sqrt{400 - 100}}{2} = 10 \pm 5\sqrt{3}$

Checking constraints ($0 < x < 15$):

• $x = 10$: valid ✓
• $x = 10 + 5\sqrt{3} \approx 18.7$: exceeds 15 → reject ✗
• $x = 10 - 5\sqrt{3} \approx 1.34$: valid ✓

Box cutting formula: V = x(L - 2x)^2 where L is the sheet side and x is the cut-out size; Always state the constraint 0 < x < L/2 before solving

Pause, copy the box-cutting formula into your book: $V = x(L-2x)^2$ where $L$ is the sheet side and $x$ is the cut-out size; always state the domain constraint $0 < x < L/2$ before solving.

We just saw the box-cutting problem: $V = x(L-2x)^2$ with constraint $0 < x < L/2$. That raises a question: in any polynomial modelling problem, what general steps do you follow from the problem description to the final answer? This card answers it → (1) define the variable; (2) write the polynomial; (3) solve; (4) check constraints; (5) interpret the result in context.

Polynomials appear in many real-world contexts beyond geometry. When a polynomial models a quantity (profit, height, population), the roots correspond to key moments, zero crossings, in that quantity's behaviour.

Example: The polynomial $P(x) = x^3 - 6x^2 + 11x - 6$ models the profit (in thousands of dollars) of a company $x$ years after startup. When was the profit zero?

Solution:

$P(1) = 1 - 6 + 11 - 6 = 0$ → $(x-1)$ is a factor.

Divide: $P(x) = (x-1)(x^2 - 5x + 6) = (x-1)(x-2)(x-3)$

Roots: $x = 1, 2, 3$ years.

Interpretation: The profit was zero at 1, 2, and 3 years after startup.

Modelling steps: define variable → write polynomial equation → solve → check constraints → interpret; Roots of the model equation correspond to zeros of the modelled quantity

Pause, copy the polynomial modelling steps into your book: (1) define variable; (2) write polynomial equation; (3) solve; (4) check constraints (domain); (5) interpret, roots correspond to zeros of the modelled quantity.