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Applications of polynomials

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Applications of Polynomials

Polynomials model real-world quantities such as area, volume, and cost. Learn how to translate a word problem into a polynomial expression, using an area example, and how to solve it by evaluating or solving the polynomial for an unknown.

Polynomials model the real world

Polynomials aren't just symbols — they model quantities that grow or shrink in predictable ways: area, volume, cost, and distance. An application of polynomials is any situation where a real quantity is written as a polynomial expression so you can solve for an unknown.

A real-world polynomial: garden area A rectangular garden of length x+8 and width x+5 has area modeled by the polynomial (x+8)(x+5), which expands to x squared plus 13x plus 40. Garden length = x + 8 width = x + 5 Area = (x + 8)(x + 5) = x² + 13x + 40
The area of a garden with length x+8 and width x+5 is the polynomial x²+13x+40.

Area and volume

The most common application is area: multiplying two polynomial side lengths (as above) gives a polynomial area. The same idea extends to volume, where three polynomial dimensions multiply together. Setting up these expressions uses the same skill as multiplying binomial by binomial.

Setting up the polynomial

The key step is translating a word problem into an expression: identify the unknown (usually x), write each quantity in terms of x, then combine them with the correct operation. If a garden's length is 8 more than its width, and width is x, the length is x + 8.

Solving with the polynomial

Once you have the polynomial, you can substitute a known value to find the total (see evaluating polynomials), or set it equal to a target value and solve for x (see solving polynomial equations). For the garden above, if the actual area is 54 square units, you'd solve x² + 13x + 40 = 54 for x.

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