In Exercises 53–54, write a polynomial that represents the length of each rectangle. Transcription: The area of the rectangle is 0.5x³ - 0.3x² + 0.22x + 0.06 square units and its width is x + 0.2 units

Verified step by step guidance

1. Recall the formula for the area of a rectangle: $\mathrm{Area}=\mathrm{Length}\backslash times\mathrm{Width}$.
2. You are given the area as $0.5x^3\; -\; 0.3x^2\; +\; 0.22x\; +\; 0.06$ and the width as $x\; +\; 0.2$.
3. Set up the equation for length by dividing the area by the width: $\mathrm{Length}=\; \backslash frac\{0.5x^3\; -\; 0.3x^2\; +\; 0.22x\; +\; 0.06\}\{x\; +\; 0.2\}$.
4. Perform polynomial division to divide $0.5x^3\; -\; 0.3x^2\; +\; 0.22x\; +\; 0.06$ by $x\; +\; 0.2$.
5. The quotient from the division will be the polynomial expression representing the length of the rectangle.

Key Concepts

Polynomial Expressions

A polynomial is an algebraic expression made up of terms consisting of variables raised to whole-number exponents and coefficients. Understanding how to manipulate polynomials, including addition, subtraction, multiplication, and division, is essential for working with expressions like the given area and width.

Area of a Rectangle

The area of a rectangle is found by multiplying its length by its width. Given the area and width, you can find the length by dividing the area polynomial by the width polynomial, which involves polynomial division.

Polynomial Division

Polynomial division is the process of dividing one polynomial by another, similar to long division with numbers. It is used here to find the length polynomial by dividing the area polynomial by the width polynomial.