Question

A rectangle has sides measuring (5x + 4) units and (3x + 2) units.

Part A: What is the expression that represents the area of the rectangle? Show your work to receive full credit. Use the equation editor. (4 points
Part B: What are the degree and classification of the expression obtained in Part A? (3 points)
Part C: How does Part A demonstrate the closure property for polynomials? (3 points) (10 points)

Answer 1

A)
`15x^2+22x+8`

B) Degree 2, Quartic Expression

C) The dimensions of the rectangle are polynomials. When multiplied together, the area of the rectangle is also a polynomial.

Explanation:

A) The formula for area of a rectangle is Area = L * W. The length and width are represented by (3x+2) and (5x+4). So we can say that

`Area = (5x+4)(3x+2)`

Use FOIL to multiply the the polynomials. First, Outside, Inside, Last

`Area = (5x)(3x) + (5x)(2) + (3x)(4) + (2)(4)\\Area = 15x^2 + 10x + 12x + 8\\Area = 15x^2 + 22x + 8`

The expression that represents the area of the rectangle is

`15x^2 + 22x + 8`.

B) The degree of the expression is 2, because two is the highest power of x. The classification of the expression is quadratic because the graph of the expression is a parabola.

Degrees vs. Classification

Degree 0: Zero Polynomial or Constant

Degree 1: Linear (line)

Degree 2: Quadratic (parabola)

Degree 3: Cubic

Degree 4: Quartic

...

C) Closure property for polynomials applies to addition, subtraction, and multiplication. It means that the result of multiplying two polynomials will also be a polynomial. Part A demonstrates polynomial closure under multiplication because the dimensions of the rectangle are polynomials and so is the area.