Question

The side of a square measures (3x − 6) units. Part A: What is the expression that represents the area of the square? Show your work to receive full credit. (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)

Answer 1

Given: Side of the Square = ( 3x - 6 ) units

Part A).

Area of the Square = Side × Side = ( 3x - 6 )( 3x - 6 ) = ( 3x - 6 )²

= ( 3x )² + 6² - 2 × 3x × 6

= 9x² + 36 - 36x

Part B).

Degree of a polynomial is the height power of the polynomial.

Degree of the 9x² + 36 - 36x is 2.

Since Polynomial representing Area of the square has 3 terms and degree = 2.

So the Polynomial is a Trinomial polynomial and Quadratic polynomial.

Part C).

Closure Property of polynomial is always under multiplication means when two polynomials are multiplies then the resulting product is also a polynomial.

Here, Also two linear polynomial are multiplied and we get a quadratic polynomial.

Answer 2

Given: The side of a square is (3x - 6) units.
Note that (a + b)² = a² + 2ab + b².

Part A
The area of the square is
A = (3x - 6)(3x - 6)
= (3x)² + 2(3x)(-6) + (-6)²
= 9x² - 36x + 36

Part B
The area is a 2nd-degree polynomial, or a quadratic function, or parabola.

Part C
A 2nd-degree polynomial is of the form
f(x) = ax² + bx¹ + cx⁰ = ax² + bx + c
where a,b,c are constants (coefficients of the polynomial).
The area obtained in Part A has the coefficients
a = 9, b = -36, c = 36.

The polynomial is closed because it is completely defined by multiply and addition operations.