Question

a) let ( p(x) ) represent the perimeter of the rectangle on the right.

1. determine ( p(x) ).
2. what is the degree of ( p(x) )?

b) let ( a(x) ) represent the area of the rectangle.

1. determine ( a(x) ).
2. what is the degree of ( a(x) )?

c) determine the perimeter and area of the rectangle when ( x = 6 mathrm{~cm} ).

1. after calculating the dimensions of the rectangle.
2. by evaluating the polynomials ( p(x) ) and ( a(x) ).

Explanation:

Step1: Determine the perimeter formula

The perimeter \(P(x)\) of a rectangle is \(P(x)=2\times(\text{length}+\text{width})\). Here, length \(l = 3x - 2\) and width \(w=2x + 3\). So \(P(x)=2\times((3x - 2)+(2x + 3))\).
Simplify the expression inside the parentheses: \((3x - 2)+(2x + 3)=3x+2x-2 + 3=5x + 1\). Then \(P(x)=2\times(5x + 1)=10x+2\). The degree of a polynomial \(a_nx^n+\cdots+a_1x + a_0\) is \(n\). For \(P(x)=10x + 2\), the degree is \(1\).

Step2: Determine the area formula

The area \(A(x)\) of a rectangle is \(A(x)=\text{length}\times\text{width}\). So \(A(x)=(3x - 2)\times(2x + 3)\).
Use the FOIL method: \((3x)\times(2x)+(3x)\times3+(- 2)\times(2x)+(-2)\times3\).
\(=6x^{2}+9x-4x - 6=6x^{2}+5x - 6\). The degree of \(A(x)=6x^{2}+5x - 6\) is \(2\).

Step3: Calculate dimensions when \(x = 6\)

Length \(l=3x - 2\), substitute \(x = 6\): \(l=3\times6-2=18 - 2=16\space\text{cm}\).
Width \(w=2x + 3\), substitute \(x = 6\): \(w=2\times6+3=12 + 3=15\space\text{cm}\).

Step4: Calculate perimeter when \(x = 6\)

Substitute \(x = 6\) into \(P(x)=10x+2\), \(P(6)=10\times6+2=60 + 2=62\space\text{cm}\).

Step5: Calculate area when \(x = 6\)

Substitute \(x = 6\) into \(A(x)=6x^{2}+5x - 6\). \(A(6)=6\times6^{2}+5\times6-6\).
First, \(6\times6^{2}=6\times36 = 216\), \(5\times6=30\). Then \(A(6)=216+30-6=240\space\text{cm}^2\).

Answer:

a)

1. \(P(x)=10x + 2\)
2. Degree of \(P(x)\) is \(1\)

b)

1. \(A(x)=6x^{2}+5x - 6\)
2. Degree of \(A(x)\) is \(2\)

c)

1. Length \(=16\space\text{cm}\), Width \(=15\space\text{cm}\)
2. Perimeter \(P(6)=62\space\text{cm}\), Area \(A(6)=240\space\text{cm}^2\)