Question

a. write a polynomial that describes the total area of
the set of rectangles and squares shown on the right.
a. the polynomial that describes the total area is □.
(do not simplify.)
b. simplify the polynomial.

Explanation:

Step1: Calculate the area of each figure

• Area of the square with side \(x\): \(A_1 = x\times x=x^{2}\)
• Area of the square with side \(2x\): \(A_2=(2x)\times(2x) = 4x^{2}\)
• Area of the rectangle with sides \(x\) and \(4\): \(A_3=x\times4 = 4x\)
• Area of the rectangle with sides \(x\) and \(7\): \(A_4=x\times7=7x\)

Step2: Find the total area polynomial

The total area \(A=A_1 + A_2+A_3+A_4\)
Substitute the areas: \(A=x^{2}+4x^{2}+4x + 7x\)

Step3: Simplify the polynomial

Combine like - terms:
For the \(x^{2}\) terms: \(x^{2}+4x^{2}=(1 + 4)x^{2}=5x^{2}\)
For the \(x\) terms: \(4x+7x=(4 + 7)x=11x\)
So the simplified polynomial is \(5x^{2}+11x\)

Answer:

A. The polynomial that describes the total area is \(x^{2}+4x^{2}+4x + 7x\).
B. The simplified polynomial is \(5x^{2}+11x\)