Question

Write a polynomial that represents the perimeter of the figure.

Short answer

The polynomial$8x^4+6x^2+15x+2$ represents the perimeter of the given figure.

Step-by-step solution

Step 1. Define standard form of a polynomial.

The standard form of a polynomial means, writing the polynomial in descending powers of the variable.

Step 2. Define like and unlike terms.

In algebra, like terms are the terms that have the same variables and powers.

The unlike terms are two or more terms that are not like terms.

Step 3. Define the perimeter of a figure.

Perimeter of a figure is the sum of the lengths of all the sides of the figure.

Step 4. Calculate the perimeter of the given figure.

Simplify the given difference by combining the like terms.

Add the lengths of all sides of the given figure.

The perimeter is

$\begin{array}{l}\left(5x^2\right)+\left(4x\right)+\left(4x\right)+\left(3x^3\right)+\left(x^2\right)+\left(1\right)+\left(x\right)+\left(1\right)+\left(6x\right)+\left(8x^4\right)\\ =5x^2+4x+4x+3x^3+x^2+1+x+1+6x+8x^4\end{array}$

Simplify by combining the like terms

$$\begin{gathered} 5x^2+4x+4x+3x^3+x^2+1+x+1+6x+8x^4 \\ =8x^4+5x^2+x^2+4x+4x+x+6x+1+1 \\ =8x^4+6x^2+15x+2 \end{gathered}$$

So the perimeter$=8x^4+6x^2+15x+2$

Therefore the polynomial$8x^4+6x^2+15x+2$represents the perimeter of the given figure.