Question

factoring to identify a side length

an equilateral triangle has a perimeter of \\(15x^3 + 33x^5\\) feet. what is the length of each side?

\\(x^3\\) feet
\\(5 + 11x^2\\) feet
\\(5x^2 + 11\\) feet
\\(5x^3 + 11x^5\\) feet

Explanation:

Identify the geometric relationship

An equilateral triangle has three sides of equal length. Therefore, the perimeter \(P\) of an equilateral triangle is related to its side length \(s\) by the formula:

$$P = 3s$$

Set up the division equation

To find the length of each side \(s\), we divide the total perimeter by \(3\):

$$s = \frac{P}{3}$$

Given that the perimeter is \(15x^3 + 33x^5\) feet, we substitute this expression into our equation:

$$s = \frac{15x^3 + 33x^5}{3}$$

Divide each term by the constant

Using the Polynomial Division and Factoring Polynomials concepts, we divide each term in the numerator by the denominator \(3\):

$$s = \frac{15x^3}{3} + \frac{33x^5}{3}$$

Simplifying each fraction:

$$s = 5x^3 + 11x^5$$

Thus, the length of each side is \(5x^3 + 11x^5\) feet.

Answer:

• (A) \(x^3\) feet
• (B) \(5 + 11x^2\) feet
• (C) \(5x^2 + 11\) feet
• (D) \(5x^3 + 11x^5\) feet (Correct answer)