Problem 55 - Algebra A Combined Function

Question

Solve. The perimeter of a square is \(\left(12 x^{3}+4 x-16\right)\) feet Find the length of its side.

Step-by-Step Solution

Answer

The length of each side of the square is \( 3x^3 + x - 4 \) feet.

Step 1: Understand the Problem

The problem gives us the perimeter of a square and asks us to find the length of one side. We know that the perimeter of a square is four times the length of one side.

Step 2: Equation for Perimeter

The formula for the perimeter of a square is given by: \[ P = 4s \] where \( P \) is the perimeter and \( s \) is the length of a side.

Step 3: Equate and Solve for Side Length

Since the perimeter is \( 12x^3 + 4x - 16 \), we equate it to \( 4s \). This gives us: \[ 4s = 12x^3 + 4x - 16 \]. To find \( s \), divide the entire expression by 4.

Step 4: Simplify the Expression

Divide the expression term by term: \[ s = \frac{12x^3}{4} + \frac{4x}{4} - \frac{16}{4} \]Simplifying, we get: \[ s = 3x^3 + x - 4 \]

Step 5: Conclusion

The length of one side of the square is \( 3x^3 + x - 4 \) feet.

Key Concepts

Perimeter of a Square

The perimeter of a square is a simple concept that involves the measurement of the total distance around the square. A square has four equal sides, and its perimeter can be calculated using the formula:

• \[ P = 4s \]

where \( P \) is the perimeter, and \( s \) is the length of one side. To find the perimeter, you multiply the length of one side by four.
The problem we are dealing with gives us a polynomial expression for the perimeter: \( 12x^3 + 4x - 16 \). This means that the length of the perimeter varies depending on the value of \( x \).
Once you understand the formula, you can find the side by solving for \( s \) using the expression provided.

Dividing Polynomials

Dividing polynomials is a key concept, especially useful when you have an expression for a perimeter or area and need to find a single dimension, like the side of a square. In this exercise, the polynomial \( 12x^3 + 4x - 16 \) represents four times the side length.
To find the length of one side, you need to divide this entire expression by four, which is the multiplier in the perimeter formula for a square:

• The polynomial \( 12x^3 + 4x - 16 \) is divided term by term by 4:
• \( \frac{12x^3}{4} \) simplifies to \( 3x^3 \)
• \( \frac{4x}{4} \) simplifies to \( x \)
• \( \frac{-16}{4} \) simplifies to \( -4 \)

Thus, the length of one side of the square, \( s \), is \( 3x^3 + x - 4 \). This method requires careful division of each term by the number outside the parentheses, ensuring all terms in the polynomial are simplified correctly.

Simplifying Algebraic Expressions

Simplifying algebraic expressions comes into play when dividing polynomials, as seen in the previous step. Simplification involves reducing expressions to their simplest form to make them easier to work with.
In this particular exercise, every step in dividing the polynomial by 4 requires simplification:

• \( 12x^3 \) divided by 4 simplifies to \( 3x^3 \), making this term easier to manage.
• \( 4x \) divided by 4 simplifies to \( x \), effectively reducing a potentially complex term to a more workable form.
• \( -16 \) divided by 4 simplifies to \( -4 \), again simplifying a constant term.

Simplification not only makes calculations manageable but also facilitates understanding and solving problems by expressing the solution in its simplest form. When dealing with polynomial expressions, always look to simplify each term as much as possible.