Problem 365 - Precalculus

Question

Use the given length and area of a rectangle to express the width algebraically. Length is \(x+5,\) area is \(2 x^{2}+9 x-5.\)

Step-by-Step Solution

Answer

The width is expressed as \(2x - 1\).

Step 1: Understand the relationship between area, length, and width

A rectangle's area can be calculated by the formula: \( \text{Area} = \text{Length} \times \text{Width} \). We will use this formula to solve for the width algebraically.

Step 2: Set up the equation using given values

Plug the given values into the formula: \( 2x^2 + 9x - 5 = (x+5) \times \text{Width} \).

Step 3: Solve for the Width

To find the width, divide both sides of the equation by the length \((x+5)\): \[ \text{Width} = \frac{2x^2 + 9x - 5}{x+5} \].

Step 4: Simplify the expression

Perform polynomial division on \(\frac{2x^2 + 9x - 5}{x+5}\). The polynomial division gives \(2x - 1\) as quotient with remainder 0. Thus, \(\text{Width} = 2x - 1\).

Key Concepts

Polynomial Division

Polynomial division is similar to long division with numbers, but here we divide polynomials instead. You start by looking at the leading term of the dividend and dividing it by the leading term of the divisor. This gives you the first term of the quotient. Next, you multiply the entire divisor by this term and subtract the result from the original dividend, just like in long division.

• Repeat the process with the new polynomial that you get after subtraction until the degree of the remaining polynomial (remainder) is less than the degree of the divisor.
• If the remainder is zero, the division is exact, and you have found the factors.

In the exercise, we divided the polynomial \(2x^2 + 9x - 5\) by \(x+5\), resulting in \(2x-1\) as the quotient with a remainder of zero. This tells us that \(x+5\) perfectly divides \(2x^2 + 9x - 5\). This is helpful in identifying polynomial factors which can be crucial in understanding algebraic expressions.

Rectangle Geometry

Let's explore the basics of rectangle geometry. A rectangle is a four-sided shape with opposite sides equal in length and every angle a right angle.

• The length of a rectangle refers to the longer side, while the width is the shorter side.
• The perimeter of a rectangle can be calculated as \(2 \times (\text{Length} + \text{Width})\).
• In our exercise, we used the given length of \(x+5\) to help find the width algebraically.

Understanding these foundational geometric principles can simplify solving many problems related to rectangles. By expressing one dimension (such as the width) in terms of the other known values, you can tackle complex algebraic expressions with ease. Through subtraction and division, geometrical problems like this become manageable.

Area Calculation

The area of a rectangle is the total space contained within its sides and is calculated by multiplying the length by the width. In mathematical terms, this is expressed as \(\text{Area} = \text{Length} \times \text{Width}\).

• This formula remains constant for all rectangles, making it a reliable tool for various calculations.
• The exercise involved using the area provided \(2x^2 + 9x - 5\) to solve for the width when the length is given as \(x+5\).
• By setting up an equation \((2x^2 + 9x - 5 = (x+5) \times \text{Width})\), we can rearrange to solve for the unknown dimension using algebraic manipulation.

Area calculations are fundamental in geometry, and knowing how to rearrange formulas to solve for missing values expands your mathematical toolkit. With regular practice, recognizing and applying these formulas becomes second nature.