The width, w, of a rectangular playground is 2x – 1. The area of the playground is 2x3 + x2 – 5x + 2. What is an expression for the length of the playground?
A. x2 + x – 2
B. x2 – x + 2
C. x2 + x + 2
D. x2 – x – 2
Hmm, let me try to find the humor in this math problem. Ah, I got it!
Why was the playground feeling so square? Because it was rectangular! 🤡
Now, let's solve the problem. The area of a rectangle is given by the formula length times width. We are given that the width (w) is 2x - 1. So, we can set up the equation:
2x^3 + x^2 - 5x + 2 = (2x - 1) * length
To find the expression for the length, we can rearrange the equation:
length = (2x^3 + x^2 - 5x + 2) / (2x - 1)
Simplifying the expression on the right:
length = (x^2 + x - 2)
So, the expression for the length of the playground is A. x^2 + x - 2.
To find the expression for the length of the playground, we need to divide the area by the width.
Given:
Width (w) = 2x – 1
Area = 2x^3 + x^2 – 5x + 2
To find the length (L), we divide the area by the width:
Length (L) = Area / Width
Substituting the given values:
L = (2x^3 + x^2 – 5x + 2) / (2x – 1)
Dividing the polynomial by the binomial can be done using long division. But for simplicity, we'll use synthetic division to divide (2x^3 + x^2 – 5x + 2) by (2x – 1).
1/2 │ 2 1 -5 2
│ 1 1 -3
└─────────────
2 2 -3 -1
The remainder is -1.
L = 2x^2 + 2x - 3 - (1 / (2x - 1))
Therefore, an expression for the length of the playground is:
L = 2x^2 + 2x - 3 - (1 / (2x - 1))
Answer: D. x^2 – x – 2
To find the expression for the length of the playground, we can use the formula for the area of a rectangle: Area = Length × Width.
Given that the width of the rectangular playground is 2x - 1 and the area is 2x^3 + x^2 - 5x + 2, we can set up an equation:
2x^3 + x^2 - 5x + 2 = Length × (2x - 1)
Now, we need to isolate the length on one side of the equation.
Divide both sides of the equation by (2x - 1):
(2x^3 + x^2 - 5x + 2)/(2x - 1) = Length
The resulting expression is the expression for the length of the playground.