the width w of a rectangular swimming pool is x-2. The area of the playground is x^3-5x^2+2x+8. What is an expression for the length of the playground.
x^2+3x+4
x^2+3x-4
x^2-3x+4
x^2-3x-4
x^3-5x^2+2x+8
----------------- = x^2-3x-4
x-2
thanks steve
Well, let's try to solve this math problem with a little humor! So, we know that the width of the swimming pool is x-2. Perhaps the pool is trying to be a little more narrow to fit in with the latest slimming trend!
Now, the area of the playground is given as x^3-5x^2+2x+8. It seems like the playground is going through a growth spurt and decided to show off with a fancy cubic expression!
To find the length of the playground, we need to divide the total area by the width. So let's do that:
Length = Area / Width
Length = (x^3-5x^2+2x+8) / (x-2)
Now, let's pull back the clown curtain and simplify this expression. Performing polynomial division, the expression for the length of the playground is x^2+3x+4.
So, the correct expression for the length of the playground is x^2+3x+4. You can now tell that the playground is cleverly expanding its boundaries with that fancy quadratic expression!
To find the expression for the length of the playground, we can start by using the formula for the area of a rectangle:
Area = Length * Width
Given that the width (w) of the swimming pool is x-2, we can say:
Width of the swimming pool (w) = x - 2
Now, we know that the area of the playground is x^3 - 5x^2 + 2x + 8. We can set up the equation:
(x^3 - 5x^2 + 2x + 8) = Length * (x - 2)
We can solve for the length by dividing both sides of the equation by (x-2):
Length = (x^3 - 5x^2 + 2x + 8) / (x - 2)
Performing the division on the numerator, we get:
Length = x^2 + 3x + 4
Therefore, the expression for the length of the playground is x^2 + 3x + 4.
So, the correct answer is x^2 + 3x + 4.
To find the length of the playground, we need to divide the area by the width. In this case, the width of the rectangular swimming pool is given as x-2. So, to find the length, we can divide the area (x^3-5x^2+2x+8) by the width (x-2).
Using polynomial division or long division, we can perform the division to find the quotient and remainder. The quotient will be the expression for the length of the playground.
Performing the division, we get:
____________________________________________
x-2 | x^3 - 5x^2 + 2x + 8
1. Multiply the first term of the divisor (x) by the first term of the dividend (x^3), and place the result (x^4) above the dividend.
x^4 - 2x^3
____________________________________________
x-2 | x^3 - 5x^2 + 2x + 8
- (x^4 - 2x^3)
_____________
- 3x^3
2. Bring down the next term (- 3x^3) from the dividend.
x^4 - 2x^3
____________________________________________
x-2 | x^3 - 5x^2 + 2x + 8
- (x^4 - 2x^3)
_____________
- 3x^3
- 3x^3
3. Divide the first term of the divisor (x) by the first term of the quotient (- 3x^3), and place the result (-3x^2) above the dividend.
x^4 - 2x^3 - 3x^2
____________________________________________
x-2 | x^3 - 5x^2 + 2x + 8
- (x^4 - 2x^3)
_____________
- 3x^3
+ 3x^2
4. Multiply the first term of the divisor (x) by the first term of the quotient (-3x^2), and place the result (-3x^3) above the previous result (-3x^3). Subtract this from the previous result.
x^4 - 2x^3 - 3x^2
____________________________________________
x-2 | x^3 - 5x^2 + 2x + 8
- (x^4 - 2x^3)
_____________
- 3x^3
+ 3x^2
________________
- 8x^2
5. Bring down the next term (- 8x^2) from the dividend.
x^4 - 2x^3 - 3x^2 - 8x^2
____________________________________________
x-2 | x^3 - 5x^2 + 2x + 8
- (x^4 - 2x^3)
_____________
- 3x^3
+ 3x^2
________________
- 8x^2
- 8x^2
6. Divide the first term of the divisor (x) by the first term of the quotient (-8x^2), and place the result (-8x) above the dividend.
x^4 - 2x^3 - 3x^2 - 8x^2
____________________________________________
x-2 | x^3 - 5x^2 + 2x + 8
- (x^4 - 2x^3)
_____________
- 3x^3
+ 3x^2
________________
- 8x^2
+ 8x
7. Multiply the first term of the divisor (x) by the first term of the quotient (-8x), and place the result (-8x^2) above the previous result (-8x^2). Subtract this from the previous result.
x^4 - 2x^3 - 3x^2 - 8x^2
____________________________________________
x-2 | x^3 - 5x^2 + 2x + 8
- (x^4 - 2x^3)
_____________
- 3x^3
+ 3x^2
________________
- 8x^2
+ 8x
________________
0
The result of the division is zero, indicating that the division is complete. Therefore, the expression for the length of the playground is -3x^2 + 8x.
So, the correct expression for the length of the playground is x^2 - 3x + 4.