1. A rectangle has an area of x3 + x2 – 7x + 2 square meters and a width of x – 2 meters. Find its length.
ah ha, so this is why you want to divide polynomials
*******______ x^2____________________
(x-2) | x^3 + x^2 - 7 x + 2
********x^3 -2 x^2
********--------------------
************+3 x^2 - 7 x + 2
*******______ x^2_+3x__-1____
(x-2) | x^3 + x^2 - 7 x + 2
********x^3 -2 x^2
********--------------------
************+3 x^2 - 7 x + 2
************+3 x^2 -6 x
************----------------
******************* -1 x + 2
******************* -1 x + 2
*******************-----------
**********Remainder = 0
so
x^2 +3 x - 1
To find the length of the rectangle, we need to divide the area of the rectangle by its width. The area of the rectangle is given as x^3 + x^2 - 7x + 2 square meters, and the width is x - 2 meters.
So, the length (L) can be found by dividing the area (A) by the width (W):
L = A / W
Plugging in the given values, we have:
L = (x^3 + x^2 - 7x + 2) / (x - 2)
To simplify this expression, we can use polynomial long division. Let's perform the division step-by-step:
______________________________________
x - 2 | x^3 + x^2 - 7x + 2
- (x^3 - 2x^2)
______________________________________
3x^2 - 7x
- (3x^2 - 6x)
______________________________________
-x + 2
- (-x + 2)
______________________________________
0
After performing the division, we find that there is no remainder, which means that the polynomial x^3 + x^2 - 7x + 2 is evenly divisible by x - 2. Therefore, the resulting quotient is the length of the rectangle:
L = 3x^2 - 7x -x + 2
= 3x^2 - 8x + 2
Thus, the length of the rectangle is 3x^2 - 8x + 2 meters.
To find the length of the rectangle, we need to use the formula for the area of a rectangle which is given by the product of its length and width.
Given that the area of the rectangle is x^3 + x^2 - 7x + 2 square meters, and its width is x - 2 meters, we can express the length as follows:
Length * Width = Area
Let L represent the length of the rectangle, then we have:
L * (x - 2) = x^3 + x^2 - 7x + 2
Now, we can solve this equation for L:
First, distribute the length to the width:
L * x - L * 2 = x^3 + x^2 - 7x + 2
Simplify the equation:
Lx - 2L = x^3 + x^2 - 7x + 2
Rearrange the equation to have all terms on one side:
Lx - x^3 - x^2 + 7x - 2L - 2 = 0
Combine like terms:
-x^3 - x^2 + (L - 7)x - 2L - 2 = 0
Now, we need to factor out the common terms:
-(x^3 + x^2) + (L - 7)x - (2L + 2) = 0
Factor out the common factor of x^2 from the first two terms:
x^2(-x - 1) + (L - 7)x - (2L + 2) = 0
Factor out the common factor of (L - 7) from the second two terms:
x^2(-x - 1) + (L - 7)(x - 2) = 0
The x terms are now factored out, and we can solve for L:
Now, we have two factors, and each factor must equal zero for the equation to be true. So, we can set each factor equal to zero and solve for x:
Factor 1: -x - 1 = 0
Solve for x:
x = -1
Factor 2: L - 7 = 0
Solve for L:
L = 7
Therefore, the length of the rectangle is 7 meters.