A rectangle has an area of `2x^2 -x-1` . What is the expression for the width of the rectangle if the length is `2x+1`? 

Asked on by rn20yr

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txmedteach | High School Teacher | (Level 3) Associate Educator

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To answer this, you need to remember the formula for area of a rectangle:

`A = l*w`

Here, `A` is the area, `l` is the length, and `w` is the width.

We are given the information for the area and the length, so we can substitute the given expressions into our equation:` `

`2x^2-x-1 = (2x+1)w`

To solve this equation for `w`, we must divide both sides by `2x+1`:

`(2x^2-x-1)/(2x+1) = w`

Now, we need to evaluate the expression on the left. The easiest way to do this is to factor the expression in the numerator, giving us:

`((2x+1)(x-1))/(2x+1) = w`

We can now cancel out `2x+1` from the top and bottom to yield our answer:

`x-1 = w`

Another way to do this is through polynomial long division, which is not easily displayed in the formats available to us:

`2x+1` / `2x^2-x-1`

Your first part of the quotient will be "`x`":


`(2x^2-x-1)-(2x+1)(x) = 2x^2-x-1 - 2x^2 - x = -2x-1`

The second part of the quotient will be "-1":

`-2x-1 - (2x+1)(-1) = -2x-1+2x+1 = 0`

We have a zero result after the division, indicating our quotient is `x-1`.

Either way, we have the same result: width = `x-1` .  


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