Solve the problem.

A rectangle with width 2x + 5 inches has an area of 2x4 + 9x3 - 12x2 - 79x - 60 square inches. Write a polynomial that represents its length.

I got x^3+2x^2-11x-12 inches. is this correct? I used long division.

correct!!!

To find the length of the rectangle, we need to divide the given polynomial, 2x^4 + 9x^3 - 12x^2 - 79x - 60, by the width 2x + 5.

The long division process for this would be as follows:

x^3 + 2x^2 - 11x - 12
_____________________________________________
2x + 5 | 2x^4 + 9x^3 - 12x^2 - 79x - 60
-(2x^4 + 5x^3)
------------------
4x^3 - 12x^2
-(4x^3 + 10x^2)
-----------------
-22x^2 - 79x
+ ( -22x^2 - 55x)
-------------------------
-24x - 60
-(-24x - 60)
----------------
0

After performing long division, we see that the quotient of the division is x^3 + 2x^2 - 11x - 12. Hence, the polynomial represents the length of the rectangle correctly.

To solve this problem, you need to factorize the given polynomial and connect it to the dimensions of the rectangle.

The given polynomial, 2x^4 + 9x^3 - 12x^2 - 79x - 60, represents the area of the rectangle.

To factorize the polynomial, you should look for its roots by using a method like synthetic division or factoring by grouping. Once you find the roots, you can write the polynomial as a product of its factors.

However, it seems like you used long division, which is not the appropriate method for finding the factors of a polynomial. Long division is typically used to divide one polynomial by another.

To find the polynomial that represents the length of the rectangle, let's factorize the given polynomial.

Starting with 2x^4 + 9x^3 - 12x^2 - 79x - 60, we can first look for any common factors. In this case, there are no common factors other than 1.

Next, we'll look for the roots of the polynomial by setting it equal to zero and solving for x. By using methods like factoring, synthetic division, or the rational root theorem, you can find the roots.

After finding the roots, you can write the polynomial as a product of its linear factors, like (x - a)(x - b)(x - c)...

Once you have the polynomial in factored form, you can connect it to the dimensions of the rectangle. The width of the rectangle, 2x + 5 inches, corresponds to one of the linear factors of the polynomial. The length of the rectangle will correspond to the remaining linear factors.

By factoring the given polynomial and identifying the linear factors, you can determine the polynomial that represents the length of the rectangle.