The area of a rectangular athletic field is represented by the expression48x^4+32x^3-72x square meters. Write an algebraic expression to represent one possible set of dimensions (in the sense “length times width”) of the athletic field. Include correct units with your solution.
To find the dimensions of the athletic field represented by the given expression, we need to factor it. By factoring out the greatest common factor, which is 8x, we can express the expression as a product of two binomials.
Factoring the expression 48x^4 + 32x^3 - 72x:
8x(6x^3 + 4x^2 - 9)
Now, let's consider the binomial factor (6x^3 + 4x^2 - 9). We can find its roots using a numerical method like the Rational Root Theorem or by plugging in possible integer values for x.
Using the Rational Root Theorem, we check the possible rational roots of 9 (the constant term) divided by 6 (the coefficient of the highest-degree term) to get 3/2. By evaluating the expression for x = 3/2, we find that it is not equal to zero.
Thus, no rational roots exist, and we cannot factor the binomial any further. This means that there is no simple algebraic expression for the dimensions of the athletic field.
However, we can make an estimate based on the leading term of the factored expression, which is 8x. This suggests that the length and width of the field could be expressed as (8x) and (6x^3 + 4x^2 - 9)/(8x), respectively.
Therefore, one possible set of dimensions, in terms of x, for the athletic field could be L = 8x meters and W = (6x^3 + 4x^2 - 9)/(8x) meters.