Which is a factor of $4 x^2 y+24 x^2-24 y-144$?
We are given the expression \(4x^2y + 24x^2 - 24y - 144\). To determine which of the given options is a factor, we need to factorize the polynomial completely. The expression contains four terms, suggesting that factoring by grouping is a suitable approach. We first observe that all four terms have a common numerical factor of 4.
First, we factor out the greatest common factor, which is 4:
\[4x^2y + 24x^2 - 24y - 144 = 4(x^2y + 6x^2 - 6y - 36)\]
Next, we focus on factoring the expression inside the parentheses: \(x^2y + 6x^2 - 6y - 36\). We can group the terms in pairs:
\[(x^2y + 6x^2) + (-6y - 36)\]
From the first group \((x^2y + 6x^2)\), we factor out \(x^2\), and from the second group \((-6y - 36)\), we factor out \(-6\):
\[x^2(y + 6) - 6(y + 6)\]
Now, we can see a common binomial factor, \((y + 6)\), which we factor out:
\[(x^2 - 6)(y + 6)\]
Combining everything, the fully factored expression is:
\[4(x^2 - 6)(y + 6)\]
By comparing our factored form \(4(x^2 - 6)(y + 6)\) with the given choices:
Therefore, the factor from the provided options is \(y + 6\). The final answer is \(\boxed{y + 6}\).
Factoring polynomials is essential in simplifying complex equations encountered in engineering and physics. For example, when calculating the area of a surface that changes as a function of two variables, you might end up with an expression representing the total cost or material required. Factoring that expression allows you to identify specific components (like the unit cost or dimensions) that can be adjusted individually to optimize the design, making calculations much faster and clearer.