What are the factors of $x^2-10 x-24$?
The quadratic polynomial to be factored is \(x^2 - 10x - 24\). This is a monic quadratic polynomial of the form \(x^2 + bx + c\), where \(b = -10\) and \(c = -24\). Our goal is to express this in the form \((x + p)(x + q) = x^2 + (p + q)x + pq\). By comparing coefficients, we have the following two conditions:
We are looking for two numbers \(p\) and \(q\) whose product is \(-24\) and whose sum is \(-10\).
Let's list pairs of factors of \(-24\) and their corresponding sums:
The pair that satisfies both conditions is \(p = -12\) and \(q = 2\).
Substituting \(p = -12\) and \(q = 2\) into the factored form \((x + p)(x + q)\), we obtain:
\((x - 12)(x + 2)\).
Let's verify this by expanding the expression:
\((x - 12)(x + 2) = x(x) + x(2) - 12(x) - 12(2) = x^2 + 2x - 12x - 24 = x^2 - 10x - 24\).
This expansion matches our original polynomial. Therefore, the correct factorization is \((x - 12)(x + 2)\), which corresponds to option D.
Factoring quadratic polynomials is a fundamental skill in algebra with many real-world applications. For example, if you are designing a rectangular garden and you know the area is represented by the quadratic expression \(x^2 - 10x - 24\) square meters, you can factor it to find the dimensions of the garden. Factoring allows you to express the area as a product of two binomials, representing the length and width as linear functions of \(x\), which is essential for solving geometric design problems and optimization tasks.