- Factor out the common term \(y\) from \(x^2 y - 2xy - 24y\) to obtain \(y(x^2 - 2x - 24)\).
- Analyze the remaining trinomial \(x^2 - 2x - 24\) and observe that it is not a perfect square because the constant term \(-24\) does not equal \((\frac{-2}{2})^2 = 1\).
- Conclude that the remaining factor is a \(\boxed{\text{trinomial that is not a perfect square}}\).
Explanation
- Factoring out the common factor
The problem asks us to factor the expression \(x^2 y - 2xy - 24y\) by first extracting the common factor \(y\), and then to classify the resulting trinomial.
First, we look for the greatest common factor in all three terms of the expression \(x^2 y - 2xy - 24y\). All terms have a factor of \(y\).
Factoring out \(y\), we get:
\[x^2 y - 2xy - 24y = y(x^2 - 2x - 24)\]
The remaining factor is the quadratic trinomial \(x^2 - 2x - 24\).
- Classifying the remaining trinomial
Now we examine the trinomial \(x^2 - 2x - 24\) to see how it can be classified.
A perfect-square trinomial takes the form \(a^2 \pm 2ab + b^2 = (a \pm b)^2\). For \(x^2 - 2x - 24\) to be a perfect-square trinomial, the constant term must be the square of half the coefficient of the linear term \(x\).
The coefficient of \(x\) is \(-2\). Half of this is \(-2/2 = -1\). The square of \(-1\) is \((-1)^2 = 1\).
Since our constant term is \(-24\), and not \(1\), it is not a perfect-square trinomial.
Next, we check if it is a difference of squares or a sum of cubes. A difference of squares involves two terms, not three. A sum of cubes involves terms of the form \(a^3 + b^3\), which this does not match.
Thus, \(x^2 - 2x - 24\) is a trinomial that is not a perfect square. It can be factored as \((x - 6)(x + 4)\), but since it does not fit the specific classifications of perfect-square trinomials, sum of cubes, or difference of squares, it is classified as a trinomial that is not a perfect square.
Examples
In algebra, recognizing the structure of a polynomial is essential for efficient simplification. For instance, if you are designing a computer program that calculates the area of a rectangle with dimensions represented by polynomials, identifying if an expression is a perfect square or a simple trinomial helps you choose the fastest way to compute or optimize the area formula. This skill allows you to break down complex expressions into simpler, manageable parts, making it a foundational tool in both mathematics and computer science applications.