In order to factor the expression [tex]\( x^2 y - 2xy - 24y \)[/tex], follow these detailed steps:
1. Identify the Common Factor:
We begin by examining the expression [tex]\( x^2 y - 2xy - 24y \)[/tex] to see if there is any common factor across all terms. Notice that each term contains the variable [tex]\( y \)[/tex].
2. Factor Out the Common Factor:
Since [tex]\( y \)[/tex] is present in each term, we can factor it out. This process involves dividing each term by [tex]\( y \)[/tex] and writing [tex]\( y \)[/tex] as a common factor outside the parentheses.
[tex]\[
x^2 y - 2xy - 24y = y(x^2 - 2x - 24)
\][/tex]
After factoring out [tex]\( y \)[/tex], we're left with the expression [tex]\( y(x^2 - 2x - 24) \)[/tex].
3. Analyze the Remaining Factor:
The remaining factor [tex]\( x^2 - 2x - 24 \)[/tex] is a quadratic expression. This quadratic can sometimes be further factored, but for the scope of our factorization here, we are focusing on the common factor [tex]\( y \)[/tex] and recognizing that [tex]\( x^2 - 2x - 24 \)[/tex] remains as it is in factored form.
Thus, the given expression [tex]\( x^2 y - 2xy - 24y \)[/tex] when factored by removing the common factor [tex]\( y \)[/tex] results in:
[tex]\[
y( x^2 - 2x - 24 )
\][/tex]
Therefore, after factoring out the common factor, the remaining factor is a quadratic expression [tex]\( x^2 - 2x - 24 \)[/tex].
