To factor the expression [tex]\(x^2 y - 2 x y - 24 y\)[/tex], follow these steps:
1. Factor out the common factor of [tex]\(y\)[/tex]:
The expression [tex]\(x^2 y - 2 x y - 24 y\)[/tex] has a common factor of [tex]\(y\)[/tex] in each term. We factor out [tex]\(y\)[/tex] from the entire expression:
[tex]\[
x^2 y - 2 x y - 24 y = y (x^2 - 2x - 24)
\][/tex]
2. Factor the remaining quadratic expression:
Now, we need to factor the quadratic expression [tex]\(x^2 - 2x - 24\)[/tex]. This is a trinomial of the form [tex]\(ax^2 + bx + c\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = -24\)[/tex].
To factor [tex]\(x^2 - 2x - 24\)[/tex], we look for two numbers that multiply to [tex]\(c\)[/tex] (which is [tex]\(-24\)[/tex]) and add up to [tex]\(b\)[/tex] (which is [tex]\(-2\)[/tex]). These two numbers are [tex]\(-6\)[/tex] and [tex]\(4\)[/tex] because:
[tex]\[
-6 \times 4 = -24 \quad \text{and} \quad -6 + 4 = -2
\][/tex]
Therefore, we can write:
[tex]\[
x^2 - 2x - 24 = (x - 6)(x + 4)
\][/tex]
3. Combine the factors:
Putting it all together, the original expression [tex]\(x^2 y - 2 x y - 24 y\)[/tex] factored is:
[tex]\[
x^2 y - 2 x y - 24 y = y (x^2 - 2x - 24) = y (x - 6)(x + 4)
\][/tex]
4. Classification:
The remaining factor [tex]\(x^2 - 2x - 24\)[/tex] is a quadratic trinomial, but it is neither a perfect-square trinomial, a sum of cubes, nor a difference of squares. It is simply a factored quadratic expression.
Thus, the fully factored form of the expression is:
[tex]\[
y(x - 6)(x + 4)
\][/tex]
And the correct classification of the remaining factor [tex]\(x^2 - 2x - 24\)[/tex] is a quadratic trinomial.
