91Ó°ÊÓ

When factoring trinomials, the first step is to identify if there is a Greatest Common Factor (GCF). The GCF is the largest number that divides into all the terms of the polynomials without leaving a remainder.
In the exercise, we looked at the equation \( 18x^2 - 48xy + 32y^2 \). To find the GCF of the coefficients (18, -48, and 32), we identify the largest number that can divide all of them. Here, the GCF is 2.
It’s essential to pull out the GCF because it simplifies the trinomial, making it easier to factor further. Once the GCF is identified, we divide each term by the GCF. For this example, the trinomial becomes: \( 2 (9x^2 - 24xy + 16y^2) \) which is significantly simpler to work with.

After factoring out the GCF, the next step is factoring the quadratic trinomial inside the parentheses. In our example, this is: \( 9x^2 - 24xy + 16y^2 \).
Factoring quadratics involves finding two binomials whose product results in the quadratic trinomial. To do this, we need to determine two numbers that multiply to the constant term and add to the coefficient of the linear term.
For the trinomial \( 9x^2 - 24xy + 16y^2 \), we look at the coefficients and constants. We need binomials of the form \((Ax + By)(Cx + Dy)\). Here, we recognize the trinomial can be factored into \((3x - 4y)(3x - 4y)\). This is a special case because it’s a perfect square trinomial. Thus, our simplified form is: \( (3x - 4y)^2 \).

The final step is writing the complete factored form. After factoring out the GCF and then factoring the quadratic trinomial, the final answer must combine these results.
From our steps, we have: Factored out the GCF: \( 2 (9x^2 - 24xy + 16y^2) \). Factored the quadratic trinomial: \( 2 (3x - 4y)^2 \).
Therefore, the completely factored form of the original trinomial \( 18x^2 - 48xy + 32y^2 \) is: \( 2 (3x - 4y)^2 \). This final expression is the simplest form we can provide and showcases all the factoring steps clearly.