91Ó°ÊÓ

When we factor polynomials, identifying the greatest common factor (GCF) is a crucial step. The GCF is the highest number that can divide all coefficients of the polynomial terms without leaving a remainder. For instance, in the polynomial \(15 + 12x - 3x^2\), the coefficients are 15, 12, and -3. The GCF of these numbers is 3, meaning 3 is the largest number that divides all of them.

To factor the GCF out of the polynomial, divide each term of the polynomial by the GCF. Here’s how it’s done:
\[15 \div 3 = 5, \ 12x \div 3 = 4x, \ -3x^2 \div 3 = -x^2\]
This turns our polynomial into \([-x^2 + 4x + 5]\) with the 3 factored out in front: \[-3(x^2 - 4x - 5)\].
Utilizing the GCF makes it easier to factor the remaining simpler polynomial inside the parentheses.

A quadratic expression is a polynomial of degree 2, usually written in the form \[ax^2 + bx + c\].
In our example, after factoring out the GCF, we have the quadratic expression \[x^2 - 4x - 5\] inside the parentheses.
To factor this quadratic expression, we look for two numbers that:

• Multiply to the constant term (\