91Ó°ÊÓ

To factor a polynomial, the first step is to identify the Greatest Common Factor (GCF). The GCF is the largest number that can divide each term of the polynomial without leaving a remainder.
For example, consider the polynomial given in the exercise:

• 3
• -30
• 75

The GCF of these terms is 3, as 3 is the highest number that can evenly divide 3, -30, and 75.
After determining the GCF, factor it out from each term. This simplifies the polynomial and makes it easier to see other factoring opportunities. In this exercise:

• 3z² - 30z + 75 becomes 3(z² - 10z + 25)

Factoring out the GCF is a crucial step that lays the foundation for further simplification.

Once the GCF is factored out, the next step involves recognizing patterns in the simplified polynomial. One important pattern is the Perfect Square Trinomial. A perfect square trinomial is a quadratic expression that can be written as the square of a binomial.
The general form of a perfect square trinomial is:

• z² - 2abz + b²

This can be factored as:

• (z - b)(z - b) or (z - b)²

In the given polynomial, after factoring out the GCF, we have:

• z² - 10z + 25

This fits the pattern because:

• z = z
• 2ab = -10z
• b² = 25

To verify, we note that

• b = 5
• 2a(-5) = -10z
• 5² = 25

Therefore, the trinomial can be expressed as:

• (z - 5)²

Recognizing a perfect square trinomial simplifies the process of factoring quadratic expressions.

A quadratic expression is a polynomial of the form:

• az² + bz + c

where a, b, and c are constants, and a ≠ 0. Quadratic expressions can be factored in different ways, such as recognizing a perfect square trinomial or using the quadratic formula.
In this exercise, the expression is:

• z² - 10z + 25

Factorizing it, we've seen it as a perfect square:

• (z - 5)²

but it could also be solved by various methods like completing the square or using the quadratic formula.
Factoring quadratic expressions is a fundamental skill in algebra because it makes solving equations simpler. Quadratic expressions frequently appear in physics, engineering, and many real-world scenarios, hence understanding these concepts deeply helps in applying them effectively.