91Ó°ÊÓ

The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers or terms without leaving a remainder. In algebra, the GCF of an expression is the largest polynomial that divides each term of the expression.
Let's break it down with an example:
Suppose we have two numbers, 12 and 15. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 15 are 1, 3, 5, and 15. The largest number common to both lists is 3. Hence, 3 is the GCF of 12 and 15.
In algebraic expressions, we look for the largest expression that can divide all terms of the polynomial.
In the example exercise, the terms are:

• \( q^{2}(p-4) \)
• \( 1(p-4) \)

You can see that \( (p-4) \) is common to both terms. Therefore, the GCF in the example is \( (p-4) \).

Factoring is the process of breaking down an expression into a product of its simpler components. When we write an expression in factored form, we express it as a product of its factors.
To factor an expression using the GCF, you must follow these steps:

• Identify the GCF of the terms.
• Divide each term by the GCF.
• Write the expression as a product of the GCF and the resulting quotient.

In our exercise, the original expression is \( q^{2}(p-4)+1(p-4) \). We first identified \( (p-4) \) as the GCF. Then, we factored out \( (p-4) \) and divided each term by it, resulting in \( q^{2} + 1 \).
Thus, the original expression is written in factored form as \( (p-4)(q^{2} + 1) \). Factoring makes it easier to simplify and solve equations in algebra.

Polynomials are mathematical expressions that consist of variables and coefficients, combined using addition, subtraction, and multiplication. A polynomial can have one or multiple terms, each term being a product of a constant and a variable raised to a non-negative integer power.
For example, the expression \( 3x^{2} - 5x + 7 \) is a polynomial with three terms: \( 3x^{2} \), \( -5x \), and \( 7 \).
More key points about polynomials include:

• Each term in a polynomial is called a monomial.
• The degree of a polynomial is the highest power of the variable.
• The coefficients are the numerical factors in each term.

The given exercise involved polynomial expressions. Specifically, the terms were \( q^{2}(p-4) \) and \( 1(p-4) \), both polynomials multiplied by a common factor \( (p-4) \). Polynomials play an important role in algebra and are often factored to simplify expressions or solve equations.
Understanding how to handle polynomials and express them in factored form is essential for mastering algebra.