What is the GCF of Given Polynomials?

Finding the Greatest Common Factor (GCF) of polynomials is similar to finding the GCF of integers. The GCF of two or more polynomials is the largest polynomial that divides each of the polynomials without leaving a remainder.

Steps to Find the GCF of Polynomials

1. Factor Each Polynomial
   First, factor each polynomial completely. For example, consider the polynomials $P(x) = x^3 – 3x^2 + 3x – 1$ and $Q(x) = x^2 – 1$. Factoring them:

   • $P(x) = (x – 1)^3$
   • $Q(x) = (x – 1)(x + 1)$

2. Identify Common Factors
   Next, identify the common factors between the polynomials. From our example, both polynomials have $(x – 1)$ as a common factor.

3. Determine the GCF
   The GCF is the highest degree polynomial that is a factor of each polynomial. In our example, the common factor $(x – 1)$ appears in both $P(x)$ and $Q(x)$. Therefore, the GCF of $P(x)$ and $Q(x)$ is $x – 1$

Example Problem

Let’s find the GCF of $P(x) = x^3 + 2x^2 – x – 2$ and $Q(x) = x^2 – 1$

1. Factor Each Polynomial
   • $P(x) = (x + 2)(x – 1)(x + 1)$
   • $Q(x) = (x – 1)(x + 1)$

2. Identify Common Factors
   Both $P(x)$ and $Q(x)$ have $(x – 1)$ and $(x + 1)$ as common factors.

3. Determine the GCF
   The GCF is the product of the common factors with the lowest degree. Therefore, the GCF of $P(x)$ and $Q(x)$ is $(x – 1)(x + 1) = x^2 – 1$

Conclusion

Understanding how to find the GCF of polynomials helps simplify expressions and solve polynomial equations. By factoring each polynomial, identifying common factors, and determining the highest degree common factor, you can easily find the GCF.

Citations

1. 1. Khan Academy – Greatest Common Factor of Polynomials
2. 2. Purplemath – Factoring Polynomials
3. 3. Math is Fun – Factoring Polynomials