GCF to Factor a Polynomial

Understanding the Greatest Common Factor (GCF)

Hey there! Today, we’re going to learn about something called the Greatest Common Factor, or GCF for short. It’s a super useful concept in math, especially when you’re working with algebra. The GCF helps us simplify expressions and solve equations more easily. Let’s dive into it with a simple example!

Step 1: Identify the Terms

We’ll start with two algebraic expressions:

1. (4x^4y)
2. (8x^3y)

Our mission is to find the GCF of these two terms.

Step 2: Prime Factorization

To find the GCF, we need to break down each term into its prime factors.

Factorization of the First Term

For (4x^4y):

• The number 4 can be broken down into (2 times 2).
• (x^4) means (x) is multiplied by itself four times: (x times x times x times x).
• The variable (y) stays as it is.

So, the prime factorization of (4x^4y) is:

(2 times 2 times x times x times x times x times y)

Factorization of the Second Term

For (8x^3y):

• The number 8 can be broken down into (2 times 2 times 2).
• (x^3) means (x) is multiplied by itself three times: (x times x times x).
• The variable (y) stays as it is.

So, the prime factorization of (8x^3y) is:

(2 times 2 times 2 times x times x times x times y)

Step 3: Determine the GCF

Now, let’s find the common factors in both expressions:

• Both terms have two factors of 2.
• Both terms have at least three factors of (x).
• Both terms have one factor of (y).

Putting these together, the GCF is:

(2 times 2 times x times x times x times y = 4x^3y)

Step 4: Rewrite the Original Expressions

Now that we know the GCF, we can rewrite each term using the GCF and the leftover factors.

Rewriting the First Term

For (4x^4y):

We factor out (4x^3y):

(4x^4y = 4x^3y cdot (x))

Rewriting the Second Term

For (8x^3y):

We factor out (4x^3y):

(8x^3y = 4x^3y cdot (2))

Step 5: Combine the Terms

Now, we can put it all together:

(4x^3y cdot (x + 2))

Conclusion

And there you have it! We’ve successfully factored out the GCF from the terms (4x^4y) and (8x^3y). The final expression is:

(4x^3y(x + 2))

Understanding how to find and use the GCF is a handy skill in algebra. It makes simplifying and working with expressions much easier. Keep practicing, and you’ll get the hang of it in no time!