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## Factoring Polynomials By GCF

To factor a polynomial, first check if there is a greatest common factor in the terms of the polynomial. Polynomials cannot be factored without knowledge of GCF.

If PGCF is so important for factoring polynomials, why not practice the greatest common factor first?

The greatest common factor (GCF) is the largest number that divides the two or more given numbers. For example; GCF of 8 and 6 is 2, GCF of 6 and 9 is 3, and GCF of 6 and 12 is 6.

Moreover, the algebraic expression can have **GCF**. For example; **“3a”** and “**5a”** have the GCF like **“a”** and **“3pk”** and **“6pr”** have GCF as **“3p”. **

Another example that seems important to me for this presentation is the**PCS of “2a³b²c”, “6a²b²” and “8a²b²c²”. **

**In this example, the PGCF for 2, 6, and 8 is 2. In the case of variables with these numbers, the PGCF of****“a³b²c”, “a²b²” and “a²b²c²” is “a²b²” because it is contained in all three terms. Therefore, the PGCF of the given expressions is “2a²b²”**

With the above concepts of greatest common factor in mind, let’s do the following examples, to explore factoring polynomials.

1. 3ab – 6

2. 10x² + 5xy

3. 2ab²c – 6abc

4. 6p²q + 9p²q² – 6pq²

5. 10x³ – 15x²y – 5x²yz²

Let’s consider the above issues one by one.

1. 3ab – 6

In this binomial, the PGCF between “3ab” and “- 6” is “3” because 6 = 3*2. Draw 3 from the two terms and write the remaining factors in parentheses as shown below;

= 3(ab - 2)

Note that when GCF 3 is removed, the remaining factors are written in parentheses.

2. 10x² + 5xy

GCF of “10x²” and “5xy” is “5x” and removing this common factor, the binomial can be written as below:

= 5x (2x + y)

3. 2ab²c – 6abc

= 2abc(b – 3) is the factorized form of the given binomial.

4. 6p²q + 9p²q² – 6pq²

We are given a trinomial with the greatest common factor of “3pq”. Draw “3pq” from the three terms to factor the given trinomial.

= 3pq (2p + 3pq + 2q) is the factorized form of the given trinomial.

5. 10x³ – 15x²y – 5x²yz²

Again, the given polynomial is a trinomial and the greatest common factor of its terms is “5x”. To factor the given trinomial, draw “5x” from the three terms as shown below:

= 5x (2x – 3xy – 5xyz²)

I hope the above explanation is enough to understand the factorization polynomials by removing the GCF.

Sincere friendships

Manjit Singh.

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