Factoring Polynomials By GCF

To factor a polynomial, first of all, check it out if there is any greatest common factor in the terms of the polynomial. Polynomials can’t be factored without the knowledge of GCF.

If the GCF is that important for factoring polynomials, why not we practice the greatest common factor first?

Greatest common factor (GCF) is the largest number which 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.

Also the algebraic expression can have GCF. For example; “3a” and “5a” have the GCF as “a” and “3pq” and “6pr” have GCF as “3p”.

Another example I think is important for this presentation is, theGCF of “2a³b²c”, “6a²b²” and “8a²b²c²”.

In this example, the GCF for 2, 6 and 8 is 2. In case of variables with these numbers the GCF of“a³b²c”, “a²b²” and “a²b²c²” is “a²b²” as this is contained in all the three terms. Hence the GCF of the given expressions is “2a²b²”

Having 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 factor the above problems one by one.

1. 3ab – 6

In this binomial the GCF between “3ab” and “- 6” is “3” because 6 = 3*2. Pull 3 out from both the terms and write the remaining factors inside the brackets as shown below;

= 3(ab – 2)

Take notice that, when the GCF 3 is pulled out, the remaining factors are written in the brackets.

2. 10x² + 5xy

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

= 5x (2x + y)

3. 2ab²c – 6abc

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

4. 6p²q + 9p²q² – 6pq²

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

= 3pq (2p + 3pq + 2q) is the factored 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 pull “5x” out from all the three terms as shown below:

= 5x (2x – 3xy – 5xyz²)

I hope that above explanation is enough for understanding the factoring polynomials by taking the GCF out.

Kind regards

Manjit Singh.


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