Factoring Polynomials

A polynomial is a sum or subtraction of monomials . i.e. 10x² +15x is a polynomial and
so is x³ + 3x² − 5x + 9 .

In any factorization problem, the first thing to look at is the greatest common factor.

Factoring out the Greatest Common Factor (GCF)



In this polynomial, 9 is the greatest integer that divides 18 and 27 and x² is the greatest
expression that divides x³ and x².



Some polynomials may have a GCF of 1, but appropriate grouping may lead to possible
factorization.

Factoring by Grouping



Common factor in x³ + 4x² is x² and common factor in 3x+ 12 is 3. These can be factored
as:



Factoring Trinomials

A strategy for factoring ax² + bx + c,

1. Find 2 numbers whose product is ac and whose sum gives b. Say these numbers are u
and v.
2. Re-write the trinomial such as ax² + ux+ vx + c
3. Use factoring by grouping to find the factors

Here is an example:

* x² + 3x −18

We need to find two numbers such that their product is -18 and their sum is 3. Let’s find
the factors of -18

Our desired numbers are 6 and -3.

We will now re-write our trinomial : x² + 6x − 3x −18



Let’s look at another example whose leading coefficient is not 1:

* 8x² −10x − 3

We need to find two numbers such that their product is -24 and their sum is -10. Let’s
find the factors of -24.

Our desired numbers are -12 and 2.

We will now re-write our trinomial : 8x² −12x + 2x − 3

Let’s look at another example:



Step 1. The GCF is 2x, therefore the polynomial can be written as: 2x(x² + 4x + 4)

Step 2. Now we have to factor a polynomial with three terms and it looks like a perfect
square trinomial. Therefore, (x² + 4x + 4) can be factored as (x + 2)²

Step 3. Check to see if factors can be factored further. If not, you have reached the final
answer i.e.

Long division/ Synthetic division method of factorizing

Higher order polynomials that cannot be solved using any of the above mentioned
methods, require the application of the rational roots test and long division. This concept
will be explained using the following example:



Since factoring by grouping does not work for this expression, we will use long division.

Step 1. Rational roots test: Find the factors of the coefficients of the constant term and
the highest order term and write these as p and q. In this case, p = 1,-1, 3, -3 and q = 1,-
1,2,-2. Now write These give us a list of possible zeroes of the
polynomial i.e. the polynomial will equal zero when or or or or or
or or etc.

Use trial and error method to find which factor gives us
In this case it is seen that when , i.e. when x =3,
This tells us that one factor of the polynomial is (x-3).

Step 2. Synthetic division
For this, we will only use the coefficients of the polynomial and
divide those by the factor ‘3’.

	1. Pull down the first coefficient
	2. Multiply 3 and 2 and carry it over
	3. Add -5 and 6
	4. Multiply 3 and 1 and carry it over
	5. Add -4 and 3
	6. Multiply 3 and -1 and carry it over
	7. Add 3 and -3

The coefficients of the quotient of this division are 2, 1, -1, and 0.

Since our problem was an order of 3. i.e. , our quotient will be an
order lower i.e. our quotient will be of order 2.

The quotient therefore is given by: 2x² + 1x -1.

The factorization so far is given by : (x-3)( 2x² + x -1)

Step 3. Check to see if factors can be factorized further

2x² + 1x -1 can be factored further.

We need to find two numbers such that their product is -2 and their sum is 1. Let’s find
the factors of -2.

Our desired numbers are 2 and -1.

We will now re-write our trinomial : 2x² + 2x −1x −1



The final result is therefore:

Try these examples on your own: