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Factor Polynomials by Grouping

In the Algebra 1 section, there was a page demonstrating how to approach times when you could factor polynomials by grouping when specifically quadratic polynomials were involved.

Factoring by grouping is also possible when the polynomial is of a degree higher than  2  and is not a quadratic.

But before looking at those situations in more detail, we can look at a recap of factor by grouping of a quadratic polynomial of  3  terms.

Quadratic Factoring by Grouping Recap:

We could try factor by grouping with the quadratic    3x^2 + 5x \space {\text{–}} \space 2.

Firstly the outer terms multiply together.   3 × -2 = -6

We then try to identify numbers that both multiply to give this  -6,  but also sum together to give the middle term, here  5.
-1  and  6  are such numbers that meet that satisfy this.

3x^2 + ({\text{-}}1 + 6)x \space {\text{–}} \space 2

Which is something we can rewrite a little to obtain our factors.

3x^2 \space {\text{–}} \space x + 6x \space {\text{–}} \space 2     =     (3x^2 \space {\text{–}} \space x) + (6x \space {\text{–}} \space 2)

x(3x \space {\text{–}} \space 1) + 2(3x \space {\text{–}} \space 1)     =>     (x + 2)(3x \space {\text{–}} \space 1)

But how would we go about factoring by grouping for polynomials with more terms, or of a larger degree?

Factor Polynomials by Grouping,
Common Factors

Trying to factor polynomials by grouping can be more tricky when there are more than  3  terms present, along with it being the case that factoring by grouping doesn’t always work.

But when attempting the grouping approach to factoring, the first step is to look for a ‘greatest common factor’ that can be factored out of the terms involved.

For example an expression such as   ax + bx.

A common factor between them is  x.     So we can re-write as   x(a + b).

Polynomial Greatest Common Factor


10x^4 + 6x^2 + 2x,

Here  2x  is the greatest common factor among the terms.

10x^4 + 6x^2 + 2x \space = \space 2x(5x^2 + 3x + 1)


12a^3b^2 \space {\text{–}} \space 6ab^2 + 9a^2b^5,

Here  3ab^2  is the greatest common factor among the terms.

12a^3b^2 \space {\text{–}} \space 6ab^2 + 9a^2b^5 \space = \space 3ab^2(4a^2 \space {\text{–}} \space 2 + 3ab^3)

Factor Polynomials by Grouping,
Larger Terms and Degree


Factor   f(x) = 20x^4 \space {\text{–}} \space 32x^3 + 48x^2 \space {\text{–}} \space 36x.


We can initially factor out a  4x.

f(x) = 4x(8x^3 \space {\text{–}} \space 6x^2 + 13x \space {\text{–}} \space 9)

We now have two pairs of terms inside the brackets that we can try factor by grouping with.

f(x) = 4x(\space 2x^2(4x \space {\text{–}} \space 3) + 3(4x \space {\text{–}} \space 3) \space)

f(x) = 4x(2x^2 + 3)(4x \space {\text{–}} \space 3)


Factor   g(x) = 2a^3 + a^2b \space {\text{–}} \space 6a + 3b.


There is no greatest common factor on this occasion, but we can still try factor by grouping.

g(x) = a^2(2a + b) \space {\text{–}} \space 3(2a + b)

It turns out that factor by grouping still works for this polynomial.

g(x) = (a^2 \space {\text{–}} \space 3)(2a + b)

( A good informative page featuring more complex examples on factoring by grouping
can be seen here at the jdm educational web site. )

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