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
Examples:
(1.1)
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)
(1.2)
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
(2.1)
Factor f(x) = 20x^4 \space {\text{–}} \space 32x^3 + 48x^2 \space {\text{–}} \space 36x.
Solution
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)
(2.2)
Factor g(x) = 2a^3 + a^2b \space {\text{–}} \space 6a + 3b.
Solution
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. )