Factoring

Factoring by Grouping

Factor a four-term polynomial by grouping pairs. Group the first two and last two terms, factor out the GCF from each group, then factor out the common binomial. Verify by expanding the final factored form.

Example

Group terms, factor each group, then factor the shared binomial.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

x3+2x2+3x+6x^{3} + 2x^{2} + 3x + 6

Step 2 — Group terms

Group the first pair and the second pair.

(x3+2x2)+(3x+6)( x^{3} + \hlmath{2x^{2}} )+( \hlmath{3x} + \hl{6} )

Step 3 — Factor groups

Factor each group: both contain x + 2, with outside factors x squared and 3.

x2(x+2)+3(x+2)\hlmath{x^{2}} (x+ 2 )+ \hl{3} (x+ 2 )

Step 4 — Common binomial

The common binomial is x + 2.

(x+2)(x+ \hl{2} )

Step 5 — Factored form

Factor out the common binomial: x + 2, leaving x squared + 3.

(x+2)(x2+3)(x+ \hl{2} )( x^{2} + \hl{3} )
factor-by-grouping After grouping, both pairs must share the same binomial factor. Factor that common binomial to get the product of two factors. The polynomial expansion check confirms correctness.