Grouping by Decomposition

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Example 4

Factor .

Step 1: If possible, identify the GCF.

There is no GCF.

Step 2: Find a pair of numbers with a sum of b (−1 in this case) and a product of ac (−15 × 2 = −30 in this case).

One strategy is to list numbers that multiply to get −30 and to then look within those pairs for a sum of −1, much like in example 3.

first number	second number	product	sum
1	−30	1 × −30 = −30	1 + −30 = −29
−1	30	−1 × 30 = −30	−1 + 30 = 29
2	−15	2 × −15 = −30	2 + −15 = −13
−2	15	−2 × 15 = −30	−2 + 15 = 13
3	−10	3 × −10 = −30	3 + −10 = −7
−3	10	−3 × 10 = −30	−3 + 10 = 7
5	−6	5 × −6 = −30	5 + −6 = −1
−5	6	−5 × 6 = −30	−5 + 6 = 1

Step 3: The numbers 5 and −6 add to −1 and multiply to −30. Rewrite the middle term (b-term) as two terms with coefficients 5 and −6. This step 'decomposes' the middle term into two usable terms.

Step 4: Group the first two terms and the last two terms.

Step 5: Remove the greatest common factor from each of the two groups.

Step 6: The result is a common factor, , in each group. Remove the common factor and simplify.

Verify by expanding.