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➕ Algebra 1 · Factoring

Factoring by Grouping: notes and common mistakes

Study notes

Key ideas

  • Factoring by grouping works for certain four-term polynomials: those whose two pairs leave the same binomial. If all four terms share a GCF, take it out first.
  • Pair the terms, take out each pair’s GCF, and look for the same binomial in both. Then pull it out: ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y).
  • Choose the sign of each GCF so the binomials match. Taking out a −1 turns (−x + 4) into −(x − 4).
  • If the first pairing does not work, check signs and then try rearranging. If no arrangement matches, grouping fails. That does not prove the polynomial is prime.
  • Grouping also splits the middle term of a trinomial: find m and n with m × n = a × c and m + n = b.
  • Always check by expanding, and factor completely.

Steps

  1. G: take out the GCF of all four terms, if there is one.
  2. P: pair the terms (first two, last two), keeping each sign with its term, and take out each pair’s GCF.
  3. M: make sure the leftover binomials match. Fix the sign with a negative GCF, or rearrange the terms.
  4. E: extract the common binomial, then expand to check.
  5. Look at each factor and keep factoring if you can.

Formulas

  • Grouping pattern: ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)
  • Taking out a negative: −px + pq = −p(x − q)
  • Splitting the middle term: ax2 + bx + c = ax2 + mx + nx + c, where m + n = b and m × n = a × c

Watch out

  • x2(x + 3) + 2(x + 3) is still a sum. Pull out the binomial to get a product.
  • The sign belongs to the term: −3x + 12 = −3(x − 4), not 3(x − 4).
  • Do not drop a 1 or a −1: x2(x + 5) − 1(x + 5) = (x + 5)(x2 − 1), not (x + 5)(x2).
  • When all four terms share a GCF, take it out first, or you will have a common factor left over at the end.
  • If the binomials do not match, do not force them. Check signs, try rearranging, or accept that grouping fails (which does not mean the polynomial is prime).
  • To split the middle term, the product is a × c and the sum is b. It is not the other way around.

Why it works

  • Why does grouping work? x2(x + 3) + 2(x + 3) has (x + 3) in both terms, so the distributive property run backward gives (x2 + 2)(x + 3).
  • Why does the sign matter? (x − 4) and (4 − x) are opposites, not equal. Only identical binomials can be pulled out as one factor.
  • Why may you rearrange the terms? Addition can be done in any order, as long as each sign stays with its term.

Common mistakes

✗ Mistakex2(x + 3) + 2(x + 3) is the final answer.
✓ InsteadThat is still a sum of two terms. Pull out the common binomial to get the product (x + 3)(x2 + 2).
✗ MistakeIn x3 − 4x2 − 3x + 12, the second pair is 3x + 12, so take out 3.
✓ InsteadThe sign belongs to the term: the pair is −3x + 12. Take out −3 to get −3(x − 4), which matches the first pair’s (x − 4).
✗ Mistakex3 + 5x2 − x − 5 = x2(x + 5) − (x + 5) = (x + 5)(x2) (the −1 disappears).
✓ Instead−(x + 5) means −1(x + 5), so the −1 stays: (x + 5)(x2 − 1). That factors further into (x + 5)(x + 1)(x − 1).
✗ MistakeIf the binomials do not match, just write them as the same binomial anyway.
✓ InsteadDifferent binomials mean that arrangement does not work. Check your signs, try rearranging the terms, or accept that grouping fails. Never change a binomial to force a match.
✗ MistakeTo split the middle term, find two numbers with sum a × c and product b.
✓ InsteadThe product is a × c and the sum is b. For 2x2 + 7x + 6, look for product 12 and sum 7, which gives 3 and 4.

Teach it back

Explain to a friend how to factor x3 − 9x2 − 2x + 18 by grouping, why the second pair needs a negative GCF, and how to check your answer.

Explaining it out loud shows what you really know. Did you…

  • I paired the terms as (x3 − 9x2) + (−2x + 18) and took out x2 from the first pair: x2(x − 9).
  • I took out −2 (not 2) from the second pair, so it became −2(x − 9) and the binomials matched.
  • I pulled out (x − 9) to get (x − 9)(x2 − 2).
  • I checked by expanding: (x − 9)(x2 − 2) = x3 − 2x − 9x2 + 18, which is the original polynomial.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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