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➕ Algebra 1 · Factoring

Factoring Out the Greatest Common Factor: notes and common mistakes

Study notes

Key ideas

  • Factoring is multiplying backward. Pulling out a common factor is the distributive property run backward: ab + ac = a(b + c).
  • The GCF of terms is the GCF of the coefficients, times each variable that appears in every term, raised to the lowest exponent found in the terms.
  • To factor it out, divide each term by the GCF and write the quotients inside the parentheses with their signs. A term equal to the GCF leaves a 1.
  • Check by distributing. That confirms the answer is correct, but not that it is complete.
  • If the leading coefficient is negative, factor out the negative of the GCF. If the GCF is 1, there is nothing to pull out. A common factor can also be a whole binomial.
  • An expression is completely factored when no common factor greater than 1 remains inside the parentheses (and, in later lessons, when no other method applies either).

Steps

  1. Write the polynomial in standard form (descending powers) so you can see the leading coefficient.
  2. Find the GCF of the coefficients.
  3. Find each variable that appears in every term, and use its lowest exponent.
  4. Multiply the pieces to get the GCF. If the leading coefficient is negative, use the negative of the GCF.
  5. Divide each term by the GCF and write the quotients inside parentheses, keeping signs.
  6. Check by distributing, then look inside the parentheses for any common factor that is left.

Formulas

  • Factoring (distributive property backward): ab + ac = a(b + c)
  • Factoring out a negative: −ab + ac = −a(b − c)
  • Lowest power: xm and xn with m ≤ n have GCF xm, because xn = xm · xn−m
  • Common binomial factor: a(b + c) + d(b + c) = (b + c)(a + d)

Watch out

  • Use the lowest exponent, not the highest: the GCF of 4x2 and 6x5 includes x2, not x5.
  • A letter must appear in every term to be in the GCF. A constant term has no letters.
  • Do not drop the 1. If a term equals the GCF, its quotient is 1: 3z + 9z2 = 3z(1 + 3z).
  • Dividing by a negative flips every sign: −10t + 25 = −5(2t − 5).
  • Distributing checks that your answer is correct. It does not check that you took out the greatest factor, so look inside the parentheses too.

Why it works

  • Why does it work? a(b + c) means ab + ac, so reading it backward lets you pull a shared a out of both terms.
  • Why the lowest exponent? x5 = x2 · x3, so x2 fits inside x5. But x2 ÷ (x5) would leave x−3, which is not a polynomial term, so x5 cannot be pulled out of x2.
  • Why bother? Factored form shows that a value is a multiple of the GCF, gives the dimensions of a rectangle from its area, and prepares you to solve equations, because a product is 0 only when one of its factors is 0.

Common mistakes

✗ MistakeThe GCF of x5 and x2 is x5 (use the biggest exponent).
✓ InsteadUse the lowest exponent. x2 fits inside both x5 and x2, but x5 does not fit inside x2. The GCF is x2.
✗ Mistake3a2 + 3a = 3a(a) (nothing is left of the second term).
✓ Instead3a ÷ (3a) = 1, so a 1 is left: 3a2 + 3a = 3a(a + 1). Distribute 3a(a) and you only get 3a2.
✗ Mistake−10t + 25 = −5(2t + 5) (forgetting that dividing by a negative flips the sign).
✓ Instead25 ÷ (−5) = −5, so −10t + 25 = −5(2t − 5). Check: −5 · 2t = −10t and −5 · (−5) = +25.
✗ Mistake10x + 20 = 5(2x + 4) is completely factored, because it multiplies out correctly.
✓ InsteadMultiplying out only shows the answer is correct. Inside, 2x + 4 still has the common factor 2, so the complete factoring is 10(x + 2).
✗ MistakeIf the GCF is 1, the expression cannot be factored at all.
✓ InsteadA GCF of 1 only means there is no common factor to pull out. Other methods can still work, such as x2 + 7x + 10 = (x + 2)(x + 5) in the next lesson.
✗ MistakeA letter that appears in most of the terms can go in the GCF.
✓ InsteadA letter must appear in every term. In 6a2 + 9b, neither letter is in both terms, so the GCF is 3.

Teach it back

Explain to a friend how to factor −12x3 + 18x2 so that the first term inside the parentheses is positive, and how you would check your answer.

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

  • I found the GCF of 12 and 18 (which is 6) and the lowest exponent of x (which is 2), so the GCF is 6x2.
  • Because the leading coefficient is negative, I factored out −6x2 and divided each term by it: −12x3 ÷ (−6x2) = 2x and 18x2 ÷ (−6x2) = −3.
  • I wrote −6x2(2x − 3) and checked by distributing: −6x2 · 2x = −12x3 and −6x2 · (−3) = 18x2.
  • I looked inside the parentheses and saw that 2 and 3 share no common factor, so it is completely factored.

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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