Factoring: GCF & Grouping
Learn to factor out the GCF (including a negative GCF) and factor four-term polynomials by grouping, then check every answer by re-multiplying.
What you'll do in this lesson
A voice-first session with the Crimsora tutor on Factoring: GCF & Grouping, then targeted practice and FRQs — with the tutor adapting to where you get stuck.
What this lesson covers
Multiplying polynomials takes two factors and produces one long expression. Factoring runs that machine backward: you start with the long expression and hunt for the factors that built it. In this lesson you will pull out the greatest common factor of a polynomial, handle the tricky case where the GCF is negative, and break four-term polynomials apart using grouping.
These two moves are the foundation of the rest of the unit. Before you factor a trinomial or recognize a difference of squares, you always check for a common factor first — skipping that step is the single most common reason a factoring answer comes out half-finished. And because factoring is just the distributive property in reverse, you can always check yourself: multiply your factors back out and see whether the original polynomial reappears.
These two moves are the foundation of the rest of the unit. Before you factor a trinomial or recognize a difference of squares, you always check for a common factor first — skipping that step is the single most common reason a factoring answer comes out half-finished. And because factoring is just the distributive property in reverse, you can always check yourself: multiply your factors back out and see whether the original polynomial reappears.
Factoring Is the Distributive Property Run Backward
When you distribute, you take and spread the across both terms to get . Factoring asks the opposite question: given , what was multiplied to produce it?
The answer comes from finding what both terms share. Write each term as a product of its pieces:
and .
Both contain a factor of and a factor of , so can be pulled out front: .
The part left inside the parentheses is what remains after each term is divided by the common factor. That is the mechanical rule: divide every term by the GCF, and the quotients become the terms inside.
Because factoring and distributing are inverse operations, checking is built in. Re-multiply and you get — the original. Make this check automatic. It takes ten seconds and catches nearly every sign slip and dropped term.
One caution about vocabulary: a factored expression is a product. If your answer still has a plus or minus sign at the top level, connecting two separate chunks, you have not finished factoring. For example, is not factored — it is a sum of two products. Factored means one multiplication statement.
The answer comes from finding what both terms share. Write each term as a product of its pieces:
and .
Both contain a factor of and a factor of , so can be pulled out front: .
The part left inside the parentheses is what remains after each term is divided by the common factor. That is the mechanical rule: divide every term by the GCF, and the quotients become the terms inside.
Because factoring and distributing are inverse operations, checking is built in. Re-multiply and you get — the original. Make this check automatic. It takes ten seconds and catches nearly every sign slip and dropped term.
One caution about vocabulary: a factored expression is a product. If your answer still has a plus or minus sign at the top level, connecting two separate chunks, you have not finished factoring. For example, is not factored — it is a sum of two products. Factored means one multiplication statement.
Finding the Greatest Common Factor of a Polynomial
The GCF of a polynomial has a number part and a variable part, and you find them separately.
For the number part, take the greatest common factor of the coefficients (ignore signs for now). For the variable part, take each variable that appears in every term, raised to the smallest exponent it has anywhere in the polynomial. A variable missing from even one term cannot be part of the GCF.
Consider .
The GCF of , , and is . The smallest power of is ; the smallest power of is . So the GCF is , andTwo places students go wrong. First, when a term is entirely used up by the GCF, they leave a blank instead of writing . In , dividing the second term by gives , so the answer is , not . Second, they pull out a common factor that is not the greatest one: is a true statement but is not fully factored, because still divides both inside terms. Always ask whether the leftover terms still share anything.
For the number part, take the greatest common factor of the coefficients (ignore signs for now). For the variable part, take each variable that appears in every term, raised to the smallest exponent it has anywhere in the polynomial. A variable missing from even one term cannot be part of the GCF.
Consider .
| Term | Coefficient | Power of | Power of |
|---|---|---|---|
| 18 | 4 | 2 | |
| 24 | 3 | 3 | |
| 30 | 3 | 1 |
Pulling Out a Negative GCF
When the leading term of a polynomial is negative, it is standard practice to factor out a negative GCF so the polynomial inside the parentheses starts with a positive term. This matters later: trinomials are far easier to factor when the leading coefficient is positive.
Take . The positive GCF is , so the negative GCF is . Divide every term by :
, , and .
So .
Notice that every sign inside flipped compared with factoring out positive . That is the whole trap. Students commonly write by changing only the first sign, or by forgetting the last one. Dividing each term one at a time, out loud, prevents this — and re-multiplying catches it if it slips through.
A related use of the negative GCF is reversing a binomial. Since , the expressions and are opposites. Factoring out of one turns it into the other, which is exactly the move that rescues a grouping problem when the two binomials come out backward from each other.
One more note: factoring out a negative is a choice, not a law. Both and are correct products. The first form is preferred because it sets up the next factoring step cleanly.
Take . The positive GCF is , so the negative GCF is . Divide every term by :
, , and .
So .
Notice that every sign inside flipped compared with factoring out positive . That is the whole trap. Students commonly write by changing only the first sign, or by forgetting the last one. Dividing each term one at a time, out loud, prevents this — and re-multiplying catches it if it slips through.
A related use of the negative GCF is reversing a binomial. Since , the expressions and are opposites. Factoring out of one turns it into the other, which is exactly the move that rescues a grouping problem when the two binomials come out backward from each other.
One more note: factoring out a negative is a choice, not a law. Both and are correct products. The first form is preferred because it sets up the next factoring step cleanly.
Factoring Four-Term Polynomials by Grouping
When a polynomial has four terms and no single GCF for all four, try grouping. The strategy is to split the polynomial into two pairs, factor the GCF out of each pair, and hope the two leftover binomials match.
Factor .
Group the first two and the last two: . The GCF of the first pair is , giving . The GCF of the second pair is , giving . Now the expression reads . Both terms share the common binomial factor , so pull it out front: .
Subtraction in the third position is where errors cluster. In , rewrite the second group with its sign attached: . Factoring (not ) out of the second group gives , matching from the first group. The result is . If you had factored out you would get , and the binomials would not match.
If the two binomials come out as opposites, such as and , factor out of one group to flip it. If they simply do not match at all, try pairing the terms in a different order — grouping the first with the third and the second with the fourth often works.
Always look for an overall GCF before grouping. Removing it first keeps the numbers small and prevents an incompletely factored answer.
Factor .
Group the first two and the last two: . The GCF of the first pair is , giving . The GCF of the second pair is , giving . Now the expression reads . Both terms share the common binomial factor , so pull it out front: .
Subtraction in the third position is where errors cluster. In , rewrite the second group with its sign attached: . Factoring (not ) out of the second group gives , matching from the first group. The result is . If you had factored out you would get , and the binomials would not match.
If the two binomials come out as opposites, such as and , factor out of one group to flip it. If they simply do not match at all, try pairing the terms in a different order — grouping the first with the third and the second with the fourth often works.
Always look for an overall GCF before grouping. Removing it first keeps the numbers small and prevents an incompletely factored answer.
Checking by Re-Multiplying and Knowing When to Stop
Every factoring answer can be verified, so there is no reason to hand in an unchecked one. To check, distribute your factors back out and compare with the original polynomial term by term.
Check : multiply to get , then reorder as . It matches, so the factoring is correct.
Check a GCF answer the same way. Does equal ? Distributing gives . It matches.
Knowing when you are done is the other half. A polynomial is completely factored when none of the remaining factors can be broken down further. After pulling out a GCF, look at what is left inside the parentheses: if it is a trinomial or a difference of squares, upcoming lessons in this unit will let you keep going. After grouping, check each of the two binomial factors the same way.
Some polynomials are prime over the integers — they have no factors besides and themselves. If the coefficients share no common factor and grouping fails in every arrangement, the polynomial may simply be prime, and saying so is a complete answer.
A quick habit that saves grief: count terms first. Two terms suggests GCF or a special form; three suggests GCF then trinomial factoring; four suggests GCF then grouping.
Check : multiply to get , then reorder as . It matches, so the factoring is correct.
Check a GCF answer the same way. Does equal ? Distributing gives . It matches.
Knowing when you are done is the other half. A polynomial is completely factored when none of the remaining factors can be broken down further. After pulling out a GCF, look at what is left inside the parentheses: if it is a trinomial or a difference of squares, upcoming lessons in this unit will let you keep going. After grouping, check each of the two binomial factors the same way.
Some polynomials are prime over the integers — they have no factors besides and themselves. If the coefficients share no common factor and grouping fails in every arrangement, the polynomial may simply be prime, and saying so is a complete answer.
A quick habit that saves grief: count terms first. Two terms suggests GCF or a special form; three suggests GCF then trinomial factoring; four suggests GCF then grouping.
Key terms
- Factor (verb).
- To rewrite a sum or difference as a product of two or more expressions, undoing the distributive property.
- Greatest common factor (GCF).
- The largest monomial that divides evenly into every term of a polynomial: the GCF of the coefficients times each shared variable raised to its smallest exponent.
- Negative GCF.
- A common factor taken out with a negative sign, used when the leading term is negative; every term inside the parentheses changes sign compared with factoring out the positive version.
- Factoring by grouping.
- A method for four-term polynomials: split into two pairs, factor a GCF from each pair, then factor out the shared binomial.
- Common binomial factor.
- A two-term expression such as that appears as a factor in more than one part of an expression and can be pulled out front.
- Completely factored.
- Written as a product in which no remaining factor can be factored further over the integers.
- Prime polynomial.
- A polynomial that cannot be written as a product of two lower-degree polynomials with integer coefficients.
- Opposite binomials.
- Two binomials like and that differ by a factor of ; factoring out of one converts it into the other.
Worked example
Factor completely: . Then check your answer by re-multiplying.
Step 1: Look for an overall GCF of all four terms. The coefficients are , , , and , which share a factor of . The last term has no , so no variable is common. Factor out :Step 2: The expression inside has four terms, so group in pairs, keeping the signs attached:Step 3: Factor a GCF from each pair. From pull out to get . From pull out (negative, so the leftover binomial matches) to get . Now:Step 4: Both inner terms share the binomial . Pull it out:Step 5: Is it completely factored? The factor is linear, and is not a difference of squares over the integers because is not a perfect square. Done.
Step 6: Check. Multiply . Multiply by : . This matches the original, so the factoring is correct.
Step 6: Check. Multiply . Multiply by : . This matches the original, so the factoring is correct.
Practice questions
Which expression shows with its negative GCF factored out?
Answer:
The coefficients , , and share a factor of , and every term has at least one , so the GCF is ; since the leading term is negative, use . Divide each term: , , and . Note that every sign flipped. The choice changes only some of the signs. The choice is a true product but is not the greatest common factor. The last choice is a correct product too, but it leaves a negative leading term inside, which is not the requested form.
Factor completely, and show the check.
Answer:
All four terms share no common factor other than (, , , have no common divisor), so go straight to grouping. Pair them: . From the first pair factor : . From the second pair factor : . The binomials match, so factor out to get . Check by multiplying: , , , . Combining gives , the original polynomial.
A student factors as and then says the polynomial cannot be factored by grouping because the binomials are different. What went wrong, and what is the correct factorization?
Answer: The second group should be factored with a negative GCF, giving ; the polynomial factors as .
The student's line is algebraically true, but choosing as the GCF of produced , which is the opposite of rather than a mismatch. Factoring out instead gives , so the expression becomes and the shared binomial appears. Pulling it out gives . Checking: , , , , which reassembles the original. Whenever two binomials are opposites, factoring out of one group fixes the mismatch.
FAQ
- What if the terms have no common factor other than 1?
- Then the GCF is and there is nothing useful to pull out, so move on to another method. With four terms, try grouping. With three terms, use trinomial factoring. With two terms, check for a difference of squares. If nothing works in any arrangement, the polynomial may be prime over the integers, and stating that is a complete answer.
- Do I always have to factor out a negative when the first term is negative?
- It is not required by the rules of algebra — both versions are valid products — but it is the standard convention and it makes life easier. Leaving a positive leading coefficient inside the parentheses lets you apply trinomial and special-form methods without extra sign juggling. Just remember that switching to a negative GCF flips the sign of every single term inside.
- What do I do if grouping gives me two binomials that do not match?
- First check whether they are opposites, like and ; if so, factor out of one group to flip it. If they are genuinely different, rearrange the four terms and pair them differently — grouping the first with the third and the second with the fourth is the usual second attempt. If no arrangement works, the polynomial does not factor by grouping.
- How do I know when a polynomial is completely factored?
- Examine each factor you have written and ask whether it can be broken down further. A monomial GCF like is finished. A linear binomial like is finished. But a trinomial, a difference of squares, or a binomial whose terms still share a factor can go further. Re-multiplying confirms your factoring is correct; scanning each factor confirms it is complete.
Learn this with a teacher, not a page
The Crimsora tutor teaches Factoring: GCF & Grouping live — explaining on a whiteboard, asking you questions, and adapting to where you get stuck.