➕ Algebra 1 · Factoring
Factoring Out the Greatest Common Factor: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Mutual friends
Compare your friend list with a classmate’s. The friends you both have are your mutual friends. Now treat primes as friends. Write 20 = 2 · 2 · 5 and 50 = 2 · 5 · 5. The mutual friends of the two numbers are one 2 and one 5. The number 20 has two 2’s but 50 has only one, so only one 2 is shared. The number 50 has two 5’s but 20 has only one, so only one 5 is shared.
How it connects: The GCF is the product of the shared primes: 2 · 5 = 10, so the GCF of 20 and 50 is 10. This is the lowest-count rule: each prime (and each variable) is used as many times as the number that has the fewest copies. The limit: a GCF is a product of the shared factors, not a list, and a friend can only be shared as many times as both lists have it.
Fruit cups
You are making identical fruit cups, with nothing left over, from 32 strawberries and 48 blueberries. The most cups you can make is 16, because 16 is the largest number that divides both 32 and 48. Each cup gets 2 strawberries and 3 blueberries.
How it connects: 32 + 48 = 16(2 + 3). The GCF, 16, is the number of cups, and the parentheses hold what is in one cup. Fewer cups also work. Eight cups of 4 strawberries and 6 blueberries gives 8(4 + 6), but that is not completely factored, because each of those cups could still be split into two identical smaller cups. The limit: counting fruit uses whole numbers only. With letters, one “cup” can hold variable amounts, like 4x2.
A chain is only as strong as its weakest link
This proverb means that something is only as strong as its weakest part. Think of 10x9, 15x6 and 5x2 as stacks of x’s: 9 of them, 6 of them and 2 of them. If you want to take the same number of x’s out of every stack, you can take at most 2, the size of the shortest stack. The weakest link sets the limit.
How it connects: The GCF uses the lowest exponent: x2 is the most you can pull out of all three terms, so 10x9 + 15x6 + 5x2 = 5x2(2x7 + 3x4 + 1). The limit: the weakest-link rule is only for the exponents of the letters. For the numbers, use the greatest common factor of the coefficients, which is not always the smallest coefficient. The GCF of 6, 4 and 8 is 2, not 4.
Find common ground
To “find common ground” means to discover what two sides already agree on. Say one friend wants noodles with chicken and another wants noodles with tofu. The common ground is noodles. Set that aside and only the topping is left to decide.
How it connects: In 6x2 + 9x, the common ground is 3x, because both terms already contain it. Pull it out and only what differs is left inside: 3x(2x + 3). The limit: common ground has to be shared by every term. In 6x2 + 9 the letter x is missing from the second term, so x is not common ground, and only the 3 can be pulled out: 3(2x2 + 3).
Instant replay
In many sports, officials can review video to check a call. The replay does not change the play. It only checks whether the call matched what happened.
How it connects: Distributing your answer back out is your instant replay. If 5y2(2y + 3) multiplies out to 10y3 + 15y2, the original, the factoring is correct. The limit: a replay confirms only that your answer equals the original. It cannot tell you whether you took out the greatest common factor. 3(4x + 6) replays correctly to 12x + 18, yet it is not completely factored, because 4x + 6 still has the common factor 2: 6(2x + 3).
The promoted 3
Why did the 3 get promoted to the front office? In 6x + 9 it was the one thing both terms had in common, so it got pulled out front: 3(2x + 3).
How it connects: Pulling the 3 out front does not change the value: 3 · 2x + 3 · 3 = 6x + 9. The limit: only a factor that every term has can move up front. In 6x + 10 the 3 does not divide 10, so the 3 stays put. (The GCF there is 2.)
Memory help
PULL it out
Pick the biggest number that divides every coefficient. Use each variable that appears in every term, at its lowest exponent. Leave the leftovers inside: divide each term by the GCF. Look back: distribute to check, then look inside for anything still common. Say it as “PULL it out front.”
The GCF rhyme
Biggest number they all share, lowest power of letters found in every term there. Pull it out and leave the rest; multiply back to pass the test.
The overlap picture
Draw two overlapping circles, one for each number. Write each number’s prime factors in its circle, and put the primes both numbers share in the overlap. Multiply what is in the overlap to get the GCF. For 24 and 36 the overlap holds 2, 2, 3, and 2 · 2 · 3 = 12. (If a prime appears three times in one number and twice in the other, only two copies go in the overlap.)
The three-column table
Make a table with the columns Term, ÷ GCF, and Leftover. Write each original term in the first column and its quotient in the third. The leftovers, with their signs and in the same order, go inside the parentheses. A row that leaves 1 shows you the 1 you must not forget.
The vanishing 1
5m3 + 5m2 has GCF 5m2, and the second term gives 5m2 ÷ (5m2) = 1. So the answer is 5m2(m + 1), not 5m2(m). Distribute to see why: 5m2 · m is only 5m3, and the second term is gone. If a term equals the GCF, a 1 stays behind.
Lowest power, not highest
4x2 + 6x5 has GCF 2x2, not 2x5. Only x2 fits inside both terms, since x5 does not fit inside x2. And a letter missing from any term is out: 6a2 + 9b has GCF 3 and nothing else.