Factoring is another way of saying: "Rewrite this so everything is all multiplied together." You usually start out with two or more terms and have to determine how to rewrite them so they're all multiplied together in some way or another. And, oh yes, the two expressions have to be equal!
Factoring is the opposite of distributing; it's "undistributing." In distribution, you multiply a series of terms by a common factor. Now, by factoring, you seek to find what a series of terms have in common and then take it away, dividing the common factor out from each term. Think of each term as a numerator and then find the same denominator for each. By factoring out, the common factor is put outside a parenthesis or bracket and all the results of the divisions are left inside.
1. Determine a common factor.
In the term 16a - 8b + 40c2, 2 is a common factor.
2. Divide or "undistribute" each term by the common factor and write the results of the division in parenthesis with the factor out in front.
In this example, this looks like 16a - 8b + 40c2 = 2 (8a - 4b +20c2)
3. Determine whether you can factor out any other terms.
The terms left in the parenthesis are still too big. They all still have something in common: 4. Factoring out 4, you get
• 2 (4 [2a - b + 5c2])
4. Simplify your answer.
If you factor out a 4 after factoring out the 2, then the product of 4 and 2, which is 8, is the total amount you factored out:
8 (2a - b + 5c2)
It's nice when you recognize going in that you can factor out a larger number, such as 8, but if it takes a couple steps to get to it, that's fine.
 | Reviewing the definitions of the following words and keeping them in mind can help. |
- Term: Group of number(s) and/or variable(s) connected to one another by multiplication or division and separated from other terms by addition or subtraction.
- Factor: Any of the values involved in a multiplication problem that when multiplied together produces a result.
- Coefficient: Number that multiplies a variable and tells how many of the variable.
- Constant: Number or variable that never changes in value.
For example, in the expression 5xy + 4z - 6, three terms are separated by the plus and minus signs. In the first term, 5xy, three factors are all multiplied together. The 5 is usually referred to as the coefficient. The second term has two factors, 4 and z, and the third term has only one — a constant.
| An expression can be written as the product of the largest number that divides all the terms evenly times the results of the divisions. |
ab + ac + ad = a(b + c + d)
The following examples put this to practice:
- Stephen has 6 cats; Brad has 18 hamsters; Carlos has 16 parakeets; Donald has 4 dogs. These pet owners want to take their pets to various nursing homes to visit the residents, but they want to divide the animals into similar groups. How can they do this?
The sum of numbers representing the animals is 6 + 18 + 16 + 4, each of which can be divided evenly by 2. The 6 and 18 can be divided by 6, but the 16 and 4 cannot be divided by 6. The 16 and 4 can be divided by 4, but the six and 18 cannot be divided by 4.
Two is the biggest number that divides each evenly.
So these gentlemen can take 2 groups of animals to the nursing homes: 2(3 cats + 9 hamsters + 8 parakeets + 2 dogs). What a nice thing for them to do!
- Each of the terms in this example has a coefficient that three divides evenly. The GCF (greatest common factor) of the numbers is three, so a factor larger than three that can divide all the terms evenly is unavailable.
Factor 9x +15y - 12z = 30:
9x + 15y - 12z +30 = 3 (3x + 5y - 4z +10)
The terms in the parenthesis are the results of dividing each term by 3. Those terms don't have anything in common.
- Each term in the following example is divisible by 6.
Factor 18a2 - 24b - 36c +42
18a2 - 24b -36c +42 + 6 (3a2 -4b -6c +7)
| Relatively prime means that two terms have no prime factors in common. If the only factor that two numbers share in common is 1, then they're considered relatively prime. |
For example, 1 is the only number that divides into both 18 and 15. Although neither 18 nor 25 is a prime number, they are relatively prime.
- The proper way to factor the following expression would be to write the prime factorization of each of the numbers and look for the GCF (greatest common factor). What's really more practical and quicker in the end is to look for the biggest factor that you can easily recognize. Factor it out and then see if the numbers in the parentheses need to be factored again. Repeat the division until the terms in the parentheses are relatively prime.
450x + 540y - 486z + 216
450x + 540y - 486z + 216 = 2 (225x + 270y - 243z +108)
The numbers in the parentheses are a mixture of odd and even, so you can't divide by 2 again. The numbers in the parentheses are all divisible by 3, but there's an even better choice.
You may have noticed that the digits in the numbers in all the terms add up to 9. That's the rule for divisibility by 9, so 9 can divide each term evenly. Thus,
2 (225x + 270y - 243z +108) = 2 [9 (25x + 30y -27z + 12)]
Now multiply the 2 and 9 together to get
450x + 540y - 486z + 216 =
18 (25x + 30y -27z + 12)
You could have divided 18 into each term in the first place, but not many people know the multiplication table of 18.
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