Factoring is one area of math where students have the most problems. Factoring is the reverse of multiplying polynomials. Here are some basic examples.

 

Example

 

Factor 8 – 6 x

 

Find out what all the terms have in common. In this case it is 2, since 2 goes into 8 and – 6 x (2 is a factor of 6). That means rewrite it as 2 times something.

 

2 (4 – 3 x)

 

Answer!

 

Note- To find out what is in the parentheses, use the following steps. Write the 2 (which is the greatest factor, i. e. the biggest number that goes into 8 and –6) then look at the original problem and look at the first term, 8. Ask the question “ What multiplied by 2 will result in 8(the first term)?” Since the answer is 4, write 4. Next, ask “ What multiplied by 2 will get -6x?” The answer is –3x. So the answer is 2 (4 – 3 x).

 

 

Example

 

Factor 4 y – 12

 

The largest number that goes into 4 and –12 is 4 so factor out a 4.

 

4(y – 3)

 

Answer!

 

Note- Anytime there are terms that have similar variables and are asked to factor, factor out the variables also.

 

 

 

Example

 

Factor 3x² + 15 x

 

Look at the numbers (3 and 15). Ask “What is the largest number that goes into both of them?” It would be 3. Factor a 3 out. Notice that every term has an x in it. To find out how many x’s to factor, find the number of x’s that each term has in common. Here the answer is 1, since the first term, 3x², has 2 x’s (the exponent is 2) and the second term, 15 x, has only 1. i.e. the 2 terms only have 1 x in common. Therefore, factor out a 3 and an x (3x).

3 x (x + 5)

 

Answer!

 

 

 

 

 

Example

 

Factor 4a⁵ + 10 a³ - 12 a² b

 

Look at the numbers (4, 10, and -12). Consider all 3 terms, not just 2 of them. The largest number that goes into all of them is 2. Next, see how many a’s each term has in common. The first has 5, the second 3, and the last 2. So each term has 2 a’s in common. The last term is the only one with b so no other term has any b’s in common with it. Therefore, factor out 2a².

 

2a² (?)

 

Look at the first term, 4a⁵, what times 2 will result in 4? The answer is 2. How many more a’s are needed? Since the term (4a⁵) has 5, 3 more a’s are needed.

2a² (2a³ ?)

 

That takes care of the first term now there are two more to deal with. Treat them the same way. For the second term, 10 a³, 2 times 5 is 10 and 1 more a is needed.

 

2a² (2a³ + 5a ?)

 

For the last term, 2 times -6 is -12. No more a’s are needed but 1 b is.

 

2a² (2a³ + 5a – 6b)

 

Answer!

 

 

 

 

 

Example

 

Factor -5y⁹ z¹⁵- 20x y⁸z⁷ - 10 y⁶ z¹¹

 

Look at the numbers (-5, -20, -10). In this case it is easiest to factor out a –5. The reason to factor out a –5 instead of just 5 is because it makes it neater on the inside of the parentheses (In the answer the first term will then be positive). Every term has 6 y’s in common and 7z’s. The middle term is the only one with an x, so do not factor an x.

 

-5y⁶ z⁷(?)

 

Look at the first term, -5y⁹ z¹⁵, -5 times +1 is –5. Also, 3 more y’s and 8 more z’s are needed.

 

-5y⁶ z⁷(y³ z⁸ + ?)

 

+1 times anything is itself so it is left out. On the second term, - 20x y⁸z⁷, -5 times 4 is –20. 1 x and 2 y’s are needed.

 

-5y⁶ z⁷(y³ z⁸ + 4 x y² + ?)

 

Get the last term.

-5y⁶ z⁷(y³ z⁸ + 4 x y² + 2z⁴)

 

Answer!

 

 

 

 

Factoring is used a lot in simplifying problems.

 

Example

Simplify

 

Factor the top. 3xy(y + 2).

 

 

 

Group the bottom with parentheses to look like one term.

 

 

 

There is a (y + 2) on top and a (y + 2) on bottom. They can cancel out and it leaves a 3xy on top.

 

Answer!