What are the factors in an algebraic expression?

Numbers have factors:

And expressions (like x2+4x+3) also have factors:

Factoring

Factoring (called "Factorising" in the UK) is the process of finding the factors:

Factoring: Finding what to multiply together to get an expression.

It is like "splitting" an expression into a multiplication of simpler expressions.

Both 2y and 6 have a common factor of 2:

So we can factor the whole expression into:

2y+6 = 2(y+3)

So 2y+6 has been "factored into" 2 and y+3

Factoring is also the opposite of Expanding:

Common Factor

In the previous example we saw that 2y and 6 had a common factor of 2

But to do the job properly we need the highest common factor, including any variables

Firstly, 3 and 12 have a common factor of 3.

So we could have:

3y2+12y = 3(y2+4y)

But we can do better!

3y2 and 12y also share the variable y.

Together that makes 3y:

• 3y2 is 3y × y
• 12y is 3y × 4

So we can factor the whole expression into:

3y2+12y = 3y(y+4)

Check: 3y(y+4) = 3y × y + 3y × 4 = 3y2+12y

More Complicated Factoring

Factoring Can Be Hard !

The examples have been simple so far, but factoring can be very tricky.

Because we have to figure what got multiplied to produce the expression we are given!

It can be hard to figure out!

Experience Helps

With more experience factoring becomes easier.

Hmmm... there don't seem to be any common factors.

But knowing the Special Binomial Products gives us a clue called the "difference of squares":

Because 4x2 is (2x)2, and 9 is (3)2,

So we have:

4x2 − 9 = (2x)2 − (3)2

And that can be produced by the difference of squares formula:

(a+b)(a−b) = a2 − b2

Where a is 2x, and b is 3.

So let us try doing that:

(2x+3)(2x−3) = (2x)2 − (3)2 = 4x2 − 9

Yes!

So the factors of 4x2 − 9 are (2x+3) and (2x−3):

Answer: 4x2 − 9 = (2x+3)(2x−3)

How can you learn to do that? By getting lots of practice, and knowing "Identities"!

Remember these Identities

Here is a list of common "Identities" (including the "difference of squares" used above).

It is worth remembering these, as they can make factoring easier.

a2 − b2	=	(a+b)(a−b)
a2 + 2ab + b2	=	(a+b)(a+b)
a2 − 2ab + b2	=	(a−b)(a−b)
a3 + b3	=	(a+b)(a2−ab+b2)
a3 − b3	=	(a−b)(a2+ab+b2)
a3+3a2b+3ab2+b3	=	(a+b)3
a3−3a2b+3ab2−b3	=	(a−b)3

There are many more like those, but those are the most useful ones.

Advice

The factored form is usually best.

When trying to factor, follow these steps:

• "Factor out" any common terms
• See if it fits any of the identities, plus any more you may know
• Keep going till you can't factor any more

There are also Computer Algebra Systems (called "CAS") such as Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce and many more that are good at factoring.

More Examples

Experience does help, so here are more examples to help you on the way:

An exponent of 4? Maybe we could try an exponent of 2:

w4 − 16 = (w2)2 − 42

Yes, it is the difference of squares

w4 − 16 = (w2 + 4)(w2 − 4)

And "(w2 − 4)" is another difference of squares

w4 − 16 = (w2 + 4)(w + 2)(w − 2)

That is as far as I can go (unless I use imaginary numbers)

Remove common factor "3u":

3u4 − 24uv3 = 3u(u3 − 8v3)

Then a difference of cubes:

3u4 − 24uv3 = 3u(u3 − (2v)3)

= 3u(u−2v)(u2+2uv+4v2)

That is as far as I can go.

Try factoring the first two and second two separately:

z2(z−1) − 9(z−1)

Wow, (z-1) is on both, so let us use that:

(z2−9)(z−1)

And z2−9 is a difference of squares

(z−3)(z+3)(z−1)

That is as far as I can go.

Now get some more experience:

Copyright © 2017 MathsIsFun.com

Algebraic expressions are combinations of variables , numbers, and at least one arithmetic operation.

For example, 2 x + 4 y − 9 is an algebraic expression.

Term: Each expression is made up of terms. A term can be a signed number, a variable, or a constant multiplied by a variable or variables.

Factor: Something which is multiplied by something else. A factor can be a number, variable, term, or a longer expression. For example, the expression 7 x ( y + 3 ) has three factors: 7 , x , and ( y + 3 ) .

Coefficient: The numerical factor of a multiplication expression that contains a variable. Consider the expression in the figure above, 2 x + 4 y − 9 . In the first term, 2 x , the coefficient is 2 : in the second term, 4 y , the coefficient is 4 .

Constant: A number that cannot change its value. In the expression 2 x + 4 y − 9 , the term 9 is a constant.

Like Terms: Terms that contain the same variables such as 2 m , 6 m or 3 x y and 7 x y . If an expression has more than one constant terms, those are also like terms.

Example:

Identify the terms, like terms, coefficients, and constants in the expression.

9 m − 5 n + 2 + m − 7

First, we can rewrite the subtractions as additions.

9 m − 5 n + 2 + m − 7 = 9 m + ( − 5 n ) + 2 + m + ( − 7 )

So, the terms are 9 m , ( − 5 n ) , m , 2 , and ( − 7 ) .

Like terms are terms that contain the same variables.

9 m and 9 m are a pair of like terms . The constant terms 2 and − 7 are also like terms.

Coefficients are the numerical parts of a term that contains a variable.

So, here the coefficients are 9 , ( − 5 ) , and 1 . ( 1 is the coefficient of the term m .)

The constant terms are the terms with no variables, in this case 2 and − 7 .

Algebraic expressions must be written and interpreted carefully. The algebraic expression 5 ( x + 9 ) is not equivalent to the algebraic expression, 5 x + 9 .

See the difference between the two expressions in the table below.

In writing expressions for unknown quantities, we often use standard formulas. For example, the algebraic expression for "the distance if the rate is 50 miles per hour and the time is T hours" is D = 50 T (using the formula D = R T ).

An expression like x n is called a power. Here x is the base, and n is the exponent. The exponent is the number of times the base is used as a factor. The word phrase for this expression is " x to the n th power."

Here are some of the examples of using exponents.