Common factors

In factoring, one of the first and most important skills is taking out a common factor. In this chapter we focus only on that idea.

A common factor of several terms is a quantity that divides each term exactly. To factor out a common factor means to rewrite an expression as a product, pulling that shared piece in front using the distributive property in reverse.

Greatest common factor (GCF) of terms

When we factor, we usually want to take out the greatest common factor (GCF) of all the terms. This makes the remaining expression inside the parentheses as simple as possible.

For numerical parts, the GCF is the largest integer that divides each coefficient. For example:

• The GCF of $12$ and $18$ is $6$.
• The GCF of $15$, $25$, and $35$ is $5$.

In algebraic terms, each term usually has:

• a numerical coefficient (like $3$, $-7$, or $10$), and
• a variable part (like $x^2$, $y^3$, or $x^2y$).

To find the GCF of algebraic terms, we consider both the number and the variables.

GCF of numerical coefficients (briefly)

For coefficients, the GCF is the largest positive integer dividing all of them. For example:

• GCF of $8$ and $12$ is $4$.
• GCF of $18$, $27$, and $45$ is $9$.

If the coefficients are negative, we usually still speak of a positive GCF; signs are handled separately.

GCF of variable parts

For variable factors, the GCF uses:

• variables that appear in every term, and
• the smallest exponent on that variable among the terms.

Examples:

• GCF of $x^3$ and $x^5$ is $x^3$.
• GCF of $x^2y$ and $xy^3$ is $xy$ (because:
• for $x$: exponents are $2$ and $1$, so take $x^1 = x$,
• for $y$: exponents are $1$ and $3$, so take $y^1 = y$).

If a variable does not appear in each term, it is not part of the GCF.

For example, for $3x^2y$ and $6xy^3z$:

• numeric GCF is $3$ (between $3$ and $6$),
• variable GCF:
• $x$: exponents $2$ and $1$ $\Rightarrow$ take $x^1 = x$,
• $y$: exponents $1$ and $3$ $\Rightarrow$ take $y^1 = y$,
• $z$: appears only in one term, so not part of GCF.

So GCF is $3xy$.

Factoring out a common factor

Once you find the GCF, you:

1. Write the GCF outside parentheses.
2. Divide each term of the original expression by the GCF to find what remains inside the parentheses.

This is exactly using the distributive property in reverse:
$$
ab + ac = a(b + c).
$$

Here $a$ is the common factor.

Basic numerical examples

1. Factor $12x + 8$.

• Coefficients $12$ and $8$ have GCF $4$.
• Variable: only $x$ in the first term, none in the second, so no variable in the GCF.
• GCF is $4$.

Now divide each term by $4$:

• $12x \div 4 = 3x$,
• $8 \div 4 = 2$.

So:
$$
12x + 8 = 4(3x + 2).
$$

2. Factor $15y - 20$.

• GCF of $15$ and $20$ is $5$.
• No variable appears in both terms (the constant $-20$ has no $y$), so GCF is $5$.

Divide:

• $15y \div 5 = 3y$,
• $-20 \div 5 = -4$.

So:
$$
15y - 20 = 5(3y - 4).
$$

Including variables in the GCF

Factor $18x^2y - 12xy^3$.

• Numeric GCF of $18$ and $12$ is $6$.
• Variable GCF:
• $x^2$ and $x$ $\Rightarrow$ take $x$,
• $y$ and $y^3$ $\Rightarrow$ take $y$.

So GCF is $6xy$.

Now divide:

• $18x^2y \div 6xy = 3x$,
• $12xy^3 \div 6xy = 2y^2$.

So:
$$
18x^2y - 12xy^3 = 6xy(3x - 2y^2).
$$

You can always check by distributing $6xy$ back in.

Common factors with signs

Sometimes it is helpful to factor out a negative common factor, especially when the leading term is negative.

Taking out a positive factor

Factor $-9x + 6$.

• Numeric GCF of $9$ and $6$ is $3$.
• If we take out $3$:
  $$
  -9x + 6 = 3(-3x + 2).
  $$

This is correct and fully factored by a common factor.

Taking out a negative factor

If we prefer the first term inside the parentheses to be positive, we can factor out $-3$ instead:

• GCF is $3$, but allow a negative: use $-3$.

Divide:

• $-9x \div (-3) = 3x$,
• $6 \div (-3) = -2$.

So:
$$
-9x + 6 = -3(3x - 2).
$$

Both $3(-3x + 2)$ and $-3(3x - 2)$ are correct factorizations; which form is preferable can depend on context.

In many algebra problems, it is common to:

• factor out a negative from a polynomial if its leading coefficient (the first term) is negative,
• to make later steps (like further factoring or solving equations) simpler.

Factoring out common monomials from polynomials

A monomial is a single-term expression like $5x^2y$ or $-3ab^3$. When several terms share a common monomial factor, we treat that monomial as the GCF.

Example: factor $4x(2x - 3) + 6(2x - 3)$.

Here, the entire expression $(2x - 3)$ appears in both terms as a factor.

• Treat the two terms as:
• first term: $4x \cdot (2x - 3)$,
• second term: $6 \cdot (2x - 3)$.
• The common factor is $(2x - 3)$.

Factor it out:
$$
4x(2x - 3) + 6(2x - 3) = (2x - 3)(4x + 6).
$$

This is still factoring out a common factor, but now the factor is a binomial instead of just a monomial. The key idea is the same: any expression that is multiplied in each term can be factored out.

Another example: factor $x^2y(3x - 1) - 5y(3x - 1)$.

• Each term has $(3x - 1)$.
• Also each term has a factor $y$.

Combined common factor is $y(3x - 1)$.

Divide:

• $x^2y(3x - 1) \div y(3x - 1) = x^2$,
• $-5y(3x - 1) \div y(3x - 1) = -5$.

So:
$$
x^2y(3x - 1) - 5y(3x - 1) = y(3x - 1)(x^2 - 5).
$$

Typical mistakes to avoid

1. Forgetting to divide every term
   When you factor out a GCF, each term must be divided by it. Missing even one term gives an incorrect factorization.
2. Dropping terms inside parentheses
   After factoring out a GCF, there should be the same number of terms inside the parentheses as there were in the original expression, unless some terms simplify to
$ (which is unusual in basic common-factor problems).
3. Wrong sign inside parentheses
   When factoring out a negative, carefully check each sign. For example:
   $$
   -2x - 6 = -2(x + 3),
   $$
   not $-2(x - 3)$.
4. Including variables that are not common
   A variable can be part of the GCF only if it appears in every term with exponent at least $. For example, in x^2 + 6x + 9$, the GCF is x$ only if all terms have $x$. But the last term $ does not, so the actual GCF is just $.

Practice-style examples (with brief outlines of reasoning)

1. Factor $21x^3y^2 - 14x^2y$.

• Numeric GCF: $\text{GCF}(21, 14) = 7$.
• Variables: $x^3$ and $x^2$ $\Rightarrow x^2$; $y^2$ and $y$ $\Rightarrow y$.
• GCF is $7x^2y$.
• Divide:
• $21x^3y^2 \div 7x^2y = 3xy$,
• $14x^2y \div 7x^2y = 2$.

So:
$$
21x^3y^2 - 14x^2y = 7x^2y(3xy - 2).
$$

2. Factor $10a^2b + 15ab^2 - 5ab$.

• Numeric GCF: $\text{GCF}(10, 15, 5) = 5$.
• Variable GCF: all three terms have $a$ and $b$, with minimum exponents $a^1$, $b^1$.
• GCF is $5ab$.

Divide:

• $10a^2b \div 5ab = 2a$,
• $15ab^2 \div 5ab = 3b$,
• $-5ab \div 5ab = -1$.

So:
$$
10a^2b + 15ab^2 - 5ab = 5ab(2a + 3b - 1).
$$

3. Factor $-4x^2 - 8x$.

• Numeric GCF: $\text{GCF}(4, 8) = 4$.
• Variable GCF: $x$ (minimum exponent is $1$).
• Basic GCF: $4x$. If we take out $4x$:
  $$
  -4x^2 - 8x = 4x(-x - 2).
  $$
• Often we prefer the inside leading term positive, so factor out $-4x$ instead:
  $$
  -4x^2 - 8x = -4x(x + 2).
  $$

Both are correct; $-4x(x + 2)$ is usually the more convenient form.

Summary

• A common factor is any quantity (number, variable, or expression) that divides each term of a polynomial.
• The greatest common factor (GCF) gives the simplest factorization.
• To factor out a common factor:
1. Find the GCF of all terms (numeric and variable parts).
2. Write the expression as GCF times parentheses.
3. Inside the parentheses, put each original term divided by the GCF.
• You may factor out a negative common factor to make the leading term inside the parentheses positive.
• The same idea applies whether the common factor is a simple monomial like $3x$ or a more complex expression like $(2x - 3)$.