Basics of factoring

The word “factor” means “multiplier”. When used as a verb, “factor&rdsquo; means “to express as a product”. We should always proceed in this order (from easiest to most difficult) and repeat this time after time until the expression is fully factored.

Taking a common factor

A common factor is a multiplier that is common to two or more terms. Taking a common factor typically means taking the greatest common divisor of all terms. Consider the following example.

\begin{align} 30a^2b-5ab &= (5ab)(6a)-(5ab)(1) \tag{You may skip this step} \\ &=5ab(6a-a) \end{align}

What remains in the brackets can be obtained by dividing each term by the common factor. In other words, we can do the factoring as follows.

\begin{align} 30a^2b-5ab &= \left(\dfrac{30a^2b}{5ab}-\dfrac{5ab}{5ab}\right) \tag{You may skip this step} \\ &=5ab(6a-a) \end{align}

Factoring by grouping

With this method, we divide the expression into groups of terms, and we take a common factor from each of the groups. The target is to end up with the same set of brackets, then we can take this as a common factor in a subsequent step. Consider the expression \(3a^2+3ab-5a-5b\).

\begin{align} \boxed{3a^2+3ab}-\boxed{5a-5b} &= 3a\underline{(a+b)}-5\underline{(a+b)} \\ &=(a+b)(3a-5) \end{align}

Special quadratics

Difference of squares

This is always (as the name tells) ‘a square minus a square’, so it is always made of two terms. The general pattern is

Consider the following as an example.

\begin{align} 4x^2-9y^2&=(2x)^2-(3y)^2 \tag{You may skip this step} \\ &=(2x-3y)(2x+3y) \end{align}

Perfect square

This is the expansion of a binomial squared. The expansion is \((a\pm b)^2=a^2\pm 2ab+b^2\). We can go from right to left as much as from left to right. So, the general pattern is

The expression is always made of three terms. The first and last are always positive (because they are squares). The second can either be positive or negative; hence the \(\pm) (pronounced ‘plus or minus’). Note that the sign of the second term matches the sign inside the squared bracket.

We have to check whether a trinomial is a perfect square. This is how we do it.

• The first and last terms are positive and are perfect squares
• The second term is double the product of the square roots of the first and third terms.

Consider the following example.

\begin{align} 9x^2-24xy+16y^2 &= (3x)^2-(2)(3x)(4y)+(4y)^2 \tag{You may skip this step} \\ &= (3x-4y)^2 \end{align}

The expression \(x^2-2x-1\) is not a perfect square because the last term is negative. The expression \(x^2-3x+1\) is not a perfect square because the middle term is not double \(\sqrt{x^2}\) and \(\sqrt{1}\).

Regular quadratics

A quadratic in standard form \(ax^2+bx+c\) can be factored by trying to find two integers that

• multiply to \(ac\)
• add up to \(b\)