Sometimes expressions in standard form don’t have a linear term. Can they still be written in factored form?

Let’s take as an example. To help us write it in factored form, we can think of it as having a linear term with a coefficient of 0: .

We know that we need to find two numbers that multiply to make -9 and add up to 0. The numbers 3 and -3 meet both requirements, so the factored form is .

To check that this expression is indeed equivalent to , we can expand the factored expression by applying the distributive property: . Adding and gives 0, so the expanded expression is .

In general, a quadratic expression that is a difference of two squares and has the form can be rewritten as .

Here is a more complicated example: . This expression can be written as , so an equivalent expression in factored form is .

What about ? Can it be written in factored form?

Let’s think about this expression as . Can we find two numbers that multiply to make 9 and add up to 0? Here are factors of 9 and their sums:

• 9 and 1, sum: 10
• -9 and -1, sum: -10
• 3 and 3, sum: 6
• -3 and -3, sum: -6

For two numbers to add up to 0, they need to be opposites (a negative and a positive), but a pair of opposites cannot multiply to make positive 9, because multiplying a negative number and a positive number always gives a negative product.

Because there are no numbers that multiply to make 9 and also add up to 0, it is not possible to write in factored form using the kinds of numbers that we know about.