Someone asked about how to teach factoring trinomials where the leading coefficient is not 1, so I thought I would post it here as well.

The best method I’ve found is often referred to as the slide and divide method, for trinomials of the type ax^2 + bx + c

Anyways, take the trinomial 3x^2 + x – 10.

(Always ensure to pull out any common factors before continuing with this method).

First you slide the leading coefficient to the end and multiply it by the constant:

You get x^2 + x – 30

Then factor like normal

(x + 6)(x – 5)

You have to “put back” the number you slid out, so to speak. You do this by dividing the constant in each factor by the leading coefficient you “slid” out of the way.

(x + 6/3)(x – 5/3)

Simplify the fractional terms you end up with.

(x + 2)(x – 5/3)

Once it’s simplified, if there’s a fraction left, the denominator becomes the coefficient of the variable term.

ANSWER: (x + 2)(3x – 5)

Here’s another example without the explanations:

2x^2 – 7x + 5

x^2 – 7x + 10

(x – 5)(x – 2)

(x – 5/2)(x – 2/2)

(x – 5/2)(x – 1)

ANSWER: (2x – 5)(x – 1)

Finally, an example where you have to pull out a common factor first:

12x^10 + 42x^9 + 18x^8

6x^8(2x^2 + 7x + 3)

6x^8(x^2 + 7x + 6)

6x^8(x + 6)(x + 1)

6x^8(x + 6/2)(x + 1/2)

6x^8(x + 3)(x + 1/2)

6x^8(x + 3)(2x + 1)