7.3 Factor quadratic trinomials with leading coefficient other than  (Page 2/6)

Factor trinomials of the form ax ² + bx + c With a gcf

Now that we have organized what we’ve covered so far, we are ready to factor trinomials whose leading coefficient is not 1, trinomials of the form $a{x}^2+bx+c$ .

Remember to always check for a GCF first! Sometimes, after you factor the GCF, the leading coefficient of the trinomial becomes 1 and you can factor it by the methods in the last section. Let’s do a few examples to see how this works.

Watch out for the signs in the next two examples.

In the next example the GCF will include a variable.

Factor trinomials using trial and error

What happens when the leading coefficient is not 1 and there is no GCF? There are several methods that can be used to factor these trinomials. First we will use the Trial and Error method.

Let’s factor the trinomial $3x^2+5x+2$ .

From our earlier work we expect this will factor into two binomials.

We know the first terms of the binomial factors will multiply to give us $3x^2$ . The only factors of $3x^2$ are $1x,3x$ . We can place them in the binomials.

Check. Does $1x·3x=3x^2$ ?

We know the last terms of the binomials will multiply to 2. Since this trinomial has all positive terms, we only need to consider positive factors. The only factors of 2 are 1 and 2. But we now have two cases to consider as it will make a difference if we write 1, 2, or 2, 1.