Factoring Trinomials (x^2+bx+c): Sometimes You Should Start With b, Not c

Factoring Trinomials (x^2+bx+c): Sometimes You Should Start With b, Not c

In today’s class we ran into the trinomial

x^2+3x-208

This can be challenging for students just learning to factor because they are not familiar with the factor set of 208. Traditionally I recommend that students follow the 10-second rule:

If you cannot find the correct factor pairing within 10 seconds, you should move on to listing all factor pairs of c.

This would result in the following pairs:

1\cdot208, 2\cdot104, 4\cdot52, 8\cdot26, 13\cdot16That means that students need to try 13 potential factors (1 through 13) before they find the correct pair. Here’s an alternate approach: focus on the term 3x. Since c is negative, we know that m and n will have opposite signs. If we ignore those signs, we know that the two values must be 3 apart from each other or, in other words, have a difference of 3. I told my students to start multiplying numbers that were 3 apart from each other. We started with 10 & 13, but that product was too small. We bumped them up to 11 & 14, 12 & 15, and finally 13 & 16. Bingo!

Now consider the trinomial

x^2+29x+208

Focusing on the middle term (29x) might not seem to be much help here. However, we could start with two numbers that have a sum of 29 like 14 & 15 and work our way back.

Give it a try and let me know what you think.

George

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