I've always found teaching common factoring to be a little troublesome, especially having them get the correct quotient in the brackets.. I stumbled on a technique that seemed to work very well for one of my students.
When introducing distributive property, I usually relate it back to how students would multiply a single digit and two digit number. So, for example, I'd ask what is 2(34), they'd reply 68, and when asked how they got it, they'd usually say something like 2x3 and 2x4, so I'd show how they were really doing 2(30+4)=60+8=68, and then move on to 2(3x+4).
One of my students though, wrote out the vertical multiplication and showed me that way. Thinking about it, I wondered if that might be a better way to demonstrate the property
When graphing higher order polynomials, one can encounter a quadratic with no real roots as a factor, which results in a "floating quadratic" turn on the graph. But where do you put it? While working with a student recently, we decided to toss the equation into a graphing program to find out. The equation was y = (x+2)(2x-5)(x^2+1), in case you care. My student guessed that it would be around -1, since that's how the other factors worked, but when plotted, we actually got what looked like a cubic twist at about (0,-1).
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