This second method is my favorite way to multiply polynomials. I call it "The Table" method or "The Box" method. Even way up in Calculus III in university, my homework and test pages would have little boxes in the margins wherever I had to multiply polynomials. Many students find this method easier to follow and more organized. While tutoring, my students rarely missed terms when using this method. However, tables can be confusing in their own right, so this method may not be for everyone!
Let's begin with a simple example.
Step 1: Set up a table with one polynomial across the top (one term per column) and the second polynomial down the left side (one term per row) (including all +/- signs!).
***Important! Introduce any missing terms with zero coefficients (in red on first polynomial, below). Doing this will ensure your like terms align properly when you multiply. With enough practice, you will be able to skip this, but it is very helpful when you first begin using this method!***
In the image below, you can see I have set up my table with the first polynomial across the top and the second down the left-hand side. I included the missing term of the first polynomial, giving it a coefficient of zero (in red).
Step 2: For each cell in the table, multiply the term for that column by the term for that row.
So, begin by multiplying x times x^2, which yields x^3. This cell is marked with a green circle. Continue multiplying in this way until every cell of the table is filled, as shown in the image below. Watch the signs on each term carefully!
Step 3: Beginning with the first cell, write out the product polynomial by combining like terms on the diagonal as you go.
Notice that like terms are collected on diagonals, illustrated with the green and yellow highlights in the image below.
As you write out your answer (boxed in red above), simply add/subtract like terms along each diagonal, working from the top left corner to the bottom right.
You can help ensure you do not miss terms by marking off the cells you have already written as you go.
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Here is another, slightly more complicated example, worked out in steps below:
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As you can see, this method is very compact, even with large polynomials.
There is one more method I'll be sharing with you. Look for it Wednesday, March 18th!
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