Wednesday, March 18, 2015

Algebra: Multiplying Polynomials, Method 3

If you missed the introductory post to this series, you should go read it.

This is the last method I'll be sharing for how to multiply polynomials. I call this one the "Numerical" method because it is set-up like a standard multiplication problem with numbers. I do not find this method particularly quick (like "The Table" method), but it is organized and easy to understand because it builds upon a skill with which the majority of students are already comfortable.

Our first example is an easy one, as usual. :-)



Step 1: Write the polynomials in a column, aligning them at the right, with like terms stack (introduce missing terms with a zero coefficient), like you would a standard multiplication problem. Write the longer polynomial on top.


In this example, there are no missing terms. But, if there were, I'd introduce them (as in previous methods), with zero coefficients. Like terms should be aligned. Notice in the image above that the term (-2x) in the first polynomial is aligned in the same "column" as the (+x) in the second polynomial.

Step 2: Multiply the last term of the bottom polynomial by each term of the top polynomial, from right to left. Write these products below the solid line, aligned with the term from the top polynomial. This is just like a standard multiplication problem with numbers.



In this step, we are working with the last term of the bottom polynomial (circled in red in the image above). Make sure you include the sign along with the term when multiplying! First, this term is multiplied by the last term of the top polynomial (circled in green). The product is written below the term used from the top polynomial (underlined in green).

Continue multiplying the last term (red circle) of the bottom polynomial by each of the terms in the top polynomial, from right to left. These are shown in purple and blue, respectively.

Step 3: Multiply the next to last term of the bottom polynomial by each term of the top polynomial, from right to left. Write these products below the products from Step 2, aligning them one term from the right side (leave a space below the far-right term of the first row of products). This is just like a standard multiplication problem with numbers.

Continue in this same manner until every term of the bottom polynomial has been used. The number of rows in the "products" section (below the solid line) should be equal to the number of terms in the bottom polynomial.


Now, we are multiplying across the top polynomial using the second to last term of the bottom polynomial (circled in red in the image above). These products are aligned below the products from Step 2, but moved one "space" to the left. This ensures that like terms are aligned in the products! This should look very familiar to you as a standard multiplication problem.

In this example, the multiplying is complete. Every term of the bottom polynomial has been multiplied through the top polynomial. Notice, there are two rows of product polynomials below the solid line, which is equal to the number of terms in the bottom polynomial (2).

Step 4: Draw another solid line below the rows of product polynomials. Add these polynomials together. Like terms are added together. The final product polynomial is yielded. (If there are any terms with a zero coefficient, rewrite the product polynomial and remove them.)




This step is where like terms are combined to form the final product polynomial. Simply add the terms in each column (example, boxed in green, above). The final product polynomial is boxed in red.

This method can be difficult to explain, but it really does work exactly like multiplying numbers with multiple digits that is taught in elementary/primary school.

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And now for an example with missing terms and longer polynomials. :-)



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I hope you have found this series helpful. As always, if you have any questions, comments, or requests for future topics, leave a comment below. :-)