Friday, March 13, 2015

Algebra: Multiplying Polynomials, Method 1

Read the introduction to this series here.

I call this first method "Term-by-Term." While tutoring, I found that most students attempted to multiply polynomials in this way. When the polynomials are small (two binomials, for example), this method is fast and easy to do. However, when the polynomials are large, it is cumbersome and lengthy.



Step 1: Multiply the first term of the first polynomial by each term in the second polynomial (including +/- signs!). 

In the image below, first the operation shown by the red arrow is performed, then the operation shown by the blue arrow. The resulting products (underlined in red and blue) are written out as terms in the product polynomial.




Step 2: Multiply the second term of the first polynomial by each term in the second polynomial (including +/- signs!).

In the image below, the two operations are shown in green and purple. These products, underlined, are tacked onto the products from the previous step in the product polynomial.



Continue multiplying in this way until every term from the first polynomial has been used.

Last Step: Combine like terms and rewrite in standard polynomial format (with exponents in descending order).

In this example, there are no like terms to combine, so we simply need to write the terms in the correct order (shown boxed in red below).



You see, when multiplying two binomials, this method works very well. Some students may find it difficult to keep track of which terms they are currently multiplying. My suggestion is to try drawing the arrows, as I've done, while you multiply one term across the second polynomial. Then, erase those arrows and draw new ones for each successive term.

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Let's try multiplying two trinomials with this method...



The images below show each step.







And the final answer...



Hopefully, this example illustrated how cumbersome this method becomes with long polynomials. Each step takes up more and more space on the page. In the end, there can be dozens of terms to sort through and combine.

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The method next up in this series is much more conducive to multiplying longer polynomials. Look for it next Monday, March 16!