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Area Development of Multiplication
Rule for Polynomials

The foregoing area perspective of the distributive law points to a quick method for introducing the multiplication method for polynomials in one and if you like several variables.  This development also teaches the column method for addition of polynomials.

Area View of Product of Polynomials

Use rectangles to expand the product 

P = (50+ 6b +4b²)(8+ 6 b + 4b²+ 10b³)

where b is a whole number

Solution:  Form intermediate size rectangles 

and compute the number of subrectangles in each. The latter operation first gives

and then this 

The foregoing could have been done in one step. Now instead of add the intermediate rows by rows, we will add them along the diagonals in the powers of b are identical. 

If we have made no mistakes, the foregoing gives the result P = (50+ 6b +4b²)(8+ 6 b + 4b²+ 10b³)

  =  300  + (300 + 48)b + (200 + 36 + 32)b²  + (500+24+24)b³  +  (60+16)b⁴  +  40b⁵ 

 = 300  + 348b + 268b²  + 548b³  +  76b⁴  +  40b⁵ 

Whether or not the powers of b increase or decrease in the  result is a cosmetic convention - some prefer one way, others the other, and some either. 

Our conclusion follows. The product

P = (50+ 6b +4b²)(8+ 6 b + 4b²+ 10b³)  = 300  + 348b + 268b²  + 548b³  +  76b⁴  +  40b⁵ 

That is computation of the left hand side for a given value of b gives the same result as computation of the right. 

Multiplication Table Approach - Stepping towards a column
multiplication method for polynomials

The above geometric approach suggests a table method:

which holds for real number b as well as whole numbers since the generalized distributive law holds for real numbers as well as whole numbers (why to come later). Here the rows and columns of the table come from the terms in the factors of the product to be computed, here (50+ 6b +4b²)(8+ 6 b + 4b²+ 10b³) while the entries inside list or tabulate the products of pairs of terms, one from each factor.

Column Method for Multiplication of Polynomials

Here we modify the table approach and dedicate a column to each power of b as follows

In this modified table approach, we compute all possible products as before, but align the products in each row according to their power of b. That makes addition and collecting like powers of b (with the aid of the distributive law) simpler. We add an extra row for the sum.

So the product P = 300  + 348b + 268b²  + 548b³  +  76b⁴  +  40b⁵   as before

In retrospect, the table (and column multiplication below) will be easier to do if we arrange the powers of b in ascending (that is, increasing) order along the top row.

and align the left column at the left of the page (or page margin), so that we may write from left to right. Here again all possible products appear, but aligned in columns. 

Finally, let us introduce or switch to the table method for multiplication. 

Column Method for Products of Polynomials

8   +      6 b +   4b²  +  10b³ 
50 +      6b   +  4b²             ×
300 + 300b + 200b² + 500b³                              (product of top row with 50)
             48b  +  36b²  +   24b³  +  60b⁴                 (product of first row with 6b)
                        32b²  +   24b³   + 16b⁴   + 40b⁵  (product of first row with 4b²)
^{------------------------------------------------------------------------ +}
300 + 348b + 268b²  + 548b³  +  76b⁴  +  40b^{5 
------------------------------------------------------------------------}

Note: The ascending order appears to work best as the position of the  intermediate products shifts to the right away from the left margin where the computation begins.  Do you see how each entry in each row of the column method corresponds to an entry in the rectangular approach and the two preceding tabular approaches to the computation of the product? All is a consequence of the distributive seen algebraically or geometrically visualized.

Addition and Subtraction of Polynomials

In the above calculation, we see how to add  polynomials

(300 + 300b + 200b² + 500b³  ) + (48b  +  36b²  +   24b³  +  60b⁴ ) + (32b²  +   24b³   + 16b⁴   + 40b⁵   ) 
 
via an addition methods in which like terms are aligned in columns as follows.

300 + 300b + 200b² + 500b³                              
             48b  +  36b²  +   24b³  +  60b⁴                 
                        32b²  +   24b³   + 16b⁴   + 40b^{5  
------------------------------------------------------------------------ +}
300 + 348b + 268b²  + 548b³  +  76b⁴  +  40b^{5 
------------------------------------------------------------------------}

So one pedagogical methods is to developed the column method for multiplication first, and then introduce column methods for addition and subtraction of polynomials, second.

Multiplication and Addition Operations with Decimals

The case where the variable b is replaced by the number 10 leads to a justification of the column method for multiplication of whole numbers using their decimal representation with powers of 10 written in decreasing order rather than increasing, Some conversion need to be considered. See Distributive Law Consequences

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