Multiplying PolynomialsDate: 01/29/2001 at 17:30:52 From: Max Teodorescu Subject: Multiplying a polynomial by a polynomial Why isn't (3x+2) squared equal to 9x squared + 4? If the 3x were squared, wouldn't it be equal to 9x squared, because 3*3 = 9 and x*x = x squared? Thanks for your time. Date: 01/29/2001 at 19:05:22 From: Doctor Greenie Subject: Re: Multiplying a polynomial by a polynomial Hi, Max -- I always found it helpful to remember rules of multiplication by thinking of finding the area of rectangles. When I was very young, I learned my multiplication tables by setting up rows of marbles, or dominoes, or whatever, and counting the total number. For example: X X X X X X X X X X X X X X X X X X X X X X X X Hmm - 4 rows with 6 in each row, and altogether there are 24, so 4*6 = 24 (and also 6*4 = 24). I can do the same thing by thinking of a rectangle without actually using the dominoes: 9 +-----------+ | | | | 7 | | | | +-----------+ This rectangle is 9 units by 7 units, so its area is 9*7 = 63 square units. Now, what if I have something like 12*12? Perhaps you already know that 12*12 = 144; let's see a picture of it using a rectangle. (Actually, since the numbers we are multiplying together are the same, this will be a square.) Instead of writing the multiplication problem as "12*12", I will write it as "(10+2)*(10+2)," and I will draw my picture that way: 10 2 +-------------+---+ | | | | | | | A | C | | | | 10 | | | | | | +-------------+---+ | D | B | 2 +-------------+---+ Now if it were true that (10+2) squared were equal to (10) squared plus (2) squared, then the answer would be 100+4 = 104. Well, the (10) squared = 100 is the large square "A" in the picture, and the (2) squared = 4 is the small square "B" in the picture. But the picture for (10+2) squared includes not only the two squares A and B, but also the two rectangular regions C and D. Each of these rectangles has area 10*2 = 20, so the correct answer to 12*12 is (as you know) 100+4+20+20 = 144. It's the same thing with (3x+2) squared. To draw a picture of (3x+2) squared, we can just repeat the figure above with each 10 replaced by 3x: 3x 2 +-------------+---+ | | | | | | | A | C | | | | 3x | | | | | | +-------------+---+ | D | B | 2 +-------------+---+ Here, the area of region A is (3x) squared = 9x^2 (^2 means squared) and the area of region b is (2) squared = 4. So your answer 9x^2+4 represents the area of the two squares A and B. But (3x+2) squared also includes rectangles C and D, and the area of each of these rectangles is (3x)*(2) = 6x, so the area of the large square (and so the answer to (3x+2) squared) is 9x^2 + 4 + 6x + 6x or 9x^2 + 12x + 4 I hope this picture helps. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ |
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