polynomial multiplication: (2x + 5)^3

yoda

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Nov 12, 2006
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I got some homework problems that require me to multiply factored polynomials and write them in standard form. I thought it was easy enough until I saw that some were cubed. How do I multiply these?

ex: (2x + 5)^3

So far I figured that I would have an 8x^3 and that I can set it up like this:

(2x + 5)*(2x + 5)*(2x + 5)

But I'm lost after that. I dont want to just be told the answer, but I would appreciate it if someone could explain to me how to do a problem like this.
 
unless you know the binomial theorem, you'll just have to grind it out using multiplication ...

(2x+5)(2x+5)(2x+5) =
(4x<sup>2</sup>+20x+25)(2x+5) =
8x<sup>3</sup>+60x<sup>2</sup>+150x+125
 
This is exactly why so many of us hate "FOIL". It presents a special case as though it were a general technique, and then the student is scréwed when he encountered something else, such as a cube.

You can find lessons online which explain multiplying polynomials vertically. You don't have to go vertical, but most students seem to prefer it, especially for messy stuff like this. I'll model a similar exercise.

. . . . .Example: Expand (3x - 1)<sup>3</sup>.

The first bit is easy: (3x - 1)(3x - 1) = (3x)(3x) + (-1)(3x) + (3x)(-1) + (-1)(-1) = 9x<sup>2</sup> - 6x + 1. Here's what it looks like when done "vertically":

Code:
       3x - 1
       3x - 1
-------------
      -3x + 1
9x^2 - 3x
-------------
9x^2 - 6x + 1
Just like when you do regular vertical multiplication, you multiply the far-right term (in this case, the -1) by everything above, carrying down to the first line below (in this case, the -3x + 1). Then you multiply the next-right term (in this case, the 3x) by everything above, carrying down to the next line below (in this case, the 9x<sup>2</sup> - 3x), staggered one to the left. Carry on until you're done. Then add down to get the final result.

The next stage, multiplying the third factor of 3x - 1 by the 9x<sup>2</sup> - 6x + 1, looks like this:
Code:
        9x^2 -  6x + 1
                3x - 1
----------------------
       -9x^2 +  6x - 1
27x^3 - 18x^2 + 3x
----------------------
27x^3 - 27x^2 + 9x - 1
Follow the same process to get your product.

Eliz.
 
Wow, that makes so much more sense. Thanks a ton for the help! I really appreciate it!!
 
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