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I am doing an Algebra course (UC Irvine / Coursera), and am having a bit of trouble understanding the following property:

$$ 3(x+h)^2 $$

Seems to be distributed as such:

$$ 3(x^2+2xh+h^2) $$

.. now, due to the distributive property, I understand why the x^2 and h^2 are so, but I have no idea where the "extra" 2xh came from. I've been grappling with the fact that while the original quantity can be represented as,

$$ 3(x+h)(x+h) $$

..it's just not clear to me how the 2xh is "created".

Any advice on this would be appreciated.

NB: This is not an assignment; it's from Week 2, Lecutre "Multiple Operations with Polynomials", at about 5:30.

Thanks!

sc.

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2 Answers

up vote 3 down vote accepted

Note that addition distributes over multiplication so

$$(x + h)^2 = (x+h)(x+h) = x(x+h)+h(x+h).$$

Now note that multiplication distributes over addition, so

$$x(x + h) = x\cdot x + xh = x^2 + xh$$

and

$$h(x+h) = hx + h\cdot h = hx + h^2.$$

Combining, we have

$$(x+h)^2 = x^2 + xh + hx + h^2.$$

Finally, note that multiplication is commutative so $hx = xh$ and hence

$$(x+h)^2 = x^2 + xh + hx + h^2 = x^2 + xh + xh + h^2 = x^2 + 2xh + h^2.$$

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Great, thanks! It seems that I neglected to recall the FOIL method. I've been "FOIL'd again" ! –  swisscheese Feb 16 '13 at 14:03
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$(x+h)(x+h)=x(x+h)+h(x+h)=\\ =x\cdot x+x\cdot h+h\cdot x+h\cdot h=\\ =x^2+xh+hx+h^2=\\ =x^2+2xh+h^2$

$2xh$ is created from $xh$ and $hx$.

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Thank you for your answer - a great complement to the above answer! –  swisscheese Feb 16 '13 at 14:04
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