Expanding Binomials using the Binomial Thm? (+radicals)

**puzzledwithpolynomials** · March 13th 2009, 06:00 PM

I am just having trouble with simplifying radicals within the binomial in conjunction w/ the bionomial thm. I understand how it works thus far and all of that...but these two problems (and the radicals) are messing me up.

1. (a-2sqrt(3))^6
2. (a-sqrt(2))^5

**Soroban** · March 13th 2009, 08:30 PM

Hello, puzzledwithpolynomials!

It helps if you write out the terms completely ... then simplify.

$2)\;\;\left(a-\sqrt{2}\right)^5$

${5\choose5}(a^5)\left(\text{-}\sqrt{2}\right)^0 + {5\choose4}(a^4)\left(\text{-}\sqrt{2}\right)^1 + {5\choose3}(a^3)\left(\text{-}\sqrt{2}\right)^2 +$ ${5\choose2}(a^2)\left(\text{-}\sqrt{2}\right)^3 + {5\choose1}(a^1)\left(\text{-}\sqrt{2}\right)^4 + {5\choose0}(a^0)\left(\text{-}\sqrt{2}\right)^5$

. . $= \;\;1\cdot a^6\cdot1 \;+\; 5(a^4)\left(\text{-}\sqrt{2}\right) \;+ \;10(a^3)(2) \;+ \;10(a^2)\left(\text{-}2\sqrt{2}\right) \;+\; 5(a)(4) \;+ \;1(1)\left(\text{-}4\sqrt{2}\right)$

. . $= \;\;a^5 - 5\sqrt{2}\,a^4 + 20a^3 - 20\sqrt{2}\,a^2 + 20a - 4\sqrt{2}$

**puzzledwithpolynomials** · March 14th 2009, 11:30 AM

Thank you! I see how I should deal with these radicals.

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