(x7 - 2x6 + x5 - 2x4 + x3 - 2x2 + x - 2) ÷ (x - 2) = x6 + x4 + x2 + 1
(x8 - 2x7 + x6 - 2x5 + x4 - 2x3 + x2 - 2x + 1) ÷ (x - 2) = x7 + x5 + x3 + x + (1⁄(x - 2))
I find it strange that the answer's highest exponent is exactly one lower than the original question. The second equation starts with x raised to the 8th power. The answer starts with x raised to the seventh power. All of the equations with an odd highest exponent have no remainder while all the equations with an even highest exponent have a remainder of (1⁄(x - 2)). The weirdest part is that by alternating the coefficients between 1 and 2 we eliminate all the variables without a coefficient of 2. In the first equation x7 is the highest exponent . The answer's highest exponent is x6.
They're being divided by (x-2) in both cases, isn't that why the power is one less in the answer than in the original equation?
ReplyDeleteYes, but what I wanted you to see was that when it starts with a highest exponent that is odd and the answer has only even exponents. You start with x^7 then end with the 6th power, the 4th power, x squared, and x^0. While 0 isn't even or odd it is less by a value of 2, the same as all the others.
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