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How do you factor # x^4 + x^3 + x^2 + x + 1#?
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1 Answer
Explanation:
This quartic has four zeros, which are the non-Real Complex #5# th roots of #1# , as we can see from:
So if we wanted to factor this polynomial as a product of linear factors with Complex coefficients then we could write:
A cleaner algebraic approach is to notice that due to the symmetry of the coefficients, if #x=r# is a zero of #x^4+x^3+x^2+x+1# , then #x=1/r# is also a zero.
Hence there is a factorisation in the form:
So let's look for a factorisation:
Equating coefficients we find:
Substituting #b=-1/a# in #a+b=1# we get:
Hence:
Using the quadratic formula, we can deduce:
Since our derivation was symmetric in #a# and #b# , one of these roots can be used for #a# and the other for #b# , to find:
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Istopmotion 3 8 2 – create stop motion animated movies. If we want to factor further, use the quadratic formula on each of these quadratic factors to find the linear factors with Complex coefficients. Data rescue professional 4 3.
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