Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. x6−2x3+1

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Factor each polynomial completely. If the polynomial cannot be factored, say it is prime.  x6−2x3+1

Mary Goodson · Accepted Answer

Step 1Factorizing a polynomial is breaking a polynomial into small polynomials. If we multiply again those smaller polynomial with each other we will get the same polynomial.There are many ways of factorizing, one such is taking GCF (greatest common factor) of each term of the polynomial and taking GCF common and represent it in terms of GCF and another factor.Another important property is distributive property which is given by a(b+c)=ab+ac. It is taking common value from function and representing it in terms of common factor and another function.Step 2For the equation x6−2x3+1, let&#039;s take x3=t and put it in given equation to get equation in terms of variable tx6−2x3+1=t2−2t+1...(1)From equation(1), factor the function of t  by substituting −2t as (−t−t) and taking common values out.t2−2t+1=t2−t−t+1=t(t-1)-1(t-1)=(t-1)(t-1)   (using distributive property)=(t−1)2...(2)Step 3Resubstitute value of t=x3 in equation (2) to get factor form equation x6−2x3+1t2−2t+1=(x3−1)2x6−2x3+1=(x3−1)2   (from equation(1))Therefore factored form of x6−2x3+1 is (x3−1)2

Durst37 · Accepted Answer

(x3−1)2  (x3−2)x3+1  (x−1)2(x2+x+1)2  x6−2x3+1

Vasquez · Accepted Answer

Given P(x)=x6−2x3+1 =(x3−1)2 =(x3−1)(x3−1) P(x)=(x−1)(x2+x+1)(x−1)(x2+x+1)[since a3−b3=(a−b)(a2+ab+b)] P(x)=(x−1)2(x2+x+1)2 To find zeros consider P(x)=0  (x−1)2(x2+x+1)2=0 (x−1)2=0,(x2+x+1)2=0 x−1=0,x2+x+1=0 x=1,x=−1±3i2 Therefore, x=1 is the real root.

Factor each polynomial completely. If the polynomial cannot be factored,

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