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

Algotssleeddynf

Algotssleeddynf

Answered question

2021-12-29

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime.
x62x3+1

Answer & Explanation

Mary Goodson

Mary Goodson

Beginner2021-12-30Added 37 answers

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 2
For the equation x62x3+1, let's take x3=t and put it in given equation to get equation in terms of variable t
x62x3+1=t22t+1...(1)
From equation(1), factor the function of t by substituting −2t as (−t−t) and taking common values out.
t22t+1=t2tt+1
=t(t-1)-1(t-1)
=(t-1)(t-1) (using distributive property)
=(t1)2...(2)
Step 3
Resubstitute value of t=x3 in equation (2) to get factor form equation x62x3+1
t22t+1=(x31)2
x62x3+1=(x31)2 (from equation(1))
Therefore factored form of x62x3+1 is (x31)2
Durst37

Durst37

Beginner2021-12-31Added 37 answers

(x31)2
(x32)x3+1
(x1)2(x2+x+1)2
x62x3+1
Vasquez

Vasquez

Skilled2022-01-09Added 457 answers

Given
P(x)=x62x3+1
=(x31)2
=(x31)(x31)
P(x)=(x1)(x2+x+1)(x1)(x2+x+1)[since a3b3=(ab)(a2+ab+b)]
P(x)=(x1)2(x2+x+1)2
To find zeros consider P(x)=0

(x1)2(x2+x+1)2=0
(x1)2=0,(x2+x+1)2=0
x1=0,x2+x+1=0
x=1,x=1±3i2
Therefore, x=1 is the real root.

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