Complete Factorization Factor the polynomial completely, and find all its zeros.State the multiplicity of each zero. P(x) = x^{4}+2x^{2}+1

Rui Baldwin 2020-11-09 Answered
Complete Factorization Factor the polynomial completely, and find all its zeros.State the multiplicity of each zero.
P(x)=x4+2x2+1
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Expert Answer

2abehn
Answered 2020-11-10 Author has 88 answers
Concept used:
The multiplicity of zero of the polynomial having factor (x — c) that appears k times in the factorization of the polynomial is k.
Calculation:
The given polynomial is P(x)=x4+2x2?+1.
Factor the above polynomial to obtain the zeros.
P(x)=x4+2x2+1=((x2)2+21x+1)
=(x2+1)2=(x2+1)(x2+1)
P(x)=(x2+1)(x2+1)
=(x2i2)(x2i2)
=(x+i)(xi)(x+i)(xi)
Substitute 0 for P (x) in the polynomial P(x)=x4+2x2+1 to obtain the zeros of the polynomial.
(x+i)(xi)(x+i)(xi)=0
Further solve for the value of x as,
(xi)=0 and (x+i)=0
x=i and x=i
All zeros of the polynomial P(x)=x4+2x2+1 appears two times in the polynomial therefore, the multiplicity of zeros -i, and i is 2.
Conclusion:
Thus, the factorization of the polynomial P(x)=x4+x2+1 is P(x)=(xi)2(x+i)2,zeros of the polynomial are +i and the multiplicity of the zeros is 2.
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