Complete Factorization Factor the polunomial completely, and find its zeros.State the multiplicity of each zero. P(x)=x^{3}-64

floymdiT 2021-03-05 Answered
Complete Factorization Factor the polunomial completely, and find its zeros.State the multiplicity of each zero.
P(x)=x364
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Expert Answer

berggansS
Answered 2021-03-06 Author has 91 answers
Concept used:
The multiplicity of zero of the polynomial having factor (xc) that appears k times in the factorization of the polynomial is k.
Calculation:
The given polynomial is P(x)=x364.
Factor the above polynomial to obtain the zeros.
P(x)=x364
=(x4)(x2+4x+16)
=(x4)(x2+22x+4+12)
=(x4)(x2+22x+22+12)
Further solve the expression,
P(x)=(x4)((x+2)2(23i)2)
=(x4)(x+223i)(x+2+23i)
Substitute 0 for P(x) in the polynomial P(x)=x364 to obtain the zeros of the polynomial.
(x4)(x+223i)(x+2+23i)=0
Further solve for the value of x as,
(x4)=0,(x+223)=0, and(x+2+2y3)=0
x=4,x=2+23, and x=223
All zeros of the polynomial P(x)=x364 appears one times in the polynomial therefore, the multiplicity of zeros 4, -2 + 2 3i, and -2-2 3i is 1.
Conclusion:
Thus, the factorization of the polynomial P(x)=x364 is
P(x)=(x4)(x+223i)(x+2+23i), zeros of the polynomial are (2+23) and 4 and the multiplicity of all the zeros is 1.
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