Question

Try to complete Factorization Factor the polynomial completely , and find all its zeros. State the multiplicity of each zero. P(x)=x^{4}-625

Polynomial factorization
ANSWERED
asked 2020-10-20
Try to complete Factorization Factor the polynomial completely , and find all its zeros. State the multiplicity of each zero.
\(P(x)=x^{4}-625\)

Answers (1)

2020-10-21
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 \(P(x) = x^{4} — 625\).
Factor the above polynomial to obtain the zeros
\(P(x) = x^{4} - 625\)
\(= ((x^{2})^{2} - (25)^{2})\)
\(= (x^{2} - 25) (x^{2} +25)\)
\(= (x^{2} - 5^{2}) (x^{2} - (5i)^{2})\)
Further factorize the above expression
\(P(x) = (x^{2}-5^{2}) (x^{2} - (5i)^{2})\)
\(= (x +5) (x- 5) (x + 5i) (x- 5i)\)
Substitute 0 for P(x) in the polynomial \(P(x) = x^{4} — 625\) to obtain the zeros of the polynomial.
\((x+5)(x-5)(x+5i)(x-5i)=0\)
Further solve for the value of x as,
\((x+5)=0, (x-5)=0, (x+5i)=0\) and \((x-5i)=0\)
\(x=-5, x=5, x=-5i\) and \(x=5i\)
All the zeros in the polynomial \(P(x) = x^{4} — 625\) appear one times in the polynomial.
Therefore, the multiplicity of zeros 5i, —5i, 5 and —5 is 1.
Conclusion:
Thus, the factorization of the polynomial \(P(x) = x^{4} — 625\) is
\(P(x) = (x + 5) (x— 5) (x + 5i) (x — 5i)\), zeros of the polynomial are \(\pm5\ and\ \pm5i\) and the multiplicity of all the zeros is 1.
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