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.

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.