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

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.