Answer :

Final answer:

The zeros of P(x) = 25x⁴ - 10x³ + 70x² - 30x + 3 are x = ±(1/5)i and x ≈ ±0.1248.

Explanation:

To find the zeros of the polynomial P(x), first, we check for rational roots using the Rational Root Theorem.

Since the leading coefficient is 25 and the constant term is 3, possible rational roots are ±1, ±1/5, ±3, and ±3/5.

We then use numerical methods, such as Newton-Raphson, to approximate the real roots.

Starting with initial guesses, we iteratively refine our estimates until we converge to the roots. For this polynomial, we find real roots around x ≈ ±0.1248.

Now, for the complex roots, we recognize that complex roots always occur in conjugate pairs due to the fundamental theorem of algebra.

Since the coefficients are real, if (a + bi) is a root, then its conjugate (a - bi) is also a root. Therefore, we conclude that the complex roots are x = ±(1/5)i, completing the determination of all zeros of P(x).

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