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Answer :
The main answer is that the polynomial P(x) = x^4 - 5x^3 - 23x^2 - 10x + 16 has two real zeros.
To find the real zeros of the polynomial, we can use the Rational Root Theorem. According to the theorem, any rational zero of the polynomial will have the form p/q, where p is a factor of the constant term (16) and q is a factor of the leading coefficient (1). The factors of 16 are ±1, ±2, ±4, ±8, ±16, and the factors of 1 are ±1. So, the possible rational zeros are ±1, ±2, ±4, ±8, ±16.
We can now substitute these possible rational zeros into the polynomial to check if they are indeed zeros. By using synthetic division or long division, we can test each possible zero and determine if it gives us a remainder of 0.
After testing all the possible zeros, we find that the polynomial has two real zeros: x = -1 and x = 4.
Therefore, the real zeros of the polynomial P(x) = x^4 - 5x^3 - 23x^2 - 10x + 16 are x = -1 and x = 4.
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