Final answer:
The polynomial h(x)=5x⁴+7x⁸-x¹² has a factor of x⁴ which indicates at least one real zero at x=0 with multiplicity 4. Identifying the exact number of zeros for the rest of the polynomial requires more advanced methods and is not straightforward.
Step-by-step explanation:
When identifying the number of zeros in the polynomial h(x)=5x⁴+7x⁸-x¹², we must recognize that all terms are multiples of x. Since the polynomial is of degree 12, the highest power of x, it suggests that there could be up to 12 zeros. However, the actual number of real zeros could be less due to the potential for complex zeros or repeated zeros.
To find the number of zeros, we can factor out the common term x⁴, which gives us:
x⁴(5 + 7x⁴ - x⁸)
Here, we have a clear zero at x=0 with multiplicity 4, because x⁴ equals zero when x is zero. For the remaining polynomial (5 + 7x⁴ - x⁸), without specific techniques or theorems like the Rational Root Theorem or using complex number methods, it’s not straightforward to determine the number of real zeros. A high-level understanding or specific instructions to solve higher-order polynomial equations would be required to further identify the potential zeros