Question
he number of zeros of
is
, provided that each zero is counted according to its multiplicity.
Ask by Huff Hanson. in the United States
Mar 15,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The polynomial
has a total of 8 zeros, including their multiplicities.
Solution
-
Factor the given polynomial:
-
Notice that
is written as a product: - The factor
contributes a zero (namely, ) with multiplicity 1. - The other factor is a degree 7 polynomial,
.
- The factor
-
By the Fundamental Theorem of Algebra, a polynomial of degree 7 has 7 zeros (counting multiplicities) in the complex numbers.
-
Therefore, the total number of zeros for
is:
Thus, the number of zeros of
is
.
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Extra Insights
To find the number of zeros of the function
, we can factor out the common term, which is
:
This shows that
is one zero. Now, we need to find the zeros of the polynomial
.
Finding the roots of
can be complex, but we can analyze its degree and behavior. Since it’s a 7th degree polynomial, it can have up to 7 zeros, considering their multiplicity (and potentially complex roots).
To determine the real roots, we might need numerical methods or graphing tools. Nevertheless, without specific tools, we note that this polynomial does not change signs (as can be observed by evaluating it at various points), indicating that it may not have any real zeros apart from the one at
.
Thus, the total number of zeros of
, counting multiplicities, is:
1 (the zero at
) + 0 (potential real zeros from the polynomial) = 1.
So, the number of zeros is
.