–3f5 + 6f3 − 3f2 + 10f + 5
-3f^5 + 6f^3 - 3f^2 + 10f + 5
has no rational roots. Descartes' Rule of Signs will limit the number of positive and negative roots.
Newton's method can provide approximate roots to any desired precision.
and on and on and on ...
To simplify the given expression -3f^5 + 6f^3 - 3f^2 + 10f + 5, we combine like terms by collecting the terms with the same exponent.
The given expression contains terms with various exponents: f^5, f^3, f^2, f, and a constant term.
Let's rearrange the terms in decreasing order of the exponents:
-3f^5 + 6f^3 - 3f^2 + 10f + 5
Now, let's simplify each group of like terms:
1. Terms with f^5: There is only one term with f^5, which is -3f^5. We can't combine it with any other terms since there are none with the same exponent.
2. Terms with f^3: The only term with f^3 is 6f^3. We can't combine it with any other terms.
3. Terms with f^2: The term with f^2 is -3f^2.
4. Terms with f: The term with f is 10f.
5. Constant term: The constant term is 5.
Now we can rewrite the expression with the simplified terms:
-3f^5 + 6f^3 - 3f^2 + 10f + 5
There are no other terms to combine, so this is the simplified form of the given expression.
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