- Expand the initial expression \(3n^2(n^2+4n-5)-(2n^2-n^4+3)\) to \(3n^4+12n^3-15n^2-2n^2+n^4-3\).
- Combine like terms to obtain the simplified polynomial \(4n^4+12n^3-17n^2-3\).
- Identify that the highest power of \(n\) is 4, making it a \(4^{\text{th}}\) degree polynomial.
- Observe that the expression consists of four distinct terms, resulting in a \(4^{\text{th}}\) degree polynomial with 4 terms.
- The final classification is \(\boxed{4^{\text{th}}\text{ degree polynomial with 4 terms}}\).
Explanation
- Problem Analysis
We are given the expression \(3n^2(n^2+4n-5) - (2n^2 - n^4 + 3)\). Our objective is to simplify this expression by expanding the terms and combining like terms, then determine the degree of the resulting polynomial and count its number of terms.
- Expanding the Expression
First, we distribute the \(3n^2\) term into the first parenthesis:
\[3n^2(n^2+4n-5) = 3n^2 \cdot n^2 + 3n^2 \cdot 4n + 3n^2 \cdot (-5) = 3n^4 + 12n^3 - 15n^2.\]
Next, we remove the second set of parentheses by distributing the negative sign:
\[-(2n^2 - n^4 + 3) = -2n^2 + n^4 - 3.\]
Now, we combine the two results:
\[(3n^4 + 12n^3 - 15n^2) + (-2n^2 + n^4 - 3).\]
- Combining Like Terms
Group the like terms by the power of \(n\):
- For \(n^4\): \(3n^4 + n^4 = 4n^4\).
- For \(n^3\): \(12n^3\).
- For \(n^2\): \(-15n^2 - 2n^2 = -17n^2\).
- Constant term: \(-3\).
Combining these, the simplified polynomial is:
\[4n^4 + 12n^3 - 17n^2 - 3.\]
- Classification and Final Conclusion
- The highest power of \(n\) in the expression \(4n^4 + 12n^3 - 17n^2 - 3\) is 4, which means the polynomial is a \(4^{\text{th}}\) degree polynomial.
- The expression has four distinct non-zero terms: \(4n^4\), \(12n^3\), \(-17n^2\), and \(-3\).
Comparing this to the given options, we identify it as a \(4^{\text{th}}\) degree polynomial with 4 terms. Thus, the final answer is the third option.
Examples
In fields like physics or engineering, simplifying polynomials is a common task when modeling physical systems. For example, if the position of an object is described by a complex polynomial and you need to calculate the difference between its initial and final states or subtract external forces, you would simplify the expression to determine the net motion, which is often represented by a polynomial of a specific degree. Understanding how to correctly simplify these expressions allows for accurate prediction of system behavior.