To tackle this problem, let's break it down step-by-step.
1. Expand the polynomials:
Start by expanding the given polynomials to make the calculation of their difference easier.
The first polynomial is:
[tex]\[
3n^2(n^2 + 4n - 5)
\][/tex]
Let's expand it:
[tex]\[
3n^2 \cdot n^2 + 3n^2 \cdot 4n - 3n^2 \cdot 5
\][/tex]
[tex]\[
= 3n^4 + 12n^3 - 15n^2
\][/tex]
The second polynomial is:
[tex]\[
2n^2 - n^4 + 3
\][/tex]
2. Write the expression for the difference of the polynomials:
Now, to find the difference, subtract the second polynomial from the first polynomial:
[tex]\[
(3n^4 + 12n^3 - 15n^2) - (2n^2 - n^4 + 3)
\][/tex]
3. Distribute the negative sign and combine like terms:
Distribute the negative sign through the second polynomial:
[tex]\[
3n^4 + 12n^3 - 15n^2 - 2n^2 + n^4 - 3
\][/tex]
Now, combine the like terms:
[tex]\[
3n^4 + n^4 + 12n^3 - 15n^2 - 2n^2 - 3
\][/tex]
Combine the terms with the same degree:
[tex]\[
(3n^4 + n^4) + 12n^3 + (-15n^2 - 2n^2) - 3
\][/tex]
[tex]\[
= 4n^4 + 12n^3 - 17n^2 - 3
\][/tex]
4. Classify the resulting polynomial:
Now that we have the resulting polynomial in its simplified form:
[tex]\[
4n^4 + 12n^3 - 17n^2 - 3
\][/tex]
- The leading term is [tex]\(4n^4\)[/tex], so the degree of the polynomial is 4.
- The polynomial has four terms: [tex]\(4n^4\)[/tex], [tex]\(12n^3\)[/tex], [tex]\(-17n^2\)[/tex], and [tex]\(-3\)[/tex].
Hence, we have a [tex]\(4^{\text{th}}\)[/tex] degree polynomial with 4 terms. Therefore, the correct answer is:
B. [tex]\(4^{\text{th}}\)[/tex] degree polynomial with 4 terms
