To find the sum of the polynomials [tex]\((5x^3 - 7x^2 + x^4) + (12x^2 + 3x^3 - 2x^4)\)[/tex], follow these steps:
1. Align the polynomials by their degrees and combine like terms:
[tex]\[
\begin{aligned}
& (+x^4) + (-2x^4) \\
& (+5x^3) + (+3x^3) \\
& (-7x^2) + (+12x^2) \\
\end{aligned}
\][/tex]
2. Add the coefficients of the like terms:
- Coefficient of [tex]\(x^4\)[/tex]: [tex]\(1 - 2 = -1\)[/tex]
- Coefficient of [tex]\(x^3\)[/tex]: [tex]\(5 + 3 = 8\)[/tex]
- Coefficient of [tex]\(x^2\)[/tex]: [tex]\(-7 + 12 = 5\)[/tex]
3. Write the resulting polynomial by combining these like terms:
[tex]\[
-x^4 + 8x^3 + 5x^2
\][/tex]
4. Express the polynomial in standard form:
- Standard form means writing the polynomial in descending order of the powers of [tex]\(x\)[/tex].
- We already have it in standard form: [tex]\(-x^4 + 8x^3 + 5x^2\)[/tex].
5. Factor out any common terms (optional, but sometimes required):
- We can factor out [tex]\(x^2\)[/tex]:
[tex]\[
-x^4 + 8x^3 + 5x^2 = x^2(-x^2 + 8x + 5)
\][/tex]
Therefore, the final answer in standard form is:
[tex]\[
\boxed{x^2(-x^2 + 8x + 5)}
\][/tex]
