To add the given polynomials, we will combine the coefficients of terms with the same power of [tex]\(a\)[/tex]. Let's go through it step-by-step:
1. Identify the terms in each polynomial:
The first polynomial is:
[tex]\[
9.7a^4 + 1
\][/tex]
The second polynomial is:
[tex]\[
-6.1a^4 - 7.3a^2 + 9.8
\][/tex]
2. Group the like terms:
We have terms involving [tex]\(a^4\)[/tex], [tex]\(a^2\)[/tex], and the constant terms (which are terms with [tex]\(a^0\)[/tex]).
- For [tex]\(a^4\)[/tex]:
[tex]\[
9.7a^4 \text{ and } -6.1a^4
\][/tex]
- For [tex]\(a^2\)[/tex]:
[tex]\[
-7.3a^2 \text{ (from the second polynomial, there is no corresponding \(a^2\) term in the first polynomial)}
\][/tex]
- Constant terms (terms with [tex]\(a^0\)[/tex]):
[tex]\[
1 \text{ and } 9.8
\][/tex]
3. Add the coefficients of the like terms:
- Coefficient of [tex]\(a^4\)[/tex]:
[tex]\[
9.7 + (-6.1) = 9.7 - 6.1 = 3.6
\][/tex]
- Coefficient of [tex]\(a^2\)[/tex]:
[tex]\[
-7.3 \text{ (no additional term to combine with)}
\][/tex]
- Constant terms:
[tex]\[
1 + 9.8 = 10.8
\][/tex]
4. Combine these results to form the new polynomial:
The resulting polynomial after combining the like terms is:
[tex]\[
3.6a^4 - 7.3a^2 + 10.8
\][/tex]
Thus, the final sum of the polynomials [tex]\(\left(9.7a^4 + 1\right) + \left(-6.1a^4 - 7.3a^2 + 9.8\right)\)[/tex] is:
[tex]\[
3.6a^4 - 7.3a^2 + 10.8
\][/tex]
