To solve the given problem, we need to find the sum of the two polynomials [tex]\((d^2 + 6d + 9)\)[/tex] and [tex]\((d^3 + 6d + 9)\)[/tex]. We will add these polynomials term by term.
Let's write down each polynomial:
[tex]\[ d^2 + 6d + 9 \][/tex]
[tex]\[ d^3 + 6d + 9 \][/tex]
When we add these polynomials, we combine the like terms (terms with the same degree).
1. Combine the [tex]\(d^3\)[/tex] terms:
- The first polynomial does not have a [tex]\(d^3\)[/tex] term.
- The second polynomial has [tex]\(d^3\)[/tex].
So, the combined [tex]\(d^3\)[/tex] term is:
[tex]\[ d^3 \][/tex]
2. Combine the [tex]\(d^2\)[/tex] terms:
- The first polynomial has [tex]\(d^2\)[/tex].
- The second polynomial does not have a [tex]\(d^2\)[/tex] term.
So, the combined [tex]\(d^2\)[/tex] term is:
[tex]\[ d^2 \][/tex]
3. Combine the [tex]\(d\)[/tex] terms:
- The first polynomial has [tex]\(6d\)[/tex].
- The second polynomial has [tex]\(6d\)[/tex].
So, the combined [tex]\(d\)[/tex] term is:
[tex]\[ 6d + 6d = 12d \][/tex]
4. Combine the constant terms:
- The first polynomial has [tex]\(9\)[/tex].
- The second polynomial has [tex]\(9\)[/tex].
So, the combined constant term is:
[tex]\[ 9 + 9 = 18 \][/tex]
Putting all the combined terms together, we get the expanded polynomial in standard form:
[tex]\[
\left(d^2 + 6d + 9\right) + \left(d^3 + 6d + 9\right) = d^3 + d^2 + 12d + 18
\][/tex]
Thus, the expanded polynomial in standard form is:
[tex]\[
d^3 + d^2 + 12d + 18
\][/tex]
