To subtract the polynomials [tex]\((d^2 + 6d + 9)\)[/tex] and [tex]\((d^3 + 6d + 9)\)[/tex], we will follow the step-by-step procedure below:
1. Write both polynomials in decreasing order of their degrees:
[tex]\[
(d^2 + 6d + 9)
\][/tex]
[tex]\[
(d^3 + 6d + 9)
\][/tex]
2. Adjust the lengths by adding zero coefficients where necessary:
Since the first polynomial does not have a [tex]\(d^3\)[/tex] term, we can adjust it by adding a zero coefficient for the [tex]\(d^3\)[/tex] term:
[tex]\[
(0d^3 + d^2 + 6d + 9)
\][/tex]
The second polynomial already has a full representation in descending order:
[tex]\[
(d^3 + 6d + 9)
\][/tex]
3. Rewrite both polynomials aligned by their degrees:
[tex]\[
0d^3 + 1d^2 + 6d + 9
\][/tex]
[tex]\[
1d^3 + 0d^2 + 6d + 9
\][/tex]
4. Subtract the corresponding coefficients:
- For the [tex]\(d^3\)[/tex] term: [tex]\(0 - 1 = -1\)[/tex]
- For the [tex]\(d^2\)[/tex] term: [tex]\(1 - 0 = 1\)[/tex]
- For the [tex]\(d\)[/tex] term: [tex]\(6 - 6 = 0\)[/tex]
- For the constant term: [tex]\(9 - 9 = 0\)[/tex]
5. Combine the results:
The polynomial resulting from the subtraction is:
[tex]\[
-1d^3 + 1d^2 + 0d + 0
\][/tex]
6. Simplify:
Discard any terms with zero coefficients:
[tex]\[
-d^3 + d^2
\][/tex]
Putting it all together, the polynomial in standard form after subtraction is:
[tex]\[
-d^3 + d^2
\][/tex]
Therefore, the answer is:
[tex]\[
d^2 + 6d + 9 - (d^3 + 6d + 9) = -d^3 + d^2
\][/tex]
