Let's solve the expression step by step in detail. We need to subtract the polynomial [tex]\( \left(d^3 + 6d + 9\right) \)[/tex] from the polynomial [tex]\( \left(d^2 + 6d + 9\right) \)[/tex].
We start by writing down both polynomials:
1. [tex]\( d^2 + 6d + 9 \)[/tex]
2. [tex]\( d^3 + 6d + 9 \)[/tex]
Now perform the subtraction:
[tex]\[ (d^2 + 6d + 9) - (d^3 + 6d + 9) \][/tex]
Distribute the subtraction across each term in the second polynomial:
[tex]\[ d^2 + 6d + 9 - d^3 - 6d - 9 \][/tex]
Next, combine like terms. We will group the terms with the same degree:
1. Combine [tex]\( d^2 \)[/tex] terms: [tex]\( d^2 \)[/tex]
2. Combine [tex]\( 6d \)[/tex] terms: [tex]\( 6d - 6d = 0 \)[/tex]
3. Combine constants: [tex]\( 9 - 9 = 0 \)[/tex]
4. Lastly, don't forget the [tex]\( -d^3 \)[/tex] term.
Resulting in:
[tex]\[ d^2 - d^3 + 0 + 0 \][/tex]
Simplified, it becomes:
[tex]\[ d^2 - d^3 \][/tex]
To write the polynomial in standard form, arrange the terms in order of descending powers of [tex]\( d \)[/tex]:
[tex]\[ -d^3 + d^2 \][/tex]
This matches our target result. Therefore, the polynomial in its simplified standard form is:
[tex]\[ \boxed{-d^3 + d^2} \][/tex]
