Sure, we'll solve the polynomial multiplication step-by-step and write the result in descending powers of [tex]\(a\)[/tex].
Given the polynomials: [tex]\((a-6)\)[/tex] and [tex]\((a^2 + 6a + 36)\)[/tex].
Here are the steps to multiply them together:
1. Expand the Multiplication:
We'll multiply each term in the first polynomial by each term in the second polynomial.
[tex]\[
(a - 6) \cdot (a^2 + 6a + 36)
\][/tex]
Let's distribute [tex]\( (a - 6) \)[/tex]:
[tex]\[
a \cdot (a^2 + 6a + 36) - 6 \cdot (a^2 + 6a + 36)
\][/tex]
2. Distribute [tex]\(a\)[/tex]:
[tex]\[
a \cdot a^2 + a \cdot 6a + a \cdot 36 = a^3 + 6a^2 + 36a
\][/tex]
3. Distribute [tex]\(-6\)[/tex]:
[tex]\[
-6 \cdot a^2 - 6 \cdot 6a - 6 \cdot 36 = -6a^2 - 36a - 216
\][/tex]
4. Combine Like Terms:
Now, we add the results from both distributions:
[tex]\[
a^3 + 6a^2 + 36a - 6a^2 - 36a - 216
\][/tex]
Combine the [tex]\(a^2\)[/tex] terms and the [tex]\(a\)[/tex] terms:
[tex]\[
a^3 + (6a^2 - 6a^2) + (36a - 36a) - 216
\][/tex]
Simplifying further:
[tex]\[
a^3 - 216
\][/tex]
So, the result of multiplying [tex]\((a-6)\)[/tex] by [tex]\((a^2 + 6a + 36)\)[/tex] is:
[tex]\[
a^3 - 216
\][/tex]
Write this answer in descending powers of [tex]\(a\)[/tex], as requested:
[tex]\[
a^3 - 216
\][/tex]
