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Multiply the following polynomials, then place the answer in the proper location on the grid. Write the answer in descending powers of [tex]\( a \)[/tex].

[tex]\[ (a-6)\left(a^2+6a+36\right) \][/tex]

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Answer :

GinnyAnswer
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]

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