Answer :
Sure! Let's multiply the polynomials step-by-step.
We are given:
[tex]\[
(x + 3)\left(3x^2 + 8x + 9\right)
\][/tex]
To multiply these polynomials, we can use the distributive property (also known as the FOIL method when dealing with binomials), which involves distributing each term in the first polynomial to every term in the second polynomial. Let's break it down:
1. Distribute [tex]\(x\)[/tex] across each term in [tex]\(3x^2 + 8x + 9\)[/tex]:
[tex]\[
x \cdot (3x^2 + 8x + 9) = x \cdot 3x^2 + x \cdot 8x + x \cdot 9
\][/tex]
This simplifies to:
[tex]\[
3x^3 + 8x^2 + 9x
\][/tex]
2. Distribute [tex]\(3\)[/tex] across each term in [tex]\(3x^2 + 8x + 9\)[/tex]:
[tex]\[
3 \cdot (3x^2 + 8x + 9) = 3 \cdot 3x^2 + 3 \cdot 8x + 3 \cdot 9
\][/tex]
This simplifies to:
[tex]\[
9x^2 + 24x + 27
\][/tex]
3. Combine the results from both distributions:
[tex]\[
3x^3 + 8x^2 + 9x + 9x^2 + 24x + 27
\][/tex]
4. Combine like terms:
[tex]\[
3x^3 + (8x^2 + 9x^2) + (9x + 24x) + 27
\][/tex]
This simplifies to:
[tex]\[
3x^3 + 17x^2 + 33x + 27
\][/tex]
So, the polynomial resulting from the multiplication is:
[tex]\[
3x^3 + 17x^2 + 33x + 27
\][/tex]
From the given answer choices, the correct answer is:
[tex]\[
\boxed{3x^3 + 17x^2 + 33x + 27}
\][/tex]
Hence, the correct answer choice is:
D. [tex]\(3 x^3 + 17 x^2 + 33 x + 27\)[/tex].
We are given:
[tex]\[
(x + 3)\left(3x^2 + 8x + 9\right)
\][/tex]
To multiply these polynomials, we can use the distributive property (also known as the FOIL method when dealing with binomials), which involves distributing each term in the first polynomial to every term in the second polynomial. Let's break it down:
1. Distribute [tex]\(x\)[/tex] across each term in [tex]\(3x^2 + 8x + 9\)[/tex]:
[tex]\[
x \cdot (3x^2 + 8x + 9) = x \cdot 3x^2 + x \cdot 8x + x \cdot 9
\][/tex]
This simplifies to:
[tex]\[
3x^3 + 8x^2 + 9x
\][/tex]
2. Distribute [tex]\(3\)[/tex] across each term in [tex]\(3x^2 + 8x + 9\)[/tex]:
[tex]\[
3 \cdot (3x^2 + 8x + 9) = 3 \cdot 3x^2 + 3 \cdot 8x + 3 \cdot 9
\][/tex]
This simplifies to:
[tex]\[
9x^2 + 24x + 27
\][/tex]
3. Combine the results from both distributions:
[tex]\[
3x^3 + 8x^2 + 9x + 9x^2 + 24x + 27
\][/tex]
4. Combine like terms:
[tex]\[
3x^3 + (8x^2 + 9x^2) + (9x + 24x) + 27
\][/tex]
This simplifies to:
[tex]\[
3x^3 + 17x^2 + 33x + 27
\][/tex]
So, the polynomial resulting from the multiplication is:
[tex]\[
3x^3 + 17x^2 + 33x + 27
\][/tex]
From the given answer choices, the correct answer is:
[tex]\[
\boxed{3x^3 + 17x^2 + 33x + 27}
\][/tex]
Hence, the correct answer choice is:
D. [tex]\(3 x^3 + 17 x^2 + 33 x + 27\)[/tex].
