College

Select the correct product.

\[ (x^2+6x+9)(3x-1) \]

A. \[ 3x^3 + 19x^2 + 9x - 9 \]

B. \[ 3x^3 + 17x^2 + 21x - 9 \]

C. \[ 3x^3 - 17x^2 - 21x - 9 \]

D. \[ 3x^3 + 19x^2 + 27x + 9 \]

Answer :

Certainly! Let's find the product of the polynomials [tex]\((x^2 + 6x + 9)\)[/tex] and [tex]\((3x - 1)\)[/tex].

We start by applying the distributive property (also known as the FOIL method when dealing with binomials), where each term in the first polynomial multiplies by each term in the second polynomial.

[tex]\[
(x^2 + 6x + 9)(3x - 1)
\][/tex]

First, distribute [tex]\(3x\)[/tex] to each term in [tex]\((x^2 + 6x + 9)\)[/tex]:

[tex]\[
(3x)\cdot(x^2) = 3x^3
\][/tex]
[tex]\[
(3x)\cdot(6x) = 18x^2
\][/tex]
[tex]\[
(3x)\cdot(9) = 27x
\][/tex]

So, the contributions from [tex]\(3x\)[/tex] are:

[tex]\[
3x^3 + 18x^2 + 27x
\][/tex]

Next, distribute [tex]\(-1\)[/tex] to each term in [tex]\((x^2 + 6x + 9)\)[/tex]:

[tex]\[
(-1)\cdot(x^2) = -x^2
\][/tex]
[tex]\[
(-1)\cdot(6x) = -6x
\][/tex]
[tex]\[
(-1)\cdot(9) = -9
\][/tex]

So, the contributions from [tex]\(-1\)[/tex] are:

[tex]\[
-x^2 - 6x - 9
\][/tex]

Now, add these two sets of terms together:

[tex]\[
3x^3 + 18x^2 + 27x - x^2 - 6x - 9
\][/tex]

Combine like terms:

[tex]\[
3x^3 + (18x^2 - x^2) + (27x - 6x) - 9
\][/tex]
[tex]\[
3x^3 + 17x^2 + 21x - 9
\][/tex]

Thus, the product of [tex]\((x^2 + 6x + 9)\)[/tex] and [tex]\((3x - 1)\)[/tex] is:

[tex]\[
3x^3 + 17x^2 + 21x - 9
\][/tex]

Therefore, the correct answer is:

[tex]\[
\boxed{3x^3 + 17x^2 + 21x - 9}
\][/tex]

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