College

The product of [tex]\left(x^2 + 3x + 9\right)[/tex] and [tex](x - 3)[/tex] is:

A. [tex]x^3 - 27[/tex]

B. [tex]x^3 - 6x^2 - 18x - 27[/tex]

C. [tex]x^2 + 4x + 6[/tex]

D. [tex]-6x^4 + x^3 - 18x^2 - 27[/tex]

Answer :

To find the product of [tex]\((x^2 + 3x + 9)\)[/tex] and [tex]\((x - 3)\)[/tex], follow these steps:

1. Write down the expression to be multiplied:
[tex]\[
(x^2 + 3x + 9)(x - 3)
\][/tex]

2. Use the distributive property (also known as the FOIL method in this case) to expand the expression:
Each term in the first polynomial will multiply each term in the second polynomial.

3. Multiply each term in [tex]\(x^2 + 3x + 9\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
x^2 \cdot x = x^3
\][/tex]
[tex]\[
3x \cdot x = 3x^2
\][/tex]
[tex]\[
9 \cdot x = 9x
\][/tex]

4. Multiply each term in [tex]\(x^2 + 3x + 9\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
x^2 \cdot (-3) = -3x^2
\][/tex]
[tex]\[
3x \cdot (-3) = -9x
\][/tex]
[tex]\[
9 \cdot (-3) = -27
\][/tex]

5. Now, combine all the products:
[tex]\[
x^3 + 3x^2 + 9x - 3x^2 - 9x - 27
\][/tex]

6. Combine like terms (terms with the same power of [tex]\(x\)[/tex]):
[tex]\[
x^3 + (3x^2 - 3x^2) + (9x - 9x) - 27
\][/tex]
[tex]\[
x^3 - 27
\][/tex]

So, the product of [tex]\((x^2 + 3x + 9)\)[/tex] and [tex]\((x - 3)\)[/tex] is:
[tex]\[
x^3 - 27
\][/tex]

The correct option is:
(1) [tex]\(x^3 - 27\)[/tex]