High School

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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^2+4x+6[/tex]

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

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

Answer :

To solve the problem of finding the product of [tex]\((x^2 + 3x + 9)\)[/tex] and [tex]\((x-3)\)[/tex], we will perform polynomial multiplication step-by-step.

First, let's denote the expressions:

[tex]\[ f(x) = x^2 + 3x + 9 \][/tex]
[tex]\[ g(x) = x - 3 \][/tex]

We need to find the product [tex]\( f(x) \cdot g(x) \)[/tex].

Here's the detailed process:

1. Multiply each term in [tex]\(f(x)\)[/tex] by each term in [tex]\(g(x)\)[/tex]:

[tex]\[
(x^2 + 3x + 9)(x - 3) = x^2 \cdot (x - 3) + 3x \cdot (x - 3) + 9 \cdot (x - 3)
\][/tex]

2. Distribute each multiplication:

[tex]\[
= (x^2 \cdot x - x^2 \cdot 3) + (3x \cdot x - 3x \cdot 3) + (9 \cdot x - 9 \cdot 3)
\][/tex]

3. Perform the multiplications:

[tex]\[
= (x^3 - 3x^2) + (3x^2 - 9x) + (9x - 27)
\][/tex]

4. Combine like terms:

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

Notice that [tex]\( -3x^2 + 3x^2 \)[/tex] and [tex]\( -9x + 9x \)[/tex] cancel out.

5. Simplify the expression:

[tex]\[
= x^3 - 27
\][/tex]

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

[tex]\[ x^3 - 27 \][/tex]

Therefore, the correct answer is:

a) 1) [tex]\(x^3 - 27\)[/tex]

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