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Simplify the expression [tex]3(x+2)(x^2-x-8)[/tex].

A. [tex]3x^3 + 3x^2 - 30x - 48[/tex]
B. [tex]3x^3 + x^2 - 10x - 16[/tex]
C. [tex]3x^3 - 30x^2 - 12x - 48[/tex]
D. [tex]3x^3 - 4x^2 - 30x - 48[/tex]

Answer :

To simplify the expression [tex]\(3(x+2)(x^2-x-8)\)[/tex], let's go through the steps:

1. Start by distributing [tex]\( (x+2) \)[/tex] across each term in the polynomial [tex]\( (x^2-x-8) \)[/tex].

[tex]\((x+2) \times (x^2-x-8)\)[/tex] can be expanded as:

[tex]\((x+2) \cdot x^2\)[/tex] gives [tex]\( x^3 + 2x^2 \)[/tex].

[tex]\((x+2) \cdot (-x)\)[/tex] gives [tex]\(-x^2 - 2x\)[/tex].

[tex]\((x+2) \cdot (-8)\)[/tex] gives [tex]\(-8x - 16\)[/tex].

So, the expanded form is:
[tex]\[
x^3 + 2x^2 - x^2 - 2x - 8x - 16
\][/tex]

2. Combine like terms in the expanded polynomial:

[tex]\[
x^3 + (2x^2 - x^2) + (-2x - 8x) - 16
\][/tex]

This simplifies to:

[tex]\[
x^3 + x^2 - 10x - 16
\][/tex]

3. Now, multiply the simplified polynomial [tex]\( (x^3 + x^2 - 10x - 16) \)[/tex] by 3:

[tex]\[
3(x^3 + x^2 - 10x - 16)
\][/tex]

Multiply each term by 3:

[tex]\[
3 \cdot x^3 = 3x^3
\][/tex]
[tex]\[
3 \cdot x^2 = 3x^2
\][/tex]
[tex]\[
3 \cdot (-10x) = -30x
\][/tex]
[tex]\[
3 \cdot (-16) = -48
\][/tex]

4. Combine all the terms to write the final expression:

[tex]\[
3x^3 + 3x^2 - 30x - 48
\][/tex]

Therefore, the simplified expression is [tex]\( 3x^3 + 3x^2 - 30x - 48 \)[/tex].

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