Let's go through the step-by-step multiplication of [tex]\((x + 1)^3\)[/tex].
We start with the given expression:
[tex]\[
(x + 1)^3
\][/tex]
First, expand [tex]\((x + 1)^2\)[/tex]:
[tex]\[
(x + 1)^2 = (x + 1)(x + 1)
\][/tex]
[tex]\[
= x \cdot x + x \cdot 1 + 1 \cdot x + 1 \cdot 1
\][/tex]
[tex]\[
= x^2 + x + x + 1
\][/tex]
[tex]\[
= x^2 + 2x + 1
\][/tex]
Next, multiply this result by [tex]\((x + 1)\)[/tex]:
[tex]\[
(x + 1)(x^2 + 2x + 1)
\][/tex]
Distribute each term of [tex]\((x + 1)\)[/tex] to the trinomial [tex]\((x^2 + 2x + 1)\)[/tex]:
[tex]\[
= x \cdot (x^2 + 2x + 1) + 1 \cdot (x^2 + 2x + 1)
\][/tex]
Multiply each term in the parentheses separately:
[tex]\[
= x \cdot x^2 + x \cdot 2x + x \cdot 1 + 1 \cdot x^2 + 1 \cdot 2x + 1 \cdot 1
\][/tex]
[tex]\[
= x^3 + 2x^2 + x + x^2 + 2x + 1
\][/tex]
Combine like terms:
[tex]\[
= x^3 + (2x^2 + x^2) + (x + 2x) + 1
\][/tex]
[tex]\[
= x^3 + 3x^2 + 3x + 1
\][/tex]
Therefore, the expanded form of [tex]\((x + 1)^3\)[/tex] is:
[tex]\[
(x + 1)^3 = x^3 + 3x^2 + 3x + 1
\][/tex]
The coefficients of the expanded polynomial [tex]\(x^3 + 3x^2 + 3x + 1\)[/tex] are:
[tex]\[
\boxed{[1, 3, 3, 1]}
\][/tex]
