To find the product of the polynomials [tex]\((x^3 + 2x + 3)\)[/tex] and [tex]\((x^3 - x + 1)\)[/tex] using vertical multiplication, follow these steps.
Firstly, align the polynomials for multiplication:
[tex]\[
\begin{array}{r}
x^3 + 2x + 3 \\
\times \quad (x^3 - x + 1) \\
\end{array}
\][/tex]
We will multiply each term of the first polynomial by each term of the second polynomial. Start with the last term [tex]\(1\)[/tex] of the second polynomial:
[tex]\[
\begin{array}{r}
x^3 + 2x + 3 \\
\times \quad 1 \quad = x^3 + 2x + 3 \\
\end{array}
\][/tex]
Next, multiply each term of the first polynomial by [tex]\(-x\)[/tex] (shifting one place to the left):
[tex]\[
\begin{array}{r}
x^3 + 2x + 3 \\
\times \quad (-x) \quad = -x^4 - 2x^2 - 3x \\
\end{array}
\][/tex]
Finally, multiply each term of the first polynomial by [tex]\(x^3\)[/tex] (shifting three places to the left):
[tex]\[
\begin{array}{r}
x^3 + 2x + 3 \\
\times \quad x^3 \quad = x^6 + 2x^4 + 3x^3 \\
\end{array}
\][/tex]
Now, let's add these partial products:
[tex]\[
\begin{array}{r}
x^6 + 2x^4 + 3x^3 \\
\quad - x^4 - 2x^2 - 3x \\
\quad + x^3 + 2x + 3 \\
\end{array}
\][/tex]
Combine like terms:
1. [tex]\(x^6\)[/tex]
2. [tex]\(2x^4 - x^4 = x^4\)[/tex]
3. [tex]\(3x^3 + x^3 = 4x^3\)[/tex]
4. [tex]\(-2x^2\)[/tex]
5. [tex]\(-3x + 2x = -x\)[/tex]
6. [tex]\(3\)[/tex]
Putting all these together, the final product is:
[tex]\[
x^6 + x^4 + 4x^3 - 2x^2 - x + 3
\][/tex]
Comparing this result with the given options, the correct answer is:
B. [tex]\(x^6 + x^4 + 4x^3 - 2x^2 - x + 3\)[/tex]
