Answer :
Sure, let's find the product of the polynomials [tex]\((x^3 + 2x + 3)\)[/tex] and [tex]\((x^3 - x + 1)\)[/tex] using vertical multiplication. Here's the step-by-step process:
1. Write the polynomials in vertical format to organize the multiplication:
```
x^3 + 2x + 3
× x^3 - x + 1
```
2. Multiply each term in the second polynomial [tex]\((x^3 - x + 1)\)[/tex] with each term in the first polynomial [tex]\((x^3 + 2x + 3)\)[/tex]:
[tex]\[
\begin{array}{r}
& & x^3 & + & 2x & + & 3 \\
\times & & x^3 & - & x & + & 1 \\
\cline{2-8}
& x^6 & + & 2x^4 & + & 3x^3 \\ % x^3 * (x^3 + 2x + 3)
- & x^4 & - & 2x^2 & - & 3x \\ % -x * (x^3 + 2x + 3)
+ & x^3 & + & 2x & + & 3 \\ % 1 * (x^3 + 2x + 3)
\end{array}
\][/tex]
3. Align and add corresponding terms:
[tex]\[
\begin{array}{r}
& x^6 & + & 2x^4 & + & 3x^3 \\
- & & x^4 & - & 2x^2 & - & 3x \\
+ & & & x^3 & + & 2x & + & 3 \\
\cline{2-9}
& x^6 & + & x^4 & + & 4x^3 & - & 2x^2 & - & x & + & 3
\end{array}
\][/tex]
4. Combine like terms:
- There is only one [tex]\(x^6\)[/tex] term.
- For [tex]\(x^4\)[/tex], add [tex]\(2x^4\)[/tex] and [tex]\(-x^4\)[/tex] to get [tex]\(x^4\)[/tex].
- For [tex]\(x^3\)[/tex], add [tex]\(3x^3\)[/tex] and [tex]\(x^3\)[/tex] to get [tex]\(4x^3\)[/tex].
- There is only one [tex]\(-2x^2\)[/tex] term.
- For [tex]\(x\)[/tex], add [tex]\(-3x\)[/tex] and [tex]\(2x\)[/tex] to get [tex]\(-x\)[/tex].
- There is only one [tex]\(3\)[/tex] term.
So, the product is:
[tex]\[
x^6 + x^4 + 4x^3 - 2x^2 - x + 3
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{x^6 + x^4 + 4x^3 - 2x^2 - x + 3}
\][/tex]
So, the correct option is:
A. [tex]\(x^6 + x^4 + 4x^3 - 2x^2 - x + 3\)[/tex]
1. Write the polynomials in vertical format to organize the multiplication:
```
x^3 + 2x + 3
× x^3 - x + 1
```
2. Multiply each term in the second polynomial [tex]\((x^3 - x + 1)\)[/tex] with each term in the first polynomial [tex]\((x^3 + 2x + 3)\)[/tex]:
[tex]\[
\begin{array}{r}
& & x^3 & + & 2x & + & 3 \\
\times & & x^3 & - & x & + & 1 \\
\cline{2-8}
& x^6 & + & 2x^4 & + & 3x^3 \\ % x^3 * (x^3 + 2x + 3)
- & x^4 & - & 2x^2 & - & 3x \\ % -x * (x^3 + 2x + 3)
+ & x^3 & + & 2x & + & 3 \\ % 1 * (x^3 + 2x + 3)
\end{array}
\][/tex]
3. Align and add corresponding terms:
[tex]\[
\begin{array}{r}
& x^6 & + & 2x^4 & + & 3x^3 \\
- & & x^4 & - & 2x^2 & - & 3x \\
+ & & & x^3 & + & 2x & + & 3 \\
\cline{2-9}
& x^6 & + & x^4 & + & 4x^3 & - & 2x^2 & - & x & + & 3
\end{array}
\][/tex]
4. Combine like terms:
- There is only one [tex]\(x^6\)[/tex] term.
- For [tex]\(x^4\)[/tex], add [tex]\(2x^4\)[/tex] and [tex]\(-x^4\)[/tex] to get [tex]\(x^4\)[/tex].
- For [tex]\(x^3\)[/tex], add [tex]\(3x^3\)[/tex] and [tex]\(x^3\)[/tex] to get [tex]\(4x^3\)[/tex].
- There is only one [tex]\(-2x^2\)[/tex] term.
- For [tex]\(x\)[/tex], add [tex]\(-3x\)[/tex] and [tex]\(2x\)[/tex] to get [tex]\(-x\)[/tex].
- There is only one [tex]\(3\)[/tex] term.
So, the product is:
[tex]\[
x^6 + x^4 + 4x^3 - 2x^2 - x + 3
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{x^6 + x^4 + 4x^3 - 2x^2 - x + 3}
\][/tex]
So, the correct option is:
A. [tex]\(x^6 + x^4 + 4x^3 - 2x^2 - x + 3\)[/tex]
