Answer :
Let's find the product of the polynomials [tex]\(x^3 + 2x + 3\)[/tex] and [tex]\(x^3 - x + 1\)[/tex] using vertical multiplication. We'll multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.
### Step-by-Step Solution:
1. Arrange the Polynomials Vertically:
[tex]\[
\begin{array}{r}
x^3 + 2x + 3 \\
\times \quad(x^3 - x + 1)
\end{array}
\][/tex]
2. Multiply Each Term:
- Multiply [tex]\(x^3\)[/tex] from the first polynomial by each term in the second polynomial:
[tex]\[
x^3 \cdot x^3 = x^6
\][/tex]
[tex]\[
x^3 \cdot (-x) = -x^4
\][/tex]
[tex]\[
x^3 \cdot 1 = x^3
\][/tex]
- Multiply [tex]\(2x\)[/tex] from the first polynomial by each term in the second polynomial:
[tex]\[
2x \cdot x^3 = 2x^4
\][/tex]
[tex]\[
2x \cdot (-x) = -2x^2
\][/tex]
[tex]\[
2x \cdot 1 = 2x
\][/tex]
- Multiply [tex]\(3\)[/tex] from the first polynomial by each term in the second polynomial:
[tex]\[
3 \cdot x^3 = 3x^3
\][/tex]
[tex]\[
3 \cdot (-x) = -3x
\][/tex]
[tex]\[
3 \cdot 1 = 3
\][/tex]
3. Write Down All Results and Align Like Terms:
- Combine the results from each multiplication:
[tex]\[
x^6 + (-x^4) + x^3
\][/tex]
[tex]\[
+ 2x^4 - 2x^2 + 2x
\][/tex]
[tex]\[
+ 3x^3 - 3x + 3
\][/tex]
4. Combine Like Terms:
- Start with the highest degree [tex]\(x^6\)[/tex]:
[tex]\[
x^6 + 0x^5
\][/tex]
- Combine [tex]\(x^4\)[/tex] terms: [tex]\((-x^4 + 2x^4) = x^4\)[/tex]
- Combine [tex]\(x^3\)[/tex] terms: [tex]\((x^3 + 3x^3) = 4x^3\)[/tex]
- Combine [tex]\(x^2\)[/tex] terms: [tex]\(-2x^2\)[/tex]
- Combine [tex]\(x\)[/tex] terms: [tex]\(2x - 3x = -x\)[/tex]
- Constant term: [tex]\(3\)[/tex]
5. Final Product:
[tex]\[
x^6 + x^4 + 4x^3 - 2x^2 - x + 3
\][/tex]
Therefore, the product of [tex]\((x^3 + 2x + 3)\)[/tex] and [tex]\((x^3 - x + 1)\)[/tex] is [tex]\(x^6 + x^4 + 4x^3 - 2x^2 - x + 3\)[/tex]. So, the correct answer is option C.
### Step-by-Step Solution:
1. Arrange the Polynomials Vertically:
[tex]\[
\begin{array}{r}
x^3 + 2x + 3 \\
\times \quad(x^3 - x + 1)
\end{array}
\][/tex]
2. Multiply Each Term:
- Multiply [tex]\(x^3\)[/tex] from the first polynomial by each term in the second polynomial:
[tex]\[
x^3 \cdot x^3 = x^6
\][/tex]
[tex]\[
x^3 \cdot (-x) = -x^4
\][/tex]
[tex]\[
x^3 \cdot 1 = x^3
\][/tex]
- Multiply [tex]\(2x\)[/tex] from the first polynomial by each term in the second polynomial:
[tex]\[
2x \cdot x^3 = 2x^4
\][/tex]
[tex]\[
2x \cdot (-x) = -2x^2
\][/tex]
[tex]\[
2x \cdot 1 = 2x
\][/tex]
- Multiply [tex]\(3\)[/tex] from the first polynomial by each term in the second polynomial:
[tex]\[
3 \cdot x^3 = 3x^3
\][/tex]
[tex]\[
3 \cdot (-x) = -3x
\][/tex]
[tex]\[
3 \cdot 1 = 3
\][/tex]
3. Write Down All Results and Align Like Terms:
- Combine the results from each multiplication:
[tex]\[
x^6 + (-x^4) + x^3
\][/tex]
[tex]\[
+ 2x^4 - 2x^2 + 2x
\][/tex]
[tex]\[
+ 3x^3 - 3x + 3
\][/tex]
4. Combine Like Terms:
- Start with the highest degree [tex]\(x^6\)[/tex]:
[tex]\[
x^6 + 0x^5
\][/tex]
- Combine [tex]\(x^4\)[/tex] terms: [tex]\((-x^4 + 2x^4) = x^4\)[/tex]
- Combine [tex]\(x^3\)[/tex] terms: [tex]\((x^3 + 3x^3) = 4x^3\)[/tex]
- Combine [tex]\(x^2\)[/tex] terms: [tex]\(-2x^2\)[/tex]
- Combine [tex]\(x\)[/tex] terms: [tex]\(2x - 3x = -x\)[/tex]
- Constant term: [tex]\(3\)[/tex]
5. Final Product:
[tex]\[
x^6 + x^4 + 4x^3 - 2x^2 - x + 3
\][/tex]
Therefore, the product of [tex]\((x^3 + 2x + 3)\)[/tex] and [tex]\((x^3 - x + 1)\)[/tex] is [tex]\(x^6 + x^4 + 4x^3 - 2x^2 - x + 3\)[/tex]. So, the correct answer is option C.
