Answer :
To find the product of the given polynomials using vertical multiplication, let's work through the steps:
We have two polynomials:
- [tex]\( P(x) = x^3 + 2x + 3 \)[/tex]
- [tex]\( Q(x) = x^3 - x + 1 \)[/tex]
Here's how you can multiply them step by step:
1. Align the polynomials for multiplication:
Write them vertically, like numbers, aligning similar terms:
```
x^3 + 0x^2 + 2x + 3
× x^3 + 0x^2 - x + 1
```
2. Multiply each term in the second polynomial by each term in the first polynomial:
- Multiply by 1 (rightmost term in the second polynomial):
- [tex]\( (x^3 + 2x + 3) \times 1 = x^3 + 2x + 3 \)[/tex]
- Multiply by [tex]\(-x\)[/tex]:
- [tex]\( (x^3 + 2x + 3) \times (-x) = -x^4 - 2x^2 - 3x \)[/tex]
- Multiply by 0 (0 is in [tex]\(x^2\)[/tex] place):
- This term becomes 0.
- Multiply by [tex]\(x^3\)[/tex]:
- [tex]\( (x^3 + 2x + 3) \times x^3 = x^6 + 2x^4 + 3x^3 \)[/tex]
3. Add all the results together: Align like terms and add:
```
x^6 + 2x^4 + 3x^3
-x^4 - 2x^2 - 3x
+x^3 + 2x + 3
----------------------------
x^6 + x^4 + 4x^3 - 2x^2 - x + 3
```
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]
This matches option A, so the correct answer is:
A. [tex]\(x^6 + x^4 + 4x^3 - 2x^2 - x + 3\)[/tex]
We have two polynomials:
- [tex]\( P(x) = x^3 + 2x + 3 \)[/tex]
- [tex]\( Q(x) = x^3 - x + 1 \)[/tex]
Here's how you can multiply them step by step:
1. Align the polynomials for multiplication:
Write them vertically, like numbers, aligning similar terms:
```
x^3 + 0x^2 + 2x + 3
× x^3 + 0x^2 - x + 1
```
2. Multiply each term in the second polynomial by each term in the first polynomial:
- Multiply by 1 (rightmost term in the second polynomial):
- [tex]\( (x^3 + 2x + 3) \times 1 = x^3 + 2x + 3 \)[/tex]
- Multiply by [tex]\(-x\)[/tex]:
- [tex]\( (x^3 + 2x + 3) \times (-x) = -x^4 - 2x^2 - 3x \)[/tex]
- Multiply by 0 (0 is in [tex]\(x^2\)[/tex] place):
- This term becomes 0.
- Multiply by [tex]\(x^3\)[/tex]:
- [tex]\( (x^3 + 2x + 3) \times x^3 = x^6 + 2x^4 + 3x^3 \)[/tex]
3. Add all the results together: Align like terms and add:
```
x^6 + 2x^4 + 3x^3
-x^4 - 2x^2 - 3x
+x^3 + 2x + 3
----------------------------
x^6 + x^4 + 4x^3 - 2x^2 - x + 3
```
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]
This matches option A, so the correct answer is:
A. [tex]\(x^6 + x^4 + 4x^3 - 2x^2 - x + 3\)[/tex]
