Answer :
Sure! Let's multiply the polynomials [tex]\( p(x) = x^3 + 2x + 3 \)[/tex] and [tex]\( q(x) = x^3 - x + 1 \)[/tex] manually using vertical multiplication.
We'll multiply each term in the first polynomial by every term in the second polynomial and then add the results.
### Step 1: Multiply each term from [tex]\( p(x) \)[/tex] by [tex]\( x^3 \)[/tex] from [tex]\( q(x) \)[/tex]:
1. [tex]\( x^3 \times x^3 = x^6 \)[/tex]
2. [tex]\( 2x \times x^3 = 2x^4 \)[/tex]
3. [tex]\( 3 \times x^3 = 3x^3 \)[/tex]
Adding these results together gives:
[tex]\( x^6 + 2x^4 + 3x^3 \)[/tex]
### Step 2: Multiply each term from [tex]\( p(x) \)[/tex] by [tex]\(-x\)[/tex] from [tex]\( q(x) \)[/tex]:
1. [tex]\( x^3 \times (-x) = -x^4 \)[/tex]
2. [tex]\( 2x \times (-x) = -2x^2 \)[/tex]
3. [tex]\( 3 \times (-x) = -3x \)[/tex]
Adding these results together gives:
[tex]\( -x^4 - 2x^2 - 3x \)[/tex]
### Step 3: Multiply each term from [tex]\( p(x) \)[/tex] by [tex]\( 1 \)[/tex] from [tex]\( q(x) \)[/tex]:
1. [tex]\( x^3 \times 1 = x^3 \)[/tex]
2. [tex]\( 2x \times 1 = 2x \)[/tex]
3. [tex]\( 3 \times 1 = 3 \)[/tex]
Adding these results together gives:
[tex]\( x^3 + 2x + 3 \)[/tex]
### Step 4: Add all the results together:
Combine all the terms from Steps 1, 2, and 3:
[tex]\[
x^6 + 2x^4 + 3x^3 + (-x^4) + (-2x^2) + (-3x) + x^3 + 2x + 3
\][/tex]
Now, combine like terms:
- The [tex]\( x^6 \)[/tex] terms: [tex]\( x^6 \)[/tex]
- The [tex]\( x^4 \)[/tex] terms: [tex]\( 2x^4 - x^4 = x^4 \)[/tex]
- The [tex]\( x^3 \)[/tex] terms: [tex]\( 3x^3 + x^3 = 4x^3 \)[/tex]
- The [tex]\( x^2 \)[/tex] terms: only [tex]\(-2x^2\)[/tex]
- The [tex]\( x \)[/tex] terms: [tex]\(-3x + 2x = -x\)[/tex]
- Constant terms: [tex]\( 3 \)[/tex]
Putting it all together, the product is:
[tex]\[ x^6 + x^4 + 4x^3 - 2x^2 - x + 3 \][/tex]
So, the correct answer is:
C. [tex]\( x^6 + x^4 + 4x^3 - 2x^2 - x + 3 \)[/tex]
We'll multiply each term in the first polynomial by every term in the second polynomial and then add the results.
### Step 1: Multiply each term from [tex]\( p(x) \)[/tex] by [tex]\( x^3 \)[/tex] from [tex]\( q(x) \)[/tex]:
1. [tex]\( x^3 \times x^3 = x^6 \)[/tex]
2. [tex]\( 2x \times x^3 = 2x^4 \)[/tex]
3. [tex]\( 3 \times x^3 = 3x^3 \)[/tex]
Adding these results together gives:
[tex]\( x^6 + 2x^4 + 3x^3 \)[/tex]
### Step 2: Multiply each term from [tex]\( p(x) \)[/tex] by [tex]\(-x\)[/tex] from [tex]\( q(x) \)[/tex]:
1. [tex]\( x^3 \times (-x) = -x^4 \)[/tex]
2. [tex]\( 2x \times (-x) = -2x^2 \)[/tex]
3. [tex]\( 3 \times (-x) = -3x \)[/tex]
Adding these results together gives:
[tex]\( -x^4 - 2x^2 - 3x \)[/tex]
### Step 3: Multiply each term from [tex]\( p(x) \)[/tex] by [tex]\( 1 \)[/tex] from [tex]\( q(x) \)[/tex]:
1. [tex]\( x^3 \times 1 = x^3 \)[/tex]
2. [tex]\( 2x \times 1 = 2x \)[/tex]
3. [tex]\( 3 \times 1 = 3 \)[/tex]
Adding these results together gives:
[tex]\( x^3 + 2x + 3 \)[/tex]
### Step 4: Add all the results together:
Combine all the terms from Steps 1, 2, and 3:
[tex]\[
x^6 + 2x^4 + 3x^3 + (-x^4) + (-2x^2) + (-3x) + x^3 + 2x + 3
\][/tex]
Now, combine like terms:
- The [tex]\( x^6 \)[/tex] terms: [tex]\( x^6 \)[/tex]
- The [tex]\( x^4 \)[/tex] terms: [tex]\( 2x^4 - x^4 = x^4 \)[/tex]
- The [tex]\( x^3 \)[/tex] terms: [tex]\( 3x^3 + x^3 = 4x^3 \)[/tex]
- The [tex]\( x^2 \)[/tex] terms: only [tex]\(-2x^2\)[/tex]
- The [tex]\( x \)[/tex] terms: [tex]\(-3x + 2x = -x\)[/tex]
- Constant terms: [tex]\( 3 \)[/tex]
Putting it all together, the product is:
[tex]\[ x^6 + x^4 + 4x^3 - 2x^2 - x + 3 \][/tex]
So, the correct answer is:
C. [tex]\( x^6 + x^4 + 4x^3 - 2x^2 - x + 3 \)[/tex]
