Answer :
Sure! Let's multiply the two polynomials step by step:
We have two polynomials to multiply:
[tex]\((3x^2 - 4x + 5)\)[/tex] and [tex]\((x^2 - 3x + 2)\)[/tex].
To multiply these polynomials, we'll distribute each term in the first polynomial by each term in the second polynomial and then combine like terms.
### Step 1: Multiply each term in the first polynomial by each term in the second polynomial.
1. Multiply [tex]\(3x^2\)[/tex] by each term in the second polynomial:
- [tex]\(3x^2 \times x^2 = 3x^4\)[/tex]
- [tex]\(3x^2 \times (-3x) = -9x^3\)[/tex]
- [tex]\(3x^2 \times 2 = 6x^2\)[/tex]
2. Multiply [tex]\(-4x\)[/tex] by each term in the second polynomial:
- [tex]\(-4x \times x^2 = -4x^3\)[/tex]
- [tex]\(-4x \times (-3x) = 12x^2\)[/tex]
- [tex]\(-4x \times 2 = -8x\)[/tex]
3. Multiply [tex]\(5\)[/tex] by each term in the second polynomial:
- [tex]\(5 \times x^2 = 5x^2\)[/tex]
- [tex]\(5 \times (-3x) = -15x\)[/tex]
- [tex]\(5 \times 2 = 10\)[/tex]
### Step 2: Add all the products together.
Combine all terms from the products we calculated:
[tex]\[3x^4 - 9x^3 + 6x^2 - 4x^3 + 12x^2 - 8x + 5x^2 - 15x + 10\][/tex]
### Step 3: Combine like terms.
- [tex]\(x^4\)[/tex] terms: [tex]\(3x^4\)[/tex]
- [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 4x^3 = -13x^3\)[/tex]
- [tex]\(x^2\)[/tex] terms: [tex]\(6x^2 + 12x^2 + 5x^2 = 23x^2\)[/tex]
- [tex]\(x\)[/tex] terms: [tex]\(-8x - 15x = -23x\)[/tex]
- Constant term: [tex]\(10\)[/tex]
Putting it all together, the result is:
[tex]\[3x^4 - 13x^3 + 23x^2 - 23x + 10\][/tex]
Therefore, the answer is option A: [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex].
We have two polynomials to multiply:
[tex]\((3x^2 - 4x + 5)\)[/tex] and [tex]\((x^2 - 3x + 2)\)[/tex].
To multiply these polynomials, we'll distribute each term in the first polynomial by each term in the second polynomial and then combine like terms.
### Step 1: Multiply each term in the first polynomial by each term in the second polynomial.
1. Multiply [tex]\(3x^2\)[/tex] by each term in the second polynomial:
- [tex]\(3x^2 \times x^2 = 3x^4\)[/tex]
- [tex]\(3x^2 \times (-3x) = -9x^3\)[/tex]
- [tex]\(3x^2 \times 2 = 6x^2\)[/tex]
2. Multiply [tex]\(-4x\)[/tex] by each term in the second polynomial:
- [tex]\(-4x \times x^2 = -4x^3\)[/tex]
- [tex]\(-4x \times (-3x) = 12x^2\)[/tex]
- [tex]\(-4x \times 2 = -8x\)[/tex]
3. Multiply [tex]\(5\)[/tex] by each term in the second polynomial:
- [tex]\(5 \times x^2 = 5x^2\)[/tex]
- [tex]\(5 \times (-3x) = -15x\)[/tex]
- [tex]\(5 \times 2 = 10\)[/tex]
### Step 2: Add all the products together.
Combine all terms from the products we calculated:
[tex]\[3x^4 - 9x^3 + 6x^2 - 4x^3 + 12x^2 - 8x + 5x^2 - 15x + 10\][/tex]
### Step 3: Combine like terms.
- [tex]\(x^4\)[/tex] terms: [tex]\(3x^4\)[/tex]
- [tex]\(x^3\)[/tex] terms: [tex]\(-9x^3 - 4x^3 = -13x^3\)[/tex]
- [tex]\(x^2\)[/tex] terms: [tex]\(6x^2 + 12x^2 + 5x^2 = 23x^2\)[/tex]
- [tex]\(x\)[/tex] terms: [tex]\(-8x - 15x = -23x\)[/tex]
- Constant term: [tex]\(10\)[/tex]
Putting it all together, the result is:
[tex]\[3x^4 - 13x^3 + 23x^2 - 23x + 10\][/tex]
Therefore, the answer is option A: [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex].
